Universal Turing machine: Difference between revisions
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<syntaxhighlight lang="lisp">;; |
<syntaxhighlight lang="lisp">;;; 23.01.16 |
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(defun acl (table input state halt blank) |
(defun acl (table input state halt blank) |
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(loop for code = (cdr (assoc state table)) |
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(let ((l nil) |
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with l = nil |
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with r = input |
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with op = nil |
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with x = t |
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(loop :until (equal state halt) |
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until (eql state halt) |
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do (dolist (c code) |
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(case c |
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( |
(= (setf x (eql (car r) op))) |
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(% (when x (rplaca r op))) |
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( |
(< (when x (push (pop l) r))) |
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(> (when x (push (pop r) l))) |
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(@ (when x (setf state op) (return))) |
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(t (setf op c))) |
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(unless (car r) |
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(setf r (cons blank (cdr r))))) |
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finally (format t "Q = (~a ~{~a~}.~{~a~})~%" state (reverse l) r)))</syntaxhighlight> |
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(format t "Q = <~a, ~{~a~}.~{~a~}>~%" state (reverse l) r)))</syntaxhighlight> |
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====Tables==== |
====Tables==== |
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Revision as of 17:39, 16 January 2023
You are encouraged to solve this task according to the task description, using any language you may know.
One of the foundational mathematical constructs behind computer science is the universal Turing Machine.
(Alan Turing introduced the idea of such a machine in 1936–1937.)
Indeed one way to definitively prove that a language is turing-complete is to implement a universal Turing machine in it.
- Task
Simulate such a machine capable of taking the definition of any other Turing machine and executing it.
Of course, you will not have an infinite tape, but you should emulate this as much as is possible.
The three permissible actions on the tape are "left", "right" and "stay".
To test your universal Turing machine (and prove your programming language is Turing complete!), you should execute the following two Turing machines based on the following definitions.
Simple incrementer
- States: q0, qf
- Initial state: q0
- Terminating states: qf
- Permissible symbols: B, 1
- Blank symbol: B
- Rules:
- (q0, 1, 1, right, q0)
- (q0, B, 1, stay, qf)
The input for this machine should be a tape of 1 1 1
Three-state busy beaver
- States: a, b, c, halt
- Initial state: a
- Terminating states: halt
- Permissible symbols: 0, 1
- Blank symbol: 0
- Rules:
- (a, 0, 1, right, b)
- (a, 1, 1, left, c)
- (b, 0, 1, left, a)
- (b, 1, 1, right, b)
- (c, 0, 1, left, b)
- (c, 1, 1, stay, halt)
The input for this machine should be an empty tape.
Bonus:
5-state, 2-symbol probable Busy Beaver machine from Wikipedia
- States: A, B, C, D, E, H
- Initial state: A
- Terminating states: H
- Permissible symbols: 0, 1
- Blank symbol: 0
- Rules:
- (A, 0, 1, right, B)
- (A, 1, 1, left, C)
- (B, 0, 1, right, C)
- (B, 1, 1, right, B)
- (C, 0, 1, right, D)
- (C, 1, 0, left, E)
- (D, 0, 1, left, A)
- (D, 1, 1, left, D)
- (E, 0, 1, stay, H)
- (E, 1, 0, left, A)
The input for this machine should be an empty tape.
This machine runs for more than 47 millions steps.
11l
F run_utm(halt, state, Char blank; rules_in, [Char] &tape = [Char](); =pos = 0)
V st = state
I tape.empty
tape.append(blank)
I pos < 0
pos += tape.len
V rules = Dict(rules_in, r -> ((r[0], Char(r[1])), (Char(r[2]), r[3], r[4])))
L
print(st.ljust(4), end' ‘ ’)
L(v) tape
V i = L.index
I i == pos
print(‘[’v‘]’, end' ‘ ’)
E
print(v, end' ‘ ’)
print()
I st == halt
L.break
I (st, tape[pos]) !C rules
L.break
V (v1, dr, s1) = rules[(st, tape[pos])]
tape[pos] = v1
I dr == ‘left’
I pos > 0
pos--
E
tape.insert(0, blank)
I dr == ‘right’
pos++
I pos >= tape.len
tape.append(blank)
st = s1
print("incr machine\n")
run_utm(
halt' ‘qf’,
state' ‘q0’,
blank' Char(‘B’),
rules_in' [‘q0 1 1 right q0’.split(‘ ’, group_delimiters' 1B),
‘q0 B 1 stay qf’.split(‘ ’, group_delimiters' 1B)],
tape' &[‘1’, ‘1’, ‘1’]
)
print("\nbusy beaver\n")
run_utm(
halt' ‘halt’,
state' ‘a’,
blank' Char(‘0’),
rules_in'
[‘a 0 1 right b’.split(‘ ’, group_delimiters' 1B),
‘a 1 1 left c’.split(‘ ’, group_delimiters' 1B),
‘b 0 1 left a’.split(‘ ’, group_delimiters' 1B),
‘b 1 1 right b’.split(‘ ’, group_delimiters' 1B),
‘c 0 1 left b’.split(‘ ’, group_delimiters' 1B),
‘c 1 1 stay halt’.split(‘ ’, group_delimiters' 1B)]
)
print("\nsorting test\n")
run_utm(
halt' ‘STOP’,
state' ‘A’,
blank' Char(‘0’),
rules_in'
[‘A 1 1 right A’.split(‘ ’, group_delimiters' 1B),
‘A 2 3 right B’.split(‘ ’, group_delimiters' 1B),
‘A 0 0 left E’.split(‘ ’, group_delimiters' 1B),
‘B 1 1 right B’.split(‘ ’, group_delimiters' 1B),
‘B 2 2 right B’.split(‘ ’, group_delimiters' 1B),
‘B 0 0 left C’.split(‘ ’, group_delimiters' 1B),
‘C 1 2 left D’.split(‘ ’, group_delimiters' 1B),
‘C 2 2 left C’.split(‘ ’, group_delimiters' 1B),
‘C 3 2 left E’.split(‘ ’, group_delimiters' 1B),
‘D 1 1 left D’.split(‘ ’, group_delimiters' 1B),
‘D 2 2 left D’.split(‘ ’, group_delimiters' 1B),
‘D 3 1 right A’.split(‘ ’, group_delimiters' 1B),
‘E 1 1 left E’.split(‘ ’, group_delimiters' 1B),
‘E 0 0 right STOP’.split(‘ ’, group_delimiters' 1B)],
tape' &‘2 2 2 1 2 2 1 2 1 2 1 2 1 2’.split(‘ ’).map(Char)
)- Output:
incr machine q0 [1] 1 1 q0 1 [1] 1 q0 1 1 [1] q0 1 1 1 [B] qf 1 1 1 [1] busy beaver a [0] b 1 [0] a [1] 1 c [0] 1 1 b [0] 1 1 1 a [0] 1 1 1 1 b 1 [1] 1 1 1 b 1 1 [1] 1 1 b 1 1 1 [1] 1 b 1 1 1 1 [1] b 1 1 1 1 1 [0] a 1 1 1 1 [1] 1 c 1 1 1 [1] 1 1 halt 1 1 1 [1] 1 1 sorting test A [2] 2 2 1 2 2 1 2 1 2 1 2 1 2 B 3 [2] 2 1 2 2 1 2 1 2 1 2 1 2 B 3 2 [2] 1 2 2 1 2 1 2 1 2 1 2 B 3 2 2 [1] 2 2 1 2 1 2 1 2 1 2 B 3 2 2 1 [2] 2 1 2 1 2 1 2 1 2 B 3 2 2 1 2 [2] 1 2 1 2 1 2 1 2 B 3 2 2 1 2 2 [1] 2 1 2 1 2 1 2 B 3 2 2 1 2 2 1 [2] 1 2 1 2 1 2 B 3 2 2 1 2 2 1 2 [1] 2 1 2 1 2 B 3 2 2 1 2 2 1 2 1 [2] 1 2 1 2 B 3 2 2 1 2 2 1 2 1 2 [1] 2 1 2 B 3 2 2 1 2 2 1 2 1 2 1 [2] 1 2 B 3 2 2 1 2 2 1 2 1 2 1 2 [1] 2 B 3 2 2 1 2 2 1 2 1 2 1 2 1 [2] B 3 2 2 1 2 2 1 2 1 2 1 2 1 2 [0] C 3 2 2 1 2 2 1 2 1 2 1 2 1 [2] 0 C 3 2 2 1 2 2 1 2 1 2 1 2 [1] 2 0 D 3 2 2 1 2 2 1 2 1 2 1 [2] 2 2 0 D 3 2 2 1 2 2 1 2 1 2 [1] 2 2 2 0 D 3 2 2 1 2 2 1 2 1 [2] 1 2 2 2 0 D 3 2 2 1 2 2 1 2 [1] 2 1 2 2 2 0 D 3 2 2 1 2 2 1 [2] 1 2 1 2 2 2 0 D 3 2 2 1 2 2 [1] 2 1 2 1 2 2 2 0 D 3 2 2 1 2 [2] 1 2 1 2 1 2 2 2 0 D 3 2 2 1 [2] 2 1 2 1 2 1 2 2 2 0 D 3 2 2 [1] 2 2 1 2 1 2 1 2 2 2 0 D 3 2 [2] 1 2 2 1 2 1 2 1 2 2 2 0 D 3 [2] 2 1 2 2 1 2 1 2 1 2 2 2 0 D [3] 2 2 1 2 2 1 2 1 2 1 2 2 2 0 A 1 [2] 2 1 2 2 1 2 1 2 1 2 2 2 0 B 1 3 [2] 1 2 2 1 2 1 2 1 2 2 2 0 B 1 3 2 [1] 2 2 1 2 1 2 1 2 2 2 0 B 1 3 2 1 [2] 2 1 2 1 2 1 2 2 2 0 B 1 3 2 1 2 [2] 1 2 1 2 1 2 2 2 0 B 1 3 2 1 2 2 [1] 2 1 2 1 2 2 2 0 B 1 3 2 1 2 2 1 [2] 1 2 1 2 2 2 0 B 1 3 2 1 2 2 1 2 [1] 2 1 2 2 2 0 B 1 3 2 1 2 2 1 2 1 [2] 1 2 2 2 0 B 1 3 2 1 2 2 1 2 1 2 [1] 2 2 2 0 B 1 3 2 1 2 2 1 2 1 2 1 [2] 2 2 0 B 1 3 2 1 2 2 1 2 1 2 1 2 [2] 2 0 B 1 3 2 1 2 2 1 2 1 2 1 2 2 [2] 0 B 1 3 2 1 2 2 1 2 1 2 1 2 2 2 [0] C 1 3 2 1 2 2 1 2 1 2 1 2 2 [2] 0 C 1 3 2 1 2 2 1 2 1 2 1 2 [2] 2 0 C 1 3 2 1 2 2 1 2 1 2 1 [2] 2 2 0 C 1 3 2 1 2 2 1 2 1 2 [1] 2 2 2 0 D 1 3 2 1 2 2 1 2 1 [2] 2 2 2 2 0 D 1 3 2 1 2 2 1 2 [1] 2 2 2 2 2 0 D 1 3 2 1 2 2 1 [2] 1 2 2 2 2 2 0 D 1 3 2 1 2 2 [1] 2 1 2 2 2 2 2 0 D 1 3 2 1 2 [2] 1 2 1 2 2 2 2 2 0 D 1 3 2 1 [2] 2 1 2 1 2 2 2 2 2 0 D 1 3 2 [1] 2 2 1 2 1 2 2 2 2 2 0 D 1 3 [2] 1 2 2 1 2 1 2 2 2 2 2 0 D 1 [3] 2 1 2 2 1 2 1 2 2 2 2 2 0 A 1 1 [2] 1 2 2 1 2 1 2 2 2 2 2 0 B 1 1 3 [1] 2 2 1 2 1 2 2 2 2 2 0 B 1 1 3 1 [2] 2 1 2 1 2 2 2 2 2 0 B 1 1 3 1 2 [2] 1 2 1 2 2 2 2 2 0 B 1 1 3 1 2 2 [1] 2 1 2 2 2 2 2 0 B 1 1 3 1 2 2 1 [2] 1 2 2 2 2 2 0 B 1 1 3 1 2 2 1 2 [1] 2 2 2 2 2 0 B 1 1 3 1 2 2 1 2 1 [2] 2 2 2 2 0 B 1 1 3 1 2 2 1 2 1 2 [2] 2 2 2 0 B 1 1 3 1 2 2 1 2 1 2 2 [2] 2 2 0 B 1 1 3 1 2 2 1 2 1 2 2 2 [2] 2 0 B 1 1 3 1 2 2 1 2 1 2 2 2 2 [2] 0 B 1 1 3 1 2 2 1 2 1 2 2 2 2 2 [0] C 1 1 3 1 2 2 1 2 1 2 2 2 2 [2] 0 C 1 1 3 1 2 2 1 2 1 2 2 2 [2] 2 0 C 1 1 3 1 2 2 1 2 1 2 2 [2] 2 2 0 C 1 1 3 1 2 2 1 2 1 2 [2] 2 2 2 0 C 1 1 3 1 2 2 1 2 1 [2] 2 2 2 2 0 C 1 1 3 1 2 2 1 2 [1] 2 2 2 2 2 0 D 1 1 3 1 2 2 1 [2] 2 2 2 2 2 2 0 D 1 1 3 1 2 2 [1] 2 2 2 2 2 2 2 0 D 1 1 3 1 2 [2] 1 2 2 2 2 2 2 2 0 D 1 1 3 1 [2] 2 1 2 2 2 2 2 2 2 0 D 1 1 3 [1] 2 2 1 2 2 2 2 2 2 2 0 D 1 1 [3] 1 2 2 1 2 2 2 2 2 2 2 0 A 1 1 1 [1] 2 2 1 2 2 2 2 2 2 2 0 A 1 1 1 1 [2] 2 1 2 2 2 2 2 2 2 0 B 1 1 1 1 3 [2] 1 2 2 2 2 2 2 2 0 B 1 1 1 1 3 2 [1] 2 2 2 2 2 2 2 0 B 1 1 1 1 3 2 1 [2] 2 2 2 2 2 2 0 B 1 1 1 1 3 2 1 2 [2] 2 2 2 2 2 0 B 1 1 1 1 3 2 1 2 2 [2] 2 2 2 2 0 B 1 1 1 1 3 2 1 2 2 2 [2] 2 2 2 0 B 1 1 1 1 3 2 1 2 2 2 2 [2] 2 2 0 B 1 1 1 1 3 2 1 2 2 2 2 2 [2] 2 0 B 1 1 1 1 3 2 1 2 2 2 2 2 2 [2] 0 B 1 1 1 1 3 2 1 2 2 2 2 2 2 2 [0] C 1 1 1 1 3 2 1 2 2 2 2 2 2 [2] 0 C 1 1 1 1 3 2 1 2 2 2 2 2 [2] 2 0 C 1 1 1 1 3 2 1 2 2 2 2 [2] 2 2 0 C 1 1 1 1 3 2 1 2 2 2 [2] 2 2 2 0 C 1 1 1 1 3 2 1 2 2 [2] 2 2 2 2 0 C 1 1 1 1 3 2 1 2 [2] 2 2 2 2 2 0 C 1 1 1 1 3 2 1 [2] 2 2 2 2 2 2 0 C 1 1 1 1 3 2 [1] 2 2 2 2 2 2 2 0 D 1 1 1 1 3 [2] 2 2 2 2 2 2 2 2 0 D 1 1 1 1 [3] 2 2 2 2 2 2 2 2 2 0 A 1 1 1 1 1 [2] 2 2 2 2 2 2 2 2 0 B 1 1 1 1 1 3 [2] 2 2 2 2 2 2 2 0 B 1 1 1 1 1 3 2 [2] 2 2 2 2 2 2 0 B 1 1 1 1 1 3 2 2 [2] 2 2 2 2 2 0 B 1 1 1 1 1 3 2 2 2 [2] 2 2 2 2 0 B 1 1 1 1 1 3 2 2 2 2 [2] 2 2 2 0 B 1 1 1 1 1 3 2 2 2 2 2 [2] 2 2 0 B 1 1 1 1 1 3 2 2 2 2 2 2 [2] 2 0 B 1 1 1 1 1 3 2 2 2 2 2 2 2 [2] 0 B 1 1 1 1 1 3 2 2 2 2 2 2 2 2 [0] C 1 1 1 1 1 3 2 2 2 2 2 2 2 [2] 0 C 1 1 1 1 1 3 2 2 2 2 2 2 [2] 2 0 C 1 1 1 1 1 3 2 2 2 2 2 [2] 2 2 0 C 1 1 1 1 1 3 2 2 2 2 [2] 2 2 2 0 C 1 1 1 1 1 3 2 2 2 [2] 2 2 2 2 0 C 1 1 1 1 1 3 2 2 [2] 2 2 2 2 2 0 C 1 1 1 1 1 3 2 [2] 2 2 2 2 2 2 0 C 1 1 1 1 1 3 [2] 2 2 2 2 2 2 2 0 C 1 1 1 1 1 [3] 2 2 2 2 2 2 2 2 0 E 1 1 1 1 [1] 2 2 2 2 2 2 2 2 2 0 E 1 1 1 [1] 1 2 2 2 2 2 2 2 2 2 0 E 1 1 [1] 1 1 2 2 2 2 2 2 2 2 2 0 E 1 [1] 1 1 1 2 2 2 2 2 2 2 2 2 0 E [1] 1 1 1 1 2 2 2 2 2 2 2 2 2 0 E [0] 1 1 1 1 1 2 2 2 2 2 2 2 2 2 0 STOP 0 [1] 1 1 1 1 2 2 2 2 2 2 2 2 2 0
Ada
The specification of the universal machine
Note that due to Ada's strict type system, a machine cannot be compiled if there is not _exactly_ one rule for each state/symbol pair. Thus, the specified machine is always deterministic.
The execution of the machine, i.e., the procedure Run, allows to define a number Max_Steps, after which the execution stops -- when, e.g., the specified machine runs infinitively. The procedure also allows to optionally output the configuration of the machine before every step.
private with Ada.Containers.Doubly_Linked_Lists;
generic
type State is (<>); -- State'First is starting state
type Symbol is (<>); -- Symbol'First is blank
package Turing is
Start: constant State := State'First;
Halt: constant State := State'Last;
subtype Action_State is State range Start .. State'Pred(Halt);
Blank: constant Symbol := Symbol'First;
type Movement is (Left, Stay, Right);
type Action is record
New_State: State;
Move_To: Movement;
New_Symbol: Symbol;
end record;
type Rules_Type is array(Action_State, Symbol) of Action;
type Tape_Type is limited private;
type Symbol_Map is array(Symbol) of Character;
function To_String(Tape: Tape_Type; Map: Symbol_Map) return String;
function Position_To_String(Tape: Tape_Type; Marker: Character := '^')
return String;
function To_Tape(Str: String; Map: Symbol_Map) return Tape_Type;
procedure Single_Step(Current: in out State;
Tape: in out Tape_Type;
Rules: Rules_Type);
procedure Run(The_Tape: in out Tape_Type;
Rules: Rules_Type;
Max_Steps: Natural := Natural'Last;
Print: access procedure(Tape: Tape_Type; Current: State));
-- runs from Start State until either Halt or # Steps exceeds Max_Steps
-- if # of steps exceeds Max_Steps, Constrained_Error is raised;
-- if Print is not null, Print is called at the beginning of each step
private
package Symbol_Lists is new Ada.Containers.Doubly_Linked_Lists(Symbol);
subtype List is Symbol_Lists.List;
type Tape_Type is record
Left: List;
Here: Symbol;
Right: List;
end record;
end Turing;
The implementation of the universal machine
package body Turing is
function List_To_String(L: List; Map: Symbol_Map) return String is
LL: List := L;
use type List;
begin
if L = Symbol_Lists.Empty_List then
return "";
else
LL.Delete_First;
return Map(L.First_Element) & List_To_String(LL, Map);
end if;
end List_To_String;
function To_String(Tape: Tape_Type; Map: Symbol_Map) return String is
begin
return List_To_String(Tape.Left, Map) & Map(Tape.Here) &
List_To_String(Tape.Right, Map);
end To_String;
function Position_To_String(Tape: Tape_Type; Marker: Character := '^')
return String is
Blank_Map: Symbol_Map := (others => ' ');
begin
return List_To_String(Tape.Left, Blank_Map) & Marker &
List_To_String(Tape.Right, Blank_Map);
end Position_To_String;
function To_Tape(Str: String; Map: Symbol_Map) return Tape_Type is
Char_Map: array(Character) of Symbol := (others => Blank);
Tape: Tape_Type;
begin
if Str = "" then
Tape.Here := Blank;
else
for S in Symbol loop
Char_Map(Map(S)) := S;
end loop;
Tape.Here := Char_Map(Str(Str'First));
for I in Str'First+1 .. Str'Last loop
Tape.Right.Append(Char_Map(Str(I)));
end loop;
end if;
return Tape;
end To_Tape;
procedure Single_Step(Current: in out State;
Tape: in out Tape_Type;
Rules: Rules_Type) is
Act: Action := Rules(Current, Tape.Here);
use type List; -- needed to compare Tape.Left/Right to the Empty_List
begin
Current := Act.New_State; -- 1. update State
Tape.Here := Act.New_Symbol; -- 2. write Symbol to Tape
case Act.Move_To is -- 3. move Tape to the Left/Right or Stay
when Left =>
Tape.Right.Prepend(Tape.Here);
if Tape.Left /= Symbol_Lists.Empty_List then
Tape.Here := Tape.Left.Last_Element;
Tape.Left.Delete_Last;
else
Tape.Here := Blank;
end if;
when Stay =>
null; -- Stay where you are!
when Right =>
Tape.Left.Append(Tape.Here);
if Tape.Right /= Symbol_Lists.Empty_List then
Tape.Here := Tape.Right.First_Element;
Tape.Right.Delete_First;
else
Tape.Here := Blank;
end if;
end case;
end Single_Step;
procedure Run(The_Tape: in out Tape_Type;
Rules: Rules_Type;
Max_Steps: Natural := Natural'Last;
Print: access procedure (Tape: Tape_Type; Current: State)) is
The_State: State := Start;
Steps: Natural := 0;
begin
Steps := 0;
while (Steps <= Max_Steps) and (The_State /= Halt) loop
if Print /= null then
Print(The_Tape, The_State);
end if;
Steps := Steps + 1;
Single_Step(The_State, The_Tape, Rules);
end loop;
if The_State /= Halt then
raise Constraint_Error;
end if;
end Run;
end Turing;
The implementation of the simple incrementer
with Ada.Text_IO, Turing;
procedure Simple_Incrementer is
type States is (Start, Stop);
type Symbols is (Blank, One);
package UTM is new Turing(States, Symbols);
use UTM;
Map: Symbol_Map := (One => '1', Blank => '_');
Rules: Rules_Type :=
(Start => (One => (Start, Right, One),
Blank => (Stop, Stay, One)));
Tape: Tape_Type := To_Tape("111", Map);
procedure Put_Tape(Tape: Tape_Type; Current: States) is
begin
Ada.Text_IO.Put_Line(To_String(Tape, Map) & " " & States'Image(Current));
Ada.Text_IO.Put_Line(Position_To_String(Tape));
end Put_Tape;
begin
Run(Tape, Rules, 20, null); -- don't print the configuration during running
Put_Tape(Tape, Stop); -- print the final configuration
end Simple_Incrementer;
- Output:
1111 STOP ^
The implementation of the busy beaver
with Ada.Text_IO, Turing;
procedure Busy_Beaver_3 is
type States is (A, B, C, Stop);
type Symbols is range 0 .. 1;
package UTM is new Turing(States, Symbols); use UTM;
Map: Symbol_Map := (1 => '1', 0 => '0');
Rules: Rules_Type :=
(A => (0 => (New_State => B, Move_To => Right, New_Symbol => 1),
1 => (New_State => C, Move_To => Left, New_Symbol => 1)),
B => (0 => (New_State => A, Move_To => Left, New_Symbol => 1),
1 => (New_State => B, Move_To => Right, New_Symbol => 1)),
C => (0 => (New_State => B, Move_To => Left, New_Symbol => 1),
1 => (New_State => Stop, Move_To => Stay, New_Symbol => 1)));
Tape: Tape_Type := To_Tape("", Map);
procedure Put_Tape(Tape: Tape_Type; Current: States) is
begin
Ada.Text_IO.Put_Line(To_String(Tape, Map) & " " &
States'Image(Current));
Ada.Text_IO.Put_Line(Position_To_String(Tape));
end Put_Tape;
begin
Run(Tape, Rules, 20, Put_Tape'Access); -- print configuration before each step
Put_Tape(Tape, Stop); -- and print the final configuration
end Busy_Beaver_3;
- Output:
0 A
^
10 B
^
11 A
^
011 C
^
0111 B
^
01111 A
^
11111 B
^
11111 B
^
11111 B
^
11111 B
^
111110 B
^
111111 A
^
111111 C
^
111111 STOP
^
Amazing Hopper
Implementation of a Universal Turing Machine:
#include <hopper.h>
#proto UniversalTuringMachine(_X_)
main:
.ctrlc
stbegin=0,stEnd=0,state=0,ptr=0
tape=0,states=0,rules=0,long=0,tapeSize=0
file="turing/prg03.tm"
// load program, rules & states:
jsub(load Archive)
// RUN Universal Turing Machine program:
i=1
__TURING_RUN__:
_Universal Turing Machine ([i,1:end]get(rules))
++i,{long,i}gt? do{ i=1 }
jt(__TURING_RUN__)
println
exit(0)
.locals
printTape:
#hl{
print(tape[1:(ptr-1)],"\R",tape[ptr],"\OFF",tape[(ptr+1):end],"\n")
//sleep(0.1)
}
up(1)
clear mark
back
Universal Turing Machine(rules)
cont=1
clear mark
#hl{
if( rules[1] == state )
if( tape[ptr] == rules[2] )
tape[ptr] = rules[3]
ptr += rules[4]
if(ptr==0)
}
++tapeSize
{0,1,tape}, array(INSERT), ++ptr
#hl{
else if(ptr>tapeSize)
}
++tapeSize
{tapeSize,tape},array(RESIZE),
[tapeSize]{0},put(tape),clear mark
#hl{
endif
state = rules[5]
if(state == stEnd)
cont=0
endif
}
jsub(print Tape)
#hl{
endif
endif
}, {cont}
back
load Archive:
{","}tok sep
{file} stats file
[1,1:end],{file},!(5),load, mov(tape)
[2,1:3], !(5),load, mov(states)
[3:end,1:5], load, mov(rules)
clear mark
[1:end,4]get(rules),colMoving=0, mov(colMoving)
{"1","RIGHT",colMoving} transform, mov(colMoving)
{"-1","LEFT",colMoving} transform, mov(colMoving)
{"0","STAY",colMoving} transform, xtonum, put(rules)
clear mark
{0}reshape(tape)
size(tape),lengthTape=0,mov(lengthTape),[2]get(lengthTape),mov(tapeSize)
#hl{
stbegin=states[1,1]
stEnd=states[1,2]
ptr=states[1,3]
state=stbegin
}
data rules=0, size(rules), mov(datarules), [2]get(data rules), mov(long)
{""}tok sep
backALternative pseudo-function UniversalTuringMachine (fast):
Universal Turing Machine(rules)
cont=1
clear mark
[1]get(rules),{state},eq?
do{
[ptr]get(tape),[2]get(rules),eq?,
do{
[3]get(rules),[ptr]put(tape)
[4]get(rules),plus(ptr),mov(ptr)
{ptr}zero?,do{
++tapeSize
{0,1,tape}, array(INSERT), ++ptr
}
{ptr}gthan(tapeSize),do{
++tapeSize
{tapeSize,tape},array(RESIZE),
[tapeSize]{0},put(tape),clear mark
}
[5]get(rules),mov(state)
{state,stEnd},eq? do { cont=0 }
jsub(print Tape)
}
},{cont}
back
Program PRG01.TM:
row 1: tape. row 2: initial state, halt state, and initial pointer position. row 3 to end, columns 1 to 5: rules.
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 a,halt,5 a,0,1,RIGHT,b a,1,1,LEFT,c b,0,1,LEFT,a b,1,1,RIGHT,b c,0,1,LEFT,b c,1,1,STAY,halt
Program PRG02.TM:
1,1,1,0,0 q0,qf,1 q0,1,1,RIGHT,q0 q0,0,1,STAY,qf
Program PRG03.TM:
0,0,0,0,0 A,H,1 A,0,1,RIGHT,B A,1,1,LEFT,C B,0,1,RIGHT,C B,1,1,RIGHT,B C,0,1,RIGHT,D C,1,0,LEFT,E D,0,1,LEFT,A D,1,1,LEFT,D E,0,1,STAY,H E,1,0,LEFT,A
- Output:
PRG01.TM: 111111000 PRG02.TM: 11110 PRG03.TM (fragment, and veeeery slow with High-level mode :( ): 1111111111111100100100100100100100100100100100100100100100100100100100100111
APL
:Namespace Turing
⍝ Run Turing machine until it halts
∇r←RunTuring (rules init halts blank itape);state;rt;lt;next
state←init
lt←⍬
rt←,blank
:If 0≠≢itape ⋄ rt←itape ⋄ :EndIf
:While ~(⊂state)∊halts
next←((⊂state(⊃rt))≡¨↓rules[;⍳2])⌿rules
'No rule applies!'⎕SIGNAL(0=≢next)/11
(⊃rt)←⊃next[1;3]
state←⊃next[1;5]
:Select ⊃next[1;4]
:Case 'stay' ⋄ ⍝nothing
:Case 'right'
lt,⍨←⊃rt
rt←1↓rt
:If 0=≢rt ⋄ rt←,blank ⋄ :EndIf
:Case 'left'
:If 0=≢lt ⋄ lt←,blank ⋄ :EndIf
rt,⍨←⊃lt
lt←1↓lt
:Else
'Invalid action'⎕SIGNAL 11
:EndSelect
:EndWhile
r←(⌽lt),rt
∇
⍝ Display the resulting tape neatly
∇r←len Display t
r←(len⌊≢t)↑t
→(len≥≢t)/0
r,←'... (total length: ',(⍕≢t),')'
∇
⍝ Simple incrementer
∇t←∆1_SimpleIncrementer
t ←⊂'q0' '1' '1' 'right' 'q0'
t,←⊂'q0' 'B' '1' 'stay' 'qf'
t←(↑t) 'q0' (,⊂'qf') 'B' '111'
∇
⍝ Three state beaver
∇t←∆2_ThreeStateBeaver
t ←⊂'a' '0' '1' 'right' 'b'
t,←⊂'a' '1' '1' 'left' 'c'
t,←⊂'b' '0' '1' 'left' 'a'
t,←⊂'b' '1' '1' 'right' 'b'
t,←⊂'c' '0' '1' 'left' 'b'
t,←⊂'c' '1' '1' 'stay' 'halt'
t←(↑t) 'a' (,⊂'halt') '0' ''
∇
⍝ Five state beaver
∇t←∆3_FiveStateBeaver
t ←⊂'A' '0' '1' 'right' 'B'
t,←⊂'A' '1' '1' 'left' 'C'
t,←⊂'B' '0' '1' 'right' 'C'
t,←⊂'B' '1' '1' 'right' 'B'
t,←⊂'C' '0' '1' 'right' 'D'
t,←⊂'C' '1' '0' 'left' 'E'
t,←⊂'D' '0' '1' 'left' 'A'
t,←⊂'D' '1' '1' 'left' 'D'
t,←⊂'E' '0' '1' 'stay' 'H'
t,←⊂'E' '1' '0' 'left' 'A'
t←(↑t) 'A' (,⊂'H') '0' ''
∇
⍝ Run all of them and display the results
∇RunAll;m;ms
ms←('∆'=⊃¨ms)/ms←⎕NL¯3
:For m :In ms
⎕←(1↓m),': ',(32 Display RunTuring ⍎m)
:EndFor
∇
:EndNamespace
- Output:
1_SimpleIncrementer: 1111 2_ThreeStateBeaver: 111111 3_FiveStateBeaver: 10100100100100100100100100100100... (total length: 12289)
AutoHotkey
; By Uberi, http://www.autohotkey.com/board/topic/58599-turing-machine/
SetBatchLines, -1
OnExit, Exit
SaveFilePath := A_ScriptFullPath ".ini"
; Defaults are for a 2-state_3-symbol turning machine. Format:
; machine state symbol on tape, symbol on tape | tape shift (- is left, + is right, 0 is halt) | machine state
, Rule1 := "A0,1|1|B"
, Rule2 := "A1,2|-1|A"
, Rule3 := "A2,1|-1|A"
, Rule4 := "B0,2|-1|A"
, Rule5 := "B1,2|1|B"
, Rule6 := "B2,0|1|A"
; no error check is run on this input, so be sure states and symbols align with actions
IniRead, UseSaveFile, %SaveFilePath%, Global, UseSaveFile, 1 ; on exit, save state to text file so I can resume on next run
IniRead, MaxIterations, %SaveFilePath%, Global, MaxIterations, 100000 ; set as %A_Space% to run indefinitely
IniRead, Section, %SaveFilePath%, Global, Section, 2-state_3-symbol ; The name of the machine to run. Options defined:
; 2-state_3-symbol
; Simple_incrementer
; Three-state_busy_beaver
; Probable_busy_beaver_Wikipedia
IniRead, States, %SaveFilePath%, %Section%, States, A|B ; valid states
IniRead, InitialState, %SaveFilePath%, %Section%, InitialState, A ; start state
IniRead, TerminalState, %SaveFilePath%, %Section%, TerminalState, C ; end state
IniRead, Symbols, %SaveFilePath%, %Section%, Symbols, 0,1,2 ; valid symbols
IniRead, DefaultCell, %SaveFilePath%, %Section%, DefaultCell, 0 ; the default symbol of any cell not defined on input tape
IniRead, ProgramCode, %SaveFilePath%, %Section%, ProgramCode, 10101|01010 ; start tape
Iniread, RuleCount, %SaveFilePath%, %Section%, RuleCount, 6 ; number of actions to read
Loop, %RuleCount%
{
IniRead, Temp1, %SaveFilePath%, %Section%, Rule%A_Index%, % Rule%A_Index%
StringSplit, Temp, Temp1, `,
Action%Temp1% := Temp2
}
IniRead, Index, %SaveFilePath%, SavedState, Index, 0
IniRead, IterationCount, %SaveFilePath%, SavedState, IterationCount, 0
IniRead, State, %SaveFilePath%, SavedState, State, %InitialState%
If IterationCount > 0
IniRead, ProgramCode, %SaveFilePath%, SavedState, ProgramCode, %ProgramCode%
IfNotInString, ProgramCode, |
ProgramCode := "|" ProgramCode
StringSplit, Temp, ProgramCode, |
NegativeCells := Temp1, PositiveCells := Temp2
Loop, Parse, Symbols, |
Color%A_LoopField% := hex(mod((A_Index+1/(2**((A_Index-1)//7))-1)/7,1)*16777215) ; unlimited number of unique colors
Color%DefaultCell% := "White"
Gui, Color, Black
Gui, +ToolWindow +AlwaysOnTop +LastFound -Caption
WindowID := WinExist()
OnMessage(0x201, "WM_LBUTTONDOWN")
Gui, Font, s6 cWhite, Arial
Loop, 61 ; display 30 cell symbols on each side of current index
{
Temp1 := ((A_Index - 1) * 15) + 1
Gui, Add, Progress, x%Temp1% y1 w14 h40 vCell%A_Index% BackgroundWhite
Gui, Add, Text, x%Temp1% y42 w15 h10 vLabel%A_Index% Center
}
Gui, Add, Text, x2 y54 w26 h10 vState
Gui, Add, Text, x35 y54 w50 h10 vCurrentCell
Gui, Add, Text, x350 y54 w158 h10 vActions
Gui, Add, Text, x844 y54 w33 h10, Iterations:
Gui, Add, Text, x884 y54 w29 h10 vIterations Right
Gui, Font, s4 cWhite Bold, Arial
Gui, Add, Text, x450 y1 w15 h10 Center, V
GuiControl, Move, Cell31, x451 y8 w14 h33
Gui, Show, y20 w916 h64, Wolfram's 2-State 3-Symbol Turing Machine ;'
;MaxIndex := ProgramOffset + StrLen(ProgramCode), MinIndex := ProgramOffset ; not implemented
While, ((MaxIterations = "") || IterationCount <= MaxIterations) ; process until limit is reached, if any
{
Loop, 61 ; color each cell per its current symbol
{ ; must run for all displayed cells because they are not directly mapped to shifting tape
TempIndex := (Index + A_Index) - 31
GuiControl, , Label%A_Index%, %TempIndex%
CellColor := CellGet(TempIndex)
, CellColor := Color%CellColor%
GuiControl, +Background%CellColor%, Cell%A_Index%
}
CurrentCell := CellGet(Index)
GuiControl, , State, State: %State%
GuiControl, , CurrentCell, Current Cell: %CurrentCell%
GuiControl, , Iterations, %IterationCount%
If (State = TerminalState)
Break
StringSplit, Temp, Action%State%%CurrentCell%, |
GuiControl, , Actions, % "Actions: Print " . Temp1 . ", Move " . ((Temp2 = -1) ? "left" : "right") . ", " . ((State <> Temp3) ? "Switch to state " . Temp3 : "Do not switch state")
IterationCount++
, CellPut(Index,Temp1)
, Index += Temp2
, State := Temp3
;, (Index > MaxIndex) ? MaxIndex := Index : ""
;, (Index < MinIndex) ? MinIndex := Index : ""
Sleep, 0.1*1000
}
MsgBox, 64, Complete, Completed %IterationCount% iterations of the Turing machine.
Return
; Hotkeys and functions:
~Pause::Pause
GuiEscape:
GuiClose:
ExitApp
Exit:
If UseSaveFile
{
IniWrite, %Index%, %SaveFilePath%, %Section%, Index
IniWrite, %IterationCount%, %SaveFilePath%, %Section%, IterationCount
IniWrite, %State%, %SaveFilePath%, %Section%, State
IniWrite, %NegativeCells%|%PositiveCells%, %SaveFilePath%, %Section%, ProgramCode
}
ExitApp
CellGet(Index)
{
global NegativeCells, PositiveCells, DefaultCell
Temp1 := (Index < 0) ? SubStr(NegativeCells,Abs(Index),1) : SubStr(PositiveCells,Index + 1,1)
Return, (Temp1 = "") ? DefaultCell : Temp1
}
CellPut(Index,Char)
{
global NegativeCells, PositiveCells, DefaultCell
static StrGetFunc := "StrGet" ; workaround to hide function from AHK Basic (which does not have or require it)
CharType := A_IsUnicode ? "UShort" : "UChar"
, (Index < 0)
? (Index := 0 - Index
, Temp1 := Index - StrLen(NegativeCells)
, (Temp1 > 0)
? (VarSetCapacity(Pad,64) ; these three functions are quirks in AHK's memory management (not required)
, VarSetCapacity(Pad,0)
, VarSetCapacity(Pad,Temp1,Asc(DefaultCell))
, NegativeCells .= A_IsUnicode ? %StrGetFunc%(&Pad,Temp1,"CP0") : Pad)
: ""
, NumPut(Asc(Char),NegativeCells,(Index - 1) << !!A_IsUnicode,CharType) )
: (Temp1 := Index - StrLen(PositiveCells) + 1
, (Temp1 > 0)
? (VarSetCapacity(Pad,64) ; these three functions are quirks in AHK's memory management (not required)
, VarSetCapacity(Pad,0)
, VarSetCapacity(Pad,Temp1,Asc(DefaultCell))
, PositiveCells .= A_IsUnicode ? %StrGetFunc%(&Pad,Temp1,"CP0") : Pad)
: ""
, NumPut(Asc(Char),PositiveCells,Index << !!A_IsUnicode,CharType) )
}
Hex(p_Integer)
{
PtrType:=(A_PtrSize=8) ? "Ptr":"UInt"
l_Format:="`%0" . 6 . "I64X"
VarSetCapacity(l_Argument,8)
NumPut(p_Integer,l_Argument,0,"Int64")
VarSetCapacity(l_Buffer,A_IsUnicode ? 12:6,0)
DllCall(A_IsUnicode ? "msvcrt\_vsnwprintf":"msvcrt\_vsnprintf"
,"Str",l_Buffer ;-- Storage location for output
,"UInt",6 ;-- Maximum number of characters to write
,"Str",l_Format ;-- Format specification
,PtrType,&l_Argument) ;-- Argument
Return l_Buffer
}
WM_LBUTTONDOWN()
{
If (A_Gui = 1)
PostMessage, 0xA1, 2
}
Input: Set Section below to desired machine, then save as <scriptname>.ini in the same folder.
[Global] UseSaveFile=0 MaxIterations=100000 Section=2-state_3-symbol [2-state_3-symbol] States=A|B InitialState=A TerminalState=C Symbols=0|1|2 DefaultCell=0 RuleCount=6 Rule1=A0,1|1|B Rule2=A1,2|-1|A Rule3=A2,1|-1|A Rule4=B0,2|-1|A Rule5=B1,2|1|B Rule6=B2,0|1|A ProgramCode=10101|01010 [Simple_incrementer] States=q0|qf InitialState=q0 TerminalState=qf Symbols=B|1 DefaultCell=B RuleCount=2 Rule1=q01,1|1|q0 Rule2=q0B,1|0|qf ProgramCode=111 [Three-state_busy_beaver] States=a|b|c|halt InitialState=a TerminalState=halt Symbols=0|1 DefaultCell=0 RuleCount=6 Rule1=a0,1|1|b Rule2=a1,1|-1|c Rule3=b0,1|-1|a Rule4=b1,1|1|b Rule5=c0,1|-1|b Rule6=c1,1|0|halt ProgramCode= [Probable_busy_beaver_Wikipedia] States=A|B|C|D|E|H InitialState=A TerminalState=H Symbols=0|1 DefaultCell=0 RuleCount=10 Rule1=A0,1|1|B Rule2=A1,1|-1|C Rule3=B0,1|1|C Rule4=B1,1|1|B Rule5=C0,1|1|D Rule6=C1,0|-1|E Rule7=D0,1|-1|A Rule8=D1,1|-1|D Rule9=E0,1|0|H Rule10=E1,0|-1|A ProgramCode=
- Output:
An animation of the chosen machine
BASIC
Sinclair ZX81 BASIC
The universal machine
This program expects to find:
• R$(), an array of rules; • T$, an input tape (where an empty string stands for a blank tape); • B$, a character to use as a blank; • S$, an initial state; • H$, a halting state.
It will execute the Turing machine these parameters describe, animating the process and highlighting the cell that is currently being read using inverse video. (See below for a link to a screenshot.) No attempt is made to check that the description is valid or that the rule set is complete.
Non-universality: as written, the program will fail if you try to use it with a Turing machine that has more than 256 distinct states or a tape that is longer than 704 cells. (In reality, of course, the ZX81's RAM would have been exhausted some time before you reached such a Goliath.) Allowing more states would be pretty trivial, assuming you had the memory space; just use as many bytes as you need. As for supporting a longer tape, the easiest way to do it would be to comment out the PRINT statements (sacrificing the animation) and add a few lines to display one screenful at a time at the very end.
1000 PRINT AT 0,0;T$
1010 LET P=1
1020 IF P>LEN T$ THEN LET T$=T$+B$
1030 PRINT AT INT (P/32),P-(32*INT (P/32)+1);CHR$ (CODE T$(P)+128)
1040 LET R=1
1050 IF R$(R,1)=S$ AND R$(R,2)=T$(P) THEN GOTO 1080
1060 LET R=R+1
1070 GOTO 1050
1080 LET T$(P)=R$(R,3)
1090 PRINT AT INT (P/32),P-(32*INT (P/32)+1);T$(P)
1100 IF R$(R,4)="L" THEN LET P=P-1
1110 IF R$(R,4)="R" THEN LET P=P+1
1120 LET S$=R$(R,5)
1130 IF S$=H$ THEN STOP
1140 IF P=0 THEN GOTO 1160
1150 GOTO 1020
1160 LET T$=B$+T$
1170 GOTO 1000
The incrementer
Works with 1k of RAM.
10 DIM R$(2,5)
20 LET S$=CHR$ (CODE "Q"+CODE "0")
30 LET H$=CHR$ (CODE "Q"+CODE "F")
40 LET R$(1)=S$+"11R"+S$
50 LET R$(2)=S$+"B1S"+H$
60 LET B$="B"
70 LET T$="111"
- Output:
1111
The three-state beaver
Requires at least 2k of RAM.
10 DIM R$(6,5)
20 LET R$(1)="A01RB"
30 LET R$(2)="A11LC"
40 LET R$(3)="B01LA"
50 LET R$(4)="B11RB"
60 LET R$(5)="C01LB"
70 LET R$(6)="C11SH"
80 LET T$=""
90 LET S$="A"
100 LET B$="0"
110 LET H$="H"
- Output:
111111
A screenshot from part-way through the execution of this machine can be found here.
If it is true that the five-state probable beaver runs for 47m cycles, then there is no point even attempting it on a slow computer like the ZX81. I don't know exactly how long it would take: but it would be months.
C
#include <stdio.h>
#include <stdarg.h>
#include <stdlib.h>
#include <string.h>
enum {
LEFT,
RIGHT,
STAY
};
typedef struct {
int state1;
int symbol1;
int symbol2;
int dir;
int state2;
} transition_t;
typedef struct tape_t tape_t;
struct tape_t {
int symbol;
tape_t *left;
tape_t *right;
};
typedef struct {
int states_len;
char **states;
int final_states_len;
int *final_states;
int symbols_len;
char *symbols;
int blank;
int state;
int tape_len;
tape_t *tape;
int transitions_len;
transition_t ***transitions;
} turing_t;
int state_index (turing_t *t, char *state) {
int i;
for (i = 0; i < t->states_len; i++) {
if (!strcmp(t->states[i], state)) {
return i;
}
}
return 0;
}
int symbol_index (turing_t *t, char symbol) {
int i;
for (i = 0; i < t->symbols_len; i++) {
if (t->symbols[i] == symbol) {
return i;
}
}
return 0;
}
void move (turing_t *t, int dir) {
tape_t *orig = t->tape;
if (dir == RIGHT) {
if (orig && orig->right) {
t->tape = orig->right;
}
else {
t->tape = calloc(1, sizeof (tape_t));
t->tape->symbol = t->blank;
if (orig) {
t->tape->left = orig;
orig->right = t->tape;
}
}
}
else if (dir == LEFT) {
if (orig && orig->left) {
t->tape = orig->left;
}
else {
t->tape = calloc(1, sizeof (tape_t));
t->tape->symbol = t->blank;
if (orig) {
t->tape->right = orig;
orig->left = t->tape;
}
}
}
}
turing_t *create (int states_len, ...) {
va_list args;
va_start(args, states_len);
turing_t *t = malloc(sizeof (turing_t));
t->states_len = states_len;
t->states = malloc(states_len * sizeof (char *));
int i;
for (i = 0; i < states_len; i++) {
t->states[i] = va_arg(args, char *);
}
t->final_states_len = va_arg(args, int);
t->final_states = malloc(t->final_states_len * sizeof (int));
for (i = 0; i < t->final_states_len; i++) {
t->final_states[i] = state_index(t, va_arg(args, char *));
}
t->symbols_len = va_arg(args, int);
t->symbols = malloc(t->symbols_len);
for (i = 0; i < t->symbols_len; i++) {
t->symbols[i] = va_arg(args, int);
}
t->blank = symbol_index(t, va_arg(args, int));
t->state = state_index(t, va_arg(args, char *));
t->tape_len = va_arg(args, int);
t->tape = NULL;
for (i = 0; i < t->tape_len; i++) {
move(t, RIGHT);
t->tape->symbol = symbol_index(t, va_arg(args, int));
}
if (!t->tape_len) {
move(t, RIGHT);
}
while (t->tape->left) {
t->tape = t->tape->left;
}
t->transitions_len = va_arg(args, int);
t->transitions = malloc(t->states_len * sizeof (transition_t **));
for (i = 0; i < t->states_len; i++) {
t->transitions[i] = malloc(t->symbols_len * sizeof (transition_t *));
}
for (i = 0; i < t->transitions_len; i++) {
transition_t *tran = malloc(sizeof (transition_t));
tran->state1 = state_index(t, va_arg(args, char *));
tran->symbol1 = symbol_index(t, va_arg(args, int));
tran->symbol2 = symbol_index(t, va_arg(args, int));
tran->dir = va_arg(args, int);
tran->state2 = state_index(t, va_arg(args, char *));
t->transitions[tran->state1][tran->symbol1] = tran;
}
va_end(args);
return t;
}
void print_state (turing_t *t) {
printf("%-10s ", t->states[t->state]);
tape_t *tape = t->tape;
while (tape->left) {
tape = tape->left;
}
while (tape) {
if (tape == t->tape) {
printf("[%c]", t->symbols[tape->symbol]);
}
else {
printf(" %c ", t->symbols[tape->symbol]);
}
tape = tape->right;
}
printf("\n");
}
void run (turing_t *t) {
int i;
while (1) {
print_state(t);
for (i = 0; i < t->final_states_len; i++) {
if (t->final_states[i] == t->state) {
return;
}
}
transition_t *tran = t->transitions[t->state][t->tape->symbol];
t->tape->symbol = tran->symbol2;
move(t, tran->dir);
t->state = tran->state2;
}
}
int main () {
printf("Simple incrementer\n");
turing_t *t = create(
/* states */ 2, "q0", "qf",
/* final_states */ 1, "qf",
/* symbols */ 2, 'B', '1',
/* blank */ 'B',
/* initial_state */ "q0",
/* initial_tape */ 3, '1', '1', '1',
/* transitions */ 2,
"q0", '1', '1', RIGHT, "q0",
"q0", 'B', '1', STAY, "qf"
);
run(t);
printf("\nThree-state busy beaver\n");
t = create(
/* states */ 4, "a", "b", "c", "halt",
/* final_states */ 1, "halt",
/* symbols */ 2, '0', '1',
/* blank */ '0',
/* initial_state */ "a",
/* initial_tape */ 0,
/* transitions */ 6,
"a", '0', '1', RIGHT, "b",
"a", '1', '1', LEFT, "c",
"b", '0', '1', LEFT, "a",
"b", '1', '1', RIGHT, "b",
"c", '0', '1', LEFT, "b",
"c", '1', '1', STAY, "halt"
);
run(t);
return 0;
printf("\nFive-state two-symbol probable busy beaver\n");
t = create(
/* states */ 6, "A", "B", "C", "D", "E", "H",
/* final_states */ 1, "H",
/* symbols */ 2, '0', '1',
/* blank */ '0',
/* initial_state */ "A",
/* initial_tape */ 0,
/* transitions */ 10,
"A", '0', '1', RIGHT, "B",
"A", '1', '1', LEFT, "C",
"B", '0', '1', RIGHT, "C",
"B", '1', '1', RIGHT, "B",
"C", '0', '1', RIGHT, "D",
"C", '1', '0', LEFT, "E",
"D", '0', '1', LEFT, "A",
"D", '1', '1', LEFT, "D",
"E", '0', '1', STAY, "H",
"E", '1', '0', LEFT, "A"
);
run(t);
}
- Output:
Simple incrementer q0 [1] 1 1 q0 1 [1] 1 q0 1 1 [1] q0 1 1 1 [B] qf 1 1 1 [1] Three-state busy beaver a [0] b 1 [0] a [1] 1 c [0] 1 1 b [0] 1 1 1 a [0] 1 1 1 1 b 1 [1] 1 1 1 b 1 1 [1] 1 1 b 1 1 1 [1] 1 b 1 1 1 1 [1] b 1 1 1 1 1 [0] a 1 1 1 1 [1] 1 c 1 1 1 [1] 1 1 halt 1 1 1 [1] 1 1
C#
using System;
using System.Collections.Generic;
using System.Diagnostics;
using System.Linq;
using System.Text;
using System.Threading;
using System.Threading.Tasks;
public class TuringMachine
{
public static async Task Main() {
var fiveStateBusyBeaver = new TuringMachine("A", '0', "H").WithTransitions(
("A", '0', '1', Right, "B"),
("A", '1', '1', Left, "C"),
("B", '0', '1', Right, "C"),
("B", '1', '1', Right, "B"),
("C", '0', '1', Right, "D"),
("C", '1', '0', Left, "E"),
("D", '0', '1', Left, "A"),
("D", '1', '1', Left, "D"),
("E", '0', '1', Stay, "H"),
("E", '1', '0', Left, "A")
);
var busyBeaverTask = fiveStateBusyBeaver.TimeAsync();
var incrementer = new TuringMachine("q0", 'B', "qf").WithTransitions(
("q0", '1', '1', Right, "q0"),
("q0", 'B', '1', Stay, "qf")
)
.WithInput("111");
foreach (var _ in incrementer.Run()) PrintLine(incrementer);
PrintResults(incrementer);
var threeStateBusyBeaver = new TuringMachine("a", '0', "halt").WithTransitions(
("a", '0', '1', Right, "b"),
("a", '1', '1', Left, "c"),
("b", '0', '1', Left, "a"),
("b", '1', '1', Right, "b"),
("c", '0', '1', Left, "b"),
("c", '1', '1', Stay, "halt")
);
foreach (var _ in threeStateBusyBeaver.Run()) PrintLine(threeStateBusyBeaver);
PrintResults(threeStateBusyBeaver);
var sorter = new TuringMachine("A", '*', "X").WithTransitions(
("A", 'a', 'a', Right, "A"),
("A", 'b', 'B', Right, "B"),
("A", '*', '*', Left, "E"),
("B", 'a', 'a', Right, "B"),
("B", 'b', 'b', Right, "B"),
("B", '*', '*', Left, "C"),
("C", 'a', 'b', Left, "D"),
("C", 'b', 'b', Left, "C"),
("C", 'B', 'b', Left, "E"),
("D", 'a', 'a', Left, "D"),
("D", 'b', 'b', Left, "D"),
("D", 'B', 'a', Right, "A"),
("E", 'a', 'a', Left, "E"),
("E", '*', '*', Right, "X")
)
.WithInput("babbababaa");
sorter.Run().Last();
Console.WriteLine("Sorted: " + sorter.TapeString);
PrintResults(sorter);
sorter.Reset().WithInput("bbbababaaabba");
sorter.Run().Last();
Console.WriteLine("Sorted: " + sorter.TapeString);
PrintResults(sorter);
Console.WriteLine(await busyBeaverTask);
PrintResults(fiveStateBusyBeaver);
void PrintLine(TuringMachine tm) => Console.WriteLine(tm.TapeString + "\tState " + tm.State);
void PrintResults(TuringMachine tm) {
Console.WriteLine($"End state: {tm.State} = {(tm.Success ? "Success" : "Failure")}");
Console.WriteLine(tm.Steps + " steps");
Console.WriteLine("tape length: " + tm.TapeLength);
Console.WriteLine();
}
}
public const int Left = -1, Stay = 0, Right = 1;
private readonly Tape tape;
private readonly string initialState;
private readonly HashSet<string> terminatingStates;
private Dictionary<(string state, char read), (char write, int move, string toState)> transitions;
public TuringMachine(string initialState, char blankSymbol, params string[] terminatingStates) {
State = this.initialState = initialState;
tape = new Tape(blankSymbol);
this.terminatingStates = terminatingStates.ToHashSet();
}
public TuringMachine WithTransitions(
params (string state, char read, char write, int move, string toState)[] transitions)
{
this.transitions = transitions.ToDictionary(k => (k.state, k.read), k => (k.write, k.move, k.toState));
return this;
}
public TuringMachine Reset() {
State = initialState;
Steps = 0;
tape.Reset();
return this;
}
public TuringMachine WithInput(string input) {
tape.Input(input);
return this;
}
public int Steps { get; private set; }
public string State { get; private set; }
public bool Success => terminatingStates.Contains(State);
public int TapeLength => tape.Length;
public string TapeString => tape.ToString();
public IEnumerable<string> Run() {
yield return State;
while (Step()) yield return State;
}
public async Task<TimeSpan> TimeAsync(CancellationToken cancel = default) {
var chrono = Stopwatch.StartNew();
await RunAsync(cancel);
chrono.Stop();
return chrono.Elapsed;
}
public Task RunAsync(CancellationToken cancel = default)
=> Task.Run(() => {
while (Step()) cancel.ThrowIfCancellationRequested();
});
private bool Step() {
if (!transitions.TryGetValue((State, tape.Current), out var action)) return false;
tape.Current = action.write;
tape.Move(action.move);
State = action.toState;
Steps++;
return true;
}
private class Tape
{
private List<char> forwardTape = new List<char>(), backwardTape = new List<char>();
private int head = 0;
private char blank;
public Tape(char blankSymbol) => forwardTape.Add(blank = blankSymbol);
public void Reset() {
backwardTape.Clear();
forwardTape.Clear();
head = 0;
forwardTape.Add(blank);
}
public void Input(string input) {
Reset();
forwardTape.Clear();
forwardTape.AddRange(input);
}
public void Move(int direction) {
head += direction;
if (head >= 0 && forwardTape.Count <= head) forwardTape.Add(blank);
if (head < 0 && backwardTape.Count <= ~head) backwardTape.Add(blank);
}
public char Current {
get => head < 0 ? backwardTape[~head] : forwardTape[head];
set {
if (head < 0) backwardTape[~head] = value;
else forwardTape[head] = value;
}
}
public int Length => backwardTape.Count + forwardTape.Count;
public override string ToString() {
int h = (head < 0 ? ~head : backwardTape.Count + head) * 2 + 1;
var builder = new StringBuilder(" ", Length * 2 + 1);
if (backwardTape.Count > 0) {
builder.Append(string.Join(" ", backwardTape)).Append(" ");
if (head < 0) (builder[h + 1], builder[h - 1]) = ('(', ')');
for (int l = 0, r = builder.Length - 1; l < r; l++, r--) (builder[l], builder[r]) = (builder[r], builder[l]);
}
builder.Append(string.Join(" ", forwardTape)).Append(" ");
if (head >= 0) (builder[h - 1], builder[h + 1]) = ('(', ')');
return builder.ToString();
}
}
}
- Output:
(1)1 1 State q0 1(1)1 State q0 1 1(1) State q0 1 1 1(B) State q0 1 1 1(1) State qf End state: qf = Success 4 steps tape length: 4 (0) State a 1(0) State b (1)1 State a (0)1 1 State c (0)1 1 1 State b (0)1 1 1 1 State a 1(1)1 1 1 State b 1 1(1)1 1 State b 1 1 1(1)1 State b 1 1 1 1(1) State b 1 1 1 1 1(0) State b 1 1 1 1(1)1 State a 1 1 1(1)1 1 State c 1 1 1(1)1 1 State halt End state: halt = Success 13 steps tape length: 6 Sorted: *(a)a a a a b b b b b * End state: X = Success 72 steps tape length: 12 Sorted: *(a)a a a a a b b b b b b b * End state: X = Success 118 steps tape length: 15 00:00:07.0626971 End state: H = Success 47176870 steps tape length: 12289
C++
#include <vector>
#include <string>
#include <iostream>
#include <algorithm>
#include <fstream>
#include <iomanip>
//--------------------------------------------------------------------------------------------------
typedef unsigned int uint;
using namespace std;
const uint TAPE_MAX_LEN = 49152;
//--------------------------------------------------------------------------------------------------
struct action { char write, direction; };
//--------------------------------------------------------------------------------------------------
class tape
{
public:
tape( uint startPos = TAPE_MAX_LEN >> 1 ) : MAX_LEN( TAPE_MAX_LEN ) { _sp = startPos; reset(); }
void reset() { clear( '0' ); headPos = _sp; }
char read(){ return _t[headPos]; }
void input( string a ){ if( a == "" ) return; for( uint s = 0; s < a.length(); s++ ) _t[headPos + s] = a[s]; }
void clear( char c ) { _t.clear(); blk = c; _t.resize( MAX_LEN, blk ); }
void action( const action* a ) { write( a->write ); move( a->direction ); }
void print( int c = 10 )
{
int ml = static_cast<int>( MAX_LEN ), st = static_cast<int>( headPos ) - c, ed = static_cast<int>( headPos ) + c + 1, tx;
for( int x = st; x < ed; x++ )
{ tx = x; if( tx < 0 ) tx += ml; if( tx >= ml ) tx -= ml; cout << _t[tx]; }
cout << endl << setw( c + 1 ) << "^" << endl;
}
private:
void move( char d ) { if( d == 'N' ) return; headPos += d == 'R' ? 1 : -1; if( headPos >= MAX_LEN ) headPos = d == 'R' ? 0 : MAX_LEN - 1; }
void write( char a ) { if( a != 'N' ) { if( a == 'B' ) _t[headPos] = blk; else _t[headPos] = a; } }
string _t; uint headPos, _sp; char blk; const uint MAX_LEN;
};
//--------------------------------------------------------------------------------------------------
class state
{
public:
bool operator ==( const string o ) { return o == name; }
string name, next; char symbol, write, direction;
};
//--------------------------------------------------------------------------------------------------
class actionTable
{
public:
bool loadTable( string file )
{
reset();
ifstream mf; mf.open( file.c_str() ); if( mf.is_open() )
{
string str; state stt;
while( mf.good() )
{
getline( mf, str ); if( str[0] == '\'' ) break;
parseState( str, stt ); states.push_back( stt );
}
while( mf.good() )
{
getline( mf, str ); if( str == "" ) continue;
if( str[0] == '!' ) blank = str.erase( 0, 1 )[0];
if( str[0] == '^' ) curState = str.erase( 0, 1 );
if( str[0] == '>' ) input = str.erase( 0, 1 );
}
mf.close(); return true;
}
cout << "Could not open " << file << endl; return false;
}
bool action( char symbol, action& a )
{
vector<state>::iterator f = states.begin();
while( true )
{
f = find( f, states.end(), curState );
if( f == states.end() ) return false;
if( ( *f ).symbol == '*' || ( *f ).symbol == symbol || ( ( *f ).symbol == 'B' && blank == symbol ) )
{ a.direction = ( *f ).direction; a.write = ( *f ).write; curState = ( *f ).next; break; }
f++;
}
return true;
}
void reset() { states.clear(); blank = '0'; curState = input = ""; }
string getInput() { return input; }
char getBlank() { return blank; }
private:
void parseState( string str, state& stt )
{
string a[5]; int idx = 0;
for( string::iterator si = str.begin(); si != str.end(); si++ )
{ if( ( *si ) == ';' ) idx++; else a[idx].append( &( *si ), 1 ); }
stt.name = a[0]; stt.symbol = a[1][0]; stt.write = a[2][0]; stt.direction = a[3][0]; stt.next = a[4];
}
vector<state> states; char blank; string curState, input;
};
//--------------------------------------------------------------------------------------------------
class utm
{
public:
utm() { files[0] = "incrementer.utm"; files[1] = "busy_beaver.utm"; files[2] = "sort.utm"; }
void start()
{
while( true )
{
reset(); int t = showMenu(); if( t == 0 ) return;
if( !at.loadTable( files[t - 1] ) ) return; startMachine();
}
}
private:
void simulate()
{
char r; action a;
while( true ) { tp.print(); r = tp.read(); if( !( at.action( r, a ) ) ) break; tp.action( &a ); }
cout << endl << endl; system( "pause" );
}
int showMenu()
{
int t = -1;
while( t < 0 || t > 3 )
{
system( "cls" ); cout << "1. Incrementer\n2. Busy beaver\n3. Sort\n\n0. Quit";
cout << endl << endl << "Choose an action "; cin >> t;
}
return t;
}
void reset() { tp.reset(); at.reset(); }
void startMachine() { system( "cls" ); tp.clear( at.getBlank() ); tp.input( at.getInput() ); simulate(); }
tape tp; actionTable at; string files[7];
};
//--------------------------------------------------------------------------------------------------
int main( int a, char* args[] ){ utm mm; mm.start(); return 0; }
//--------------------------------------------------------------------------------------------------
These are the files you'll need
File explanation:
Each line contains one tuple of the form '<current state> <current symbol> <new symbol> <direction> <new state>
B = blank, H = halt, N = do nothing, * matches any current symbol
' = marks the end of the action table
! = blank symbol => eg: !0 => 0 is the blank symbol
^ starting state
> input
Incrementer
q0;1;1;R;q0 q0;B;1;H;qf ' !0 ^q0 >111
Busy beaver
A;0;1;R;B A;1;1;L;C B;0;1;L;A B;1;1;R;B C;0;1;L;B C;1;1;N;H ' !0 ^A
Sort
A;1;1;R;A A;2;3;R;B A;0;0;L;E B;1;1;R;B B;2;2;R;B B;0;0;L;C C;1;2;L;D C;2;2;L;C C;3;2;L;E D;1;1;L;D D;2;2;L;D D;3;1;R;A E;1;1;L;E E;0;0;R;H ' !0 ^A >1221221211
- Output:
Busy beaver
000000000000000000000
^
000000000100000000000
^
000000000011000000000
^
000000000001100000000
^
000000000001110000000
^
000000000001111000000
^
000000000111110000000
^
000000001111100000000
^
000000011111000000000
^
000000111110000000000
^
000001111100000000000
^
000000111111000000000
^
000000011111100000000
^
000000011111100000000
^
Clojure
(defn tape
"Creates a new tape with given blank character and tape contents"
([blank] (tape () blank () blank))
([right blank] (tape () (first right) (rest right) blank))
([left head right blank] [(reverse left) (or head blank) (into () right) blank]))
; Tape operations
(defn- left [[[l & ls] _ rs b] c] [ls (or l b) (conj rs c) b])
(defn- right [[ls _ [r & rs] b] c] [(conj ls c) (or r b) rs b])
(defn- stay [[ls _ rs b] c] [ls c rs b])
(defn- head [[_ c _ b]] (or c b))
(defn- pretty [[ls c rs b]] (concat (reverse ls) [[(or c b)]] rs))
(defn new-machine
"Returns a function that takes a tape as input, and returns the tape
after running the machine specified in `machine`."
[machine]
(let [rules (into {} (for [[s c c' a s'] (:rules machine)]
[[s c] [c' (-> a name symbol resolve) s']]))
finished? (into #{} (:terminating machine))]
(fn [input-tape]
(loop [state (:initial machine) tape input-tape]
(if (finished? state)
(pretty tape)
(let [[out action new-state] (get rules [state (head tape)])]
(recur new-state (action tape out))))))))
Tests
(def simple-incrementer
(new-machine {:initial :q0
:terminating [:qf]
:rules [[:q0 1 1 :right :q0]
[:q0 \B 1 :stay :qf]]}))
(deftest simple-incrementer-test
(is (= [1 1 1 [1]] (simple-incrementer (tape [1 1 1] \B)))))
(def three-state-two-symbol-busy-beaver
(new-machine {:initial :a
:terminating [:halt]
:rules [[:a 0 1 :right :b]
[:a 1 1 :left :c]
[:b 0 1 :left :a]
[:b 1 1 :right :b]
[:c 0 1 :left :b]
[:c 1 1 :stay :halt]]}))
(deftest three-state-two-symbol-busy-beaver-test
(is (= [1 1 1 [1] 1 1] (three-state-two-symbol-busy-beaver (tape 0)))))
(def five-state-two-symbol-busy-beaver
(new-machine {:initial :A
:terminating [:H]
:rules [[:A 0 1 :right :B]
[:A 1 1 :left :C]
[:B 0 1 :right :C]
[:B 1 1 :right :B]
[:C 0 1 :right :D]
[:C 1 0 :left :E]
[:D 0 1 :left :A]
[:D 1 1 :left :D]
[:E 0 1 :stay :H]
[:E 1 0 :left :A]]}))
(deftest five-state-two-symbol-busy-beaver-test
(let [result (flatten (five-state-two-symbol-busy-beaver (tape 0)))
freq (frequencies result)]
(is (= 4098 (get freq 1)))
(is (= 8191 (get freq 0)))))
CLU
% Bidirectional 'infinite' tape
tape = cluster [T: type] is make, left, right, get_cell, set_cell,
elements, get_size
rep = record[
blank: T,
loc: int,
data: array[T]
]
% Make a new tape with a given blank value and initial value
make = proc (blank: T, init: sequence[T]) returns (cvt)
data: array[T]
if sequence[T]$empty(init) then
data := array[T]$[blank]
else
data := sequence[T]$s2a(init)
end
return(rep${
blank: blank,
loc: 1,
data: data
})
end make
% Move the tape head left
left = proc (tap: cvt)
tap.loc := tap.loc - 1
if tap.loc < array[T]$low(tap.data) then
array[T]$addl(tap.data,tap.blank)
end
end left
% Move the tape head right
right = proc (tap: cvt)
tap.loc := tap.loc + 1
if tap.loc > array[T]$high(tap.data) then
array[T]$addh(tap.data,tap.blank)
end
end right
% Get the value of the current cell
get_cell = proc (tap: cvt) returns (T)
return(tap.data[tap.loc])
end get_cell
% Set the value of the current cell
set_cell = proc (tap: cvt, val: T)
tap.data[tap.loc] := val
end set_cell
% Retrieve all touched values, one by one, from left to right
elements = iter (tap: cvt) yields (T)
for v: T in array[T]$elements(tap.data) do
yield(v)
end
end elements
% Get the current size of the tape
get_size = proc (tap: cvt) returns (int)
return(array[T]$size(tap.data))
end get_size
end tape
% Turing machine state table
turing = cluster [T: type] is make, add_rule, run
where T has equal: proctype (T,T) returns (bool)
A_LEFT = 'L'
A_RIGHT = 'R'
A_STAY = 'S'
state = record[name: string, term: bool]
rule = struct[
cur_state: int,
read_sym, write_sym: T,
action: char,
next_state: int
]
rep = struct[
states: array[state],
rules: array[rule],
init_state: int
]
% Find the index of a state given its name
find_state = proc (states: array[state], name: string)
returns (int) signals (bad_state)
for i: int in array[state]$indexes(states) do
if states[i].name = name then return(i) end
end
signal bad_state
end find_state
% Make a new Turing machine given a list of states
make = proc (state_seq: sequence[string], init: string, term: sequence[string])
returns (cvt) signals (bad_state)
states: array[state] := array[state]$[]
for s: string in sequence[string]$elements(state_seq) do
array[state]$addh(states, state${name: s, term: false} )
end
init_n: int := find_state(states, init) resignal bad_state
for s: string in sequence[string]$elements(term) do
term_n: int := find_state(states, s) resignal bad_state
states[term_n].term := true
end
return(rep${states: states,
init_state: init_n,
rules: array[rule]$[]})
end make
% Add a rule to the Turing machine
add_rule = proc (tur: cvt,
in_state: string,
read_sym, write_sym: T,
action: string,
out_state: string)
signals (bad_state, bad_action)
cur_state: int := find_state(tur.states, in_state) resignal bad_state
next_state: int := find_state(tur.states, out_state) resignal bad_state
act: char
if action = "left" then act := A_LEFT
elseif action = "right" then act := A_RIGHT
elseif action = "stay" then act := A_STAY
else signal bad_action
end
array[rule]$addh(tur.rules,
rule${cur_state: cur_state,
read_sym: read_sym,
write_sym: write_sym,
action: act,
next_state: next_state})
end add_rule
% Find first matching rule
find_rule = proc (rules: array[rule], st: int, sym: T)
returns (rule) signals (no_rule)
for r: rule in array[rule]$elements(rules) do
if r.cur_state = st & r.read_sym = sym then
return(r)
end
end
signal no_rule
end find_rule
% Run the Turing machine on a given tape until it terminates
run = proc (tur: cvt, tap: tape[T]) signals (no_rule)
cur: int := tur.init_state
while ~tur.states[cur].term do
r: rule := find_rule(tur.rules, cur, tap.cell) resignal no_rule
tap.cell := r.write_sym
if r.action = A_LEFT then tape[T]$left(tap)
elseif r.action = A_RIGHT then tape[T]$right(tap)
end
cur := r.next_state
end
end run
end turing
% Simple incrementer
simple_incrementer = proc () returns (turing[char])
tc = turing[char]
t: tc := tc$make(
sequence[string]$["q0", "qf"],
"q0",
sequence[string]$["qf"]
)
tc$add_rule(t, "q0", '1', '1', "right", "q0")
tc$add_rule(t, "q0", 'B', '1', "stay", "qf")
return(t)
end simple_incrementer
% Three state beaver
three_state_beaver = proc () returns (turing[char])
tc = turing[char]
t: tc := tc$make(
sequence[string]$["a", "b", "c", "halt"],
"a",
sequence[string]$["halt"]
)
tc$add_rule(t, "a", '0', '1', "right", "b")
tc$add_rule(t, "a", '1', '1', "left", "c")
tc$add_rule(t, "b", '0', '1', "left", "a")
tc$add_rule(t, "b", '1', '1', "right", "b")
tc$add_rule(t, "c", '0', '1', "left", "b")
tc$add_rule(t, "c", '1', '1', "stay", "halt")
return(t)
end three_state_beaver
% Five state beaver
five_state_beaver = proc () returns (turing[char])
tc = turing[char]
t: tc := tc$make(
sequence[string]$["A", "B", "C", "D", "E", "H"],
"A",
sequence[string]$["H"]
)
tc$add_rule(t, "A", '0', '1', "right", "B")
tc$add_rule(t, "A", '1', '1', "left", "C")
tc$add_rule(t, "B", '0', '1', "right", "C")
tc$add_rule(t, "B", '1', '1', "right", "B")
tc$add_rule(t, "C", '0', '1', "right", "D")
tc$add_rule(t, "C", '1', '0', "left", "E")
tc$add_rule(t, "D", '0', '1', "left", "A")
tc$add_rule(t, "D", '1', '1', "left", "D")
tc$add_rule(t, "E", '0', '1', "stay", "H")
tc$add_rule(t, "E", '1', '0', "left", "A")
return(t)
end five_state_beaver
% Print the first 32 touched symbols on a tape
print_tape = proc (s: stream, t: tape[char])
n: int := 32
for c: char in tape[char]$elements(t) do
stream$putc(s, c)
n := n - 1
if n=0 then break end
end
if n=0 then
stream$puts(s, "... (length: " || int$unparse(t.size) || ")")
end
stream$putl(s, "")
end print_tape
% Run the three Turing machines and show the results
start_up = proc ()
turing_factory = proctype () returns (turing[char])
test = record[name: string, tf: turing_factory, tap: tape[char]]
stest = sequence[test]
sc = sequence[char]
tests: stest := stest$[
test${name: "Simple incrementer",
tf: simple_incrementer,
tap: tape[char]$make('B', sc$['1','1','1'])},
test${name: "Three-state busy beaver",
tf: three_state_beaver,
tap: tape[char]$make('0', sc$[])},
test${name: "Five-state probable busy beaver",
tf: five_state_beaver,
tap: tape[char]$make('0', sc$[])}]
po: stream := stream$primary_output()
for t: test in stest$elements(tests) do
stream$puts(po, t.name || ": ")
tm: turing[char] := t.tf()
turing[char]$run(tm, t.tap)
print_tape(po, t.tap)
end
end start_up- Output:
Simple incrementer: 1111 Three-state busy beaver: 111111 Five-state probable busy beaver: 10100100100100100100100100100100... (length: 12289)
Common Lisp
Iterative version
The infinite tape is represented by two lists:
frontcontains all cells before the current cell in reverse order (i.e. the first element infrontis the direct predecessor of the current cell)backcontains the current cell as its first element, followed by all successors.
(defun turing (initial terminal blank rules tape &optional (verbose NIL))
(labels ((combine (front back)
(if front
(combine (cdr front) (cons (car front) back))
back))
(update-tape (old-front old-back new-content move)
(cond ((eq move 'right)
(list (cons new-content old-front)
(cdr old-back)))
((eq move 'left)
(list (cdr old-front)
(list* (car old-front) new-content (cdr old-back))))
(T (list old-front
(cons new-content (cdr old-back))))))
(show-tape (front back)
(format T "~{~a~}[~a]~{~a~}~%"
(nreverse (subseq front 0 (min 10 (length front))))
(or (car back) blank)
(subseq (cdr back) 0 (min 10 (length (cdr back)))))))
(loop for back = tape then new-back
for front = '() then new-front
for state = initial then new-state
for content = (or (car back) blank)
for (new-state new-content move) = (gethash (cons state content) rules)
for (new-front new-back) = (update-tape front back new-content move)
until (equal state terminal)
do (when verbose
(show-tape front back))
finally (progn
(when verbose
(show-tape front back))
(return (combine front back))))))
Recursive version
Using the same interface and general idea as the iterative version.
(defun turing (initial terminal blank rules tape &optional (verbose NIL))
(labels ((run (state front back)
(if (equal state terminal)
(progn
(when verbose
(show-tape front back))
(combine front back))
(let ((current-content (or (car back) blank)))
(destructuring-bind
(new-state new-content move)
(gethash (cons state current-content) rules)
(when verbose
(show-tape front back))
(cond ((eq move 'right)
(run new-state
(cons new-content front)
(cdr back)))
((eq move 'left)
(run new-state
(cdr front)
(list* (car front) new-content (cdr back))))
(T (run new-state
front
(cons new-content (cdr back)))))))))
(show-tape (front back)
(format T "~{~a~}[~a]~{~a~}~%"
(nreverse (subseq front 0 (min 10 (length front))))
(or (car back) blank)
(subseq (cdr back) 0 (min 10 (length (cdr back))))))
(combine (front back)
(if front
(combine (cdr front) (cons (car front) back))
back)))
(run initial '() tape)))
Usage
;; Helper function for creating the rules table
(defun make-rules-table (rules-list)
(let ((rules (make-hash-table :test 'equal)))
(loop for (state content new-content dir new-state) in rules-list
do (setf (gethash (cons state content) rules)
(list new-state new-content dir)))
rules))
(format T "Simple incrementer~%")
(turing 'q0 'qf 'B (make-rules-table '((q0 1 1 right q0) (q0 B 1 stay qf))) '(1 1 1) T)
(format T "Three-state busy beaver~%")
(turing 'a 'halt 0
(make-rules-table '((a 0 1 right b)
(a 1 1 left c)
(b 0 1 left a)
(b 1 1 right b)
(c 0 1 left b)
(c 1 1 stay halt)))
'() T)
(format T "Sort (final tape)~%")
(format T "~{~a~}~%"
(turing 'A 'H 0
(make-rules-table '((A 1 1 right A)
(A 2 3 right B)
(A 0 0 left E)
(B 1 1 right B)
(B 2 2 right B)
(B 0 0 left C)
(C 1 2 left D)
(C 2 2 left C)
(C 3 2 left E)
(D 1 1 left D)
(D 2 2 left D)
(D 3 1 right A)
(E 1 1 left E)
(E 0 0 right H)))
'(2 1 2 2 2 1 1)))
(format T "5-state busy beaver (first 20 cells)~%")
(format T "~{~a~}...~%"
(subseq (turing 'A 'H 0
(make-rules-table '((A 0 1 right B)
(A 1 1 left C)
(B 0 1 right C)
(B 1 1 right B)
(C 0 1 right D)
(C 1 0 left E)
(D 0 1 left A)
(D 1 1 left D)
(E 0 1 stay H)
(E 1 0 left A)))
'())
0 20))
- Output:
Simple incrementer [1]11 1[1]1 11[1] 111[B] 111[1] Three-state busy beaver [0] 1[0] [1]1 [0]11 [0]111 [0]1111 1[1]111 11[1]11 111[1]1 1111[1] 11111[0] 1111[1]1 111[1]11 111[1]11 Sort (final tape) 011122220 5-state busy beaver (first 20 cells) 10100100100100100100...
Acl
Acl is a tiny a-machine control language. Instructions are
'=' : if scanned symbol = arg do '%' : print arg '<' : move left '>' : move right '@' : state <- arg <symbol> : arg <- symbol
Program
Tape is split into two stacks.
T = reverse (l) . r where r is the scanned symbol.
0
;;; 23.01.16
(defun acl (table input state halt blank)
(loop for code = (cdr (assoc state table))
with l = nil
with r = input
with op = nil
with x = t
until (eql state halt)
do (dolist (c code)
(case c
(= (setf x (eql (car r) op)))
(% (when x (rplaca r op)))
(< (when x (push (pop l) r)))
(> (when x (push (pop r) l)))
(@ (when x (setf state op) (return)))
(t (setf op c)))
(unless (car r)
(setf r (cons blank (cdr r)))))
finally (format t "Q = (~a ~{~a~}.~{~a~})~%" state (reverse l) r)))
Tables
Tables are stored in association lists.
(defconstant +incrementer+ '((q0 . (1 = > q0 @ b = 1 % qf @))))
(defconstant +three-states-buzy-beaver+ '((a . (0 = 1 % > b @ 1 = < c @))
(b . (0 = 1 % < a @ 1 = > b @))
(c . (0 = 1 % < b @ 1 = halt @))))
Execution
(acl +incrementer+ '(1 1 1) 'q0 'qf 'b) (acl +three-states-buzy-beaver+ '(0) 'a 'halt '0)
- Output:
Q = <QF, 111.1> Q = <HALT, 111.111>
cyril nocton (cyril.nocton@gmail.com) w/ google translate
Cowgol
include "cowgol.coh";
include "strings.coh";
include "malloc.coh";
###############################################################################
########################## Turing machine definition ##########################
###############################################################################
typedef Symbol is uint8; # 256 symbols ought to be enough for everyone
const LEFT := 1;
const RIGHT := 2;
const STAY := 3;
typedef Action is int(LEFT, STAY);
# Linked list
record Linked is
next: [Linked];
end record;
record DoublyLinked: Linked is
prev: [DoublyLinked];
end record;
sub FreeLinked(r: [Linked]) is
while r != 0 as [Linked] loop
var v := r.next;
Free(r as [uint8]);
r := v;
end loop;
end sub;
sub FreeDoublyLinked(r: [DoublyLinked]) is
FreeLinked(r.next);
while r != 0 as [DoublyLinked] loop
var v := r.prev;
Free(r as [uint8]);
r := v;
end loop;
end sub;
# Turing machine
typedef Turing is [TuringR];
record StateR: Linked is
tm: Turing; # turing machine this state belongs to
term: uint8; # whether state is terminating
end record;
typedef State is [StateR];
record Cell: DoublyLinked is
sym: Symbol;
end record;
record RuleR: Linked is
instate: State;
insym: Symbol;
outsym: Symbol;
action: Action;
outstate: State;
end record;
typedef Rule is [RuleR];
record TuringR is
states: State;
rules: Rule;
initial: State;
current: State;
blank: Symbol;
head: [Cell];
end record;
sub MakeCell(): (c: [Cell]) is
c := Alloc(@bytesof Cell) as [Cell];
MemZero(c as [uint8], @bytesof Cell);
end sub;
# Define a Turing machine
sub MakeTuring(blank: Symbol, init: [Symbol]): (t: Turing) is
t := Alloc(@bytesof TuringR) as Turing;
MemZero(t as [uint8], @bytesof TuringR);
t.blank := blank;
t.head := MakeCell();
t.head.sym := blank;
var c := t.head;
var d: [Cell];
while [init] != 0 loop
c.sym := [init];
init := @next init;
if [init] == 0 then break; end if;
d := Alloc(@bytesof Cell) as [Cell];
d.prev := c as [DoublyLinked];
d.next := 0 as [Linked];
c.next := d as [Linked];
c := d;
end loop;
end sub;
# Add a state to a Turing machine
const T_NONE := 0;
const T_INIT := 1;
const T_HALT := 2;
sub MakeState(t: Turing, type: uint8): (s: State) is
s := Alloc(@bytesof StateR) as State;
s.tm := t;
s.next := t.states as [Linked];
t.states := s;
if type & T_INIT != 0 then
t.initial := s;
t.current := s;
end if;
s.term := 0;
if type & T_HALT != 0 then
s.term := 1;
end if;
end sub;
# Add a rule to a Turing machine
sub MakeRule(t: Turing,
instate: State,
insym: Symbol,
outsym: Symbol,
action: Action,
outstate: State): (r: Rule) is
r := Alloc(@bytesof RuleR) as Rule;
r.instate := instate;
r.insym := insym;
r.outsym := outsym;
r.action := action;
r.outstate := outstate;
r.next := t.rules as [Linked];
t.rules := r;
end sub;
# Free a Turing machine
sub FreeTuring(t: Turing) is
FreeDoublyLinked(t.head as [DoublyLinked]);
FreeLinked(t.states as [Linked]);
FreeLinked(t.rules as [Linked]);
Free(t as [uint8]);
end sub;
# Move the head
sub MoveHead(t: Turing, a: Action) is
var c: [Cell];
case a is
when STAY: return;
when LEFT:
if t.head.prev == 0 as [DoublyLinked] then
c := Alloc(@bytesof Cell) as [Cell];
c.prev := 0 as [DoublyLinked];
c.next := t.head as [Linked];
c.sym := t.blank;
t.head.prev := c as [DoublyLinked];
end if;
t.head := t.head.prev as [Cell];
return;
when RIGHT:
if t.head.next == 0 as [Linked] then
c := Alloc(@bytesof Cell) as [Cell];
c.next := 0 as [Linked];
c.prev := t.head as [DoublyLinked];
c.sym := t.blank;
t.head.next := c as [Linked];
end if;
t.head := t.head.next as [Cell];
return;
when else:
print("Invalid action\n");
ExitWithError();
end case;
end sub;
# Step a Turing machine
sub Step(t: Turing): (halt: uint8) is
# If we're in a halt state, do nothing
if t.current.term != 0 then
halt := 1;
return;
end if;
var r := t.rules;
while r != 0 as Rule loop
# Check each rule to see if it matches the current configuration
if t.current == r.instate
and t.head.sym == r.insym then
# Found a match
t.head.sym := r.outsym;
MoveHead(t, r.action);
t.current := r.outstate;
halt := t.current.term;
return;
end if;
r := r.next as Rule;
end loop;
print("No valid rule!\n");
ExitWithError();
end sub;
# Run a Turing machine until it halts
sub Run(t: Turing) is
while Step(t) == 0 loop
end loop;
end sub;
# Print the touched part of the tape of a Turing machine
sub PrintTape(t: Turing, max: uint32) is
var c := t.head;
var len: uint32 := 0;
while c.prev != 0 as [DoublyLinked] loop
c := c.prev as [Cell];
end loop;
while c != 0 as [Cell] loop
if len < max then
print_char(c.sym as uint8);
end if;
c := c.next as [Cell];
len := len + 1;
end loop;
if len >= max then
print("... (total length: ");
print_i32(len);
print(")");
end if;
end sub;
###############################################################################
######################## Turing machines from the task ########################
###############################################################################
interface TuringFactory(): (t: Turing);
sub SimpleIncrementer implements TuringFactory is
var r: Rule;
t := MakeTuring('B', "111");
var q0 := MakeState(t, T_INIT);
var qf := MakeState(t, T_HALT);
r := MakeRule(t, q0, '1', '1', RIGHT, q0);
r := MakeRule(t, q0, 'B', '1', STAY, qf);
end sub;
sub ThreeStateBeaver implements TuringFactory is
var r: Rule;
t := MakeTuring('0', "");
var a := MakeState(t, T_INIT);
var b := MakeState(t, T_NONE);
var c := MakeState(t, T_NONE);
var halt := MakeState(t, T_HALT);
r := MakeRule(t, a, '0', '1', RIGHT, b);
r := MakeRule(t, a, '1', '1', LEFT, c);
r := MakeRule(t, b, '0', '1', LEFT, a);
r := MakeRule(t, b, '1', '1', RIGHT, b);
r := MakeRule(t, c, '0', '1', LEFT, b);
r := MakeRule(t, c, '1', '1', STAY, halt);
end sub;
sub FiveStateBeaver implements TuringFactory is
var r: Rule;
t := MakeTuring('0', "");
var A := MakeState(t, T_INIT);
var B := MakeState(t, T_NONE);
var C := MakeState(t, T_NONE);
var D := MakeState(t, T_NONE);
var E := MakeState(t, T_NONE);
var H := MakeState(t, T_HALT);
r := MakeRule(t, A, '0', '1', RIGHT, B);
r := MakeRule(t, A, '1', '1', LEFT, C);
r := MakeRule(t, B, '0', '1', RIGHT, C);
r := MakeRule(t, B, '1', '1', RIGHT, B);
r := MakeRule(t, C, '0', '1', RIGHT, D);
r := MakeRule(t, C, '1', '0', LEFT, E);
r := MakeRule(t, D, '0', '1', LEFT, A);
r := MakeRule(t, D, '1', '1', LEFT, D);
r := MakeRule(t, E, '0', '1', STAY, H);
r := MakeRule(t, E, '1', '0', LEFT, A);
end sub;
record TF is
name: [uint8];
tf: TuringFactory;
end record;
var machines: TF[] := {
{"Simple incrementer", SimpleIncrementer},
{"Three state beaver", ThreeStateBeaver},
{"Five state beaver", FiveStateBeaver}
};
var i: @indexof machines;
i := 0;
while i < @sizeof machines loop
print(machines[i].name);
print(": ");
var t := (machines[i].tf) ();
Run(t);
PrintTape(t, 32);
FreeTuring(t);
print_nl();
i := i + 1;
end loop;- Output:
Simple incrementer: 1111 Three state beaver: 111111 Five state beaver: 10100100100100100100100100100100... (total length: 12289)
D
Nearly Strongly Typed Version
This is typed a little less strongly than the Ada entry. It's fast and safe.
import std.stdio, std.algorithm, std.string, std.conv, std.array,
std.exception, std.traits, std.math, std.range;
struct UTM(State, Symbol, bool doShow=true)
if (is(State == enum) && is(Symbol == enum)) {
static assert(is(typeof({ size_t x = State.init; })),
"State must to be usable as array index.");
static assert([EnumMembers!State].equal(EnumMembers!State.length.iota),
"State must be a plain enum.");
static assert(is(typeof({ size_t x = Symbol.init; })),
"Symbol must to be usable as array index.");
static assert([EnumMembers!Symbol].equal(EnumMembers!Symbol.length.iota),
"Symbol must be a plain enum.");
enum Direction { right, left, stay }
private const TuringMachine tm;
private TapeHead head;
alias SymbolMap = string[EnumMembers!Symbol.length];
// The first index of this 'rules' matrix is a subtype of State
// because it can't contain H, but currently D can't enforce this,
// statically unlike Ada language.
Rule[EnumMembers!Symbol.length][EnumMembers!State.length - 1] mRules;
static struct Rule {
Symbol toWrite;
Direction direction;
State nextState;
this(in Symbol toWrite_, in Direction direction_, in State nextState_)
pure nothrow @safe @nogc {
this.toWrite = toWrite_;
this.direction = direction_;
this.nextState = nextState_;
}
}
// This is kept separated from the rest so it can be inialized
// one field at a time in the main function, yet it will become
// const.
static struct TuringMachine {
Symbol blank;
State initialState;
Rule[Symbol][State] rules;
Symbol[] input;
SymbolMap symbolMap;
}
static struct TapeHead {
immutable Symbol blank;
Symbol[] tapeLeft, tapeRight;
int position;
const SymbolMap sMap;
size_t nSteps;
this(in ref TuringMachine t) pure nothrow @safe {
this.blank = EnumMembers!Symbol[0];
//tapeRight = t.input.empty ? [this.blank] : t.input.dup;
if (t.input.empty)
this.tapeRight = [this.blank];
else
this.tapeRight = t.input.dup;
this.position = 0;
this.sMap = t.symbolMap;
}
pure nothrow @safe @nogc invariant {
assert(this.tapeRight.length > 0);
if (this.position >= 0)
assert(this.position < this.tapeRight.length);
else
assert(this.position.abs <= this.tapeLeft.length);
}
Symbol readSymb() const pure nothrow @safe @nogc {
if (this.position >= 0)
return this.tapeRight[this.position];
else
return this.tapeLeft[this.position.abs - 1];
}
void showSymb() const @safe {
this.write;
}
void writeSymb(in Symbol symbol) @safe {
static if (doShow)
showSymb;
if (this.position >= 0)
this.tapeRight[this.position] = symbol;
else
this.tapeLeft[this.position.abs - 1] = symbol;
}
void goRight() pure nothrow @safe {
this.position++;
if (position > 0 && position == tapeRight.length)
tapeRight ~= blank;
}
void goLeft() pure nothrow @safe {
this.position--;
if (position < 0 && (position.abs - 1) == tapeLeft.length)
tapeLeft ~= blank;
}
void move(in Direction dir) pure nothrow @safe {
nSteps++;
final switch (dir) with (Direction) {
case left: goLeft; break;
case right: goRight; break;
case stay: /*Do nothing*/ break;
}
}
string toString() const @safe {
immutable pos = tapeLeft.length.signed + this.position + 4;
return format("...%-(%)...", tapeLeft.retro.chain(tapeRight)
.map!(s => sMap[s])) ~
'\n' ~
format("%" ~ pos.text ~ "s", "^") ~
'\n';
}
}
void show() const @safe {
head.showSymb;
}
this(in ref TuringMachine tm_) @safe {
static assert(__traits(compiles, State.H), "State needs a 'H' (Halt).");
immutable errMsg = "Invalid input.";
auto runningStates = remove!(s => s == State.H)([EnumMembers!State]);
enforce(!runningStates.empty, errMsg);
enforce(tm_.rules.length == EnumMembers!State.length - 1, errMsg);
enforce(State.H !in tm_.rules, errMsg);
enforce(runningStates.canFind(tm_.initialState), errMsg);
// Create a matrix to reduce running time.
foreach (immutable State st, const rset; tm_.rules)
foreach (immutable Symbol sy, immutable rule; rset)
mRules[st][sy] = rule;
this.tm = tm_;
head = TapeHead(this.tm);
State state = tm.initialState;
while (state != State.H) {
immutable next = mRules[state][head.readSymb];
head.writeSymb(next.toWrite);
head.move(next.direction);
state = next.nextState;
}
static if (doShow)
show;
writeln("Performed ", head.nSteps, " steps.");
}
}
void main() @safe {
"Incrementer:".writeln;
enum States1 : ubyte { A, H }
enum Symbols1 : ubyte { s0, s1 }
alias M1 = UTM!(States1, Symbols1);
M1.TuringMachine tm1;
with (tm1) with (States1) with (Symbols1) with (M1.Direction) {
alias R = M1.Rule;
initialState = A;
rules = [A: [s0: R(s1, stay, H), s1: R(s1, right, A)]];
input = [s1, s1, s1];
symbolMap = ["0", "1"];
}
M1(tm1);
// http://en.wikipedia.org/wiki/Busy_beaver
"\nBusy Beaver machine (3-state, 2-symbol):".writeln;
enum States2 : ubyte { A, B, C, H }
alias Symbols2 = Symbols1;
alias M2 = UTM!(States2, Symbols2);
M2.TuringMachine tm2;
with (tm2) with (States2) with (Symbols2) with (M2.Direction) {
alias R = M2.Rule;
initialState = A;
rules = [A: [s0: R(s1, right, B), s1: R(s1, left, C)],
B: [s0: R(s1, left, A), s1: R(s1, right, B)],
C: [s0: R(s1, left, B), s1: R(s1, stay, H)]];
symbolMap = ["0", "1"];
}
M2(tm2);
"\nSorting stress test (12212212121212):".writeln;
enum States3 : ubyte { A, B, C, D, E, H }
enum Symbols3 : ubyte { s0, s1, s2, s3 }
alias M3 = UTM!(States3, Symbols3, false);
M3.TuringMachine tm3;
with (tm3) with (States3) with (Symbols3) with (M3.Direction) {
alias R = M3.Rule;
initialState = A;
rules = [A: [s1: R(s1, right, A),
s2: R(s3, right, B),
s0: R(s0, left, E)],
B: [s1: R(s1, right, B),
s2: R(s2, right, B),
s0: R(s0, left, C)],
C: [s1: R(s2, left, D),
s2: R(s2, left, C),
s3: R(s2, left, E)],
D: [s1: R(s1, left, D),
s2: R(s2, left, D),
s3: R(s1, right, A)],
E: [s1: R(s1, left, E),
s0: R(s0, stay, H)]];
input = [s1, s2, s2, s1, s2, s2, s1,
s2, s1, s2, s1, s2, s1, s2];
symbolMap = ["0", "1", "2", "3"];
}
M3(tm3).show;
"\nPossible best Busy Beaver machine (5-state, 2-symbol):".writeln;
alias States4 = States3;
alias Symbols4 = Symbols1;
alias M4 = UTM!(States4, Symbols4, false);
M4.TuringMachine tm4;
with (tm4) with (States4) with (Symbols4) with (M4.Direction) {
alias R = M4.Rule;
initialState = A;
rules = [A: [s0: R(s1, right, B), s1: R(s1, left, C)],
B: [s0: R(s1, right, C), s1: R(s1, right, B)],
C: [s0: R(s1, right, D), s1: R(s0, left, E)],
D: [s0: R(s1, left, A), s1: R(s1, left, D)],
E: [s0: R(s1, stay, H), s1: R(s0, left, A)]];
symbolMap = ["0", "1"];
}
M4(tm4);
}
- Output:
Incrementer:
...111...
^
...111...
^
...111...
^
...1110...
^
...1111...
^
Performed 4 steps.
Busy Beaver machine (3-state, 2-symbol):
...0...
^
...10...
^
...11...
^
...011...
^
...0111...
^
...01111...
^
...11111...
^
...11111...
^
...11111...
^
...11111...
^
...111110...
^
...111111...
^
...111111...
^
...111111...
^
Performed 13 steps.
Sorting stress test (12212212121212):
Performed 118 steps.
...0111111222222220...
^
Possible best Busy Beaver machine (5-state, 2-symbol):
Performed 47176870 steps.
The total run-time is about 0.31 seconds.
Simple Version
While the precedent version is Ada-like, this is more like a script.
import std.stdio, std.typecons, std.algorithm, std.string, std.array;
void turing(Sy, St)(in St state, Sy[int] tape, in int pos,
in Tuple!(Sy, int, St)[Sy][St] rules) {
if (state.empty) return;
const r = rules[state][tape[pos] = tape.get(pos, Sy.init)];
writefln("%-(%s%)", tape.keys.sort()
.map!(i => format(i == pos ? "(%s)" : " %s ", tape[i])));
tape[pos] = r[0];
turing(r[2], tape, pos + r[1], rules);
}
void main() {
turing("a", null, 0,
["a": [0: tuple(1, 1, "b"), 1: tuple(1, -1, "c")],
"b": [0: tuple(1, -1, "a"), 1: tuple(1, 1, "b")],
"c": [0: tuple(1, -1, "b"), 1: tuple(1, 0, "")]]);
}
- Output:
(0) 1 (0) (1) 1 (0) 1 1 (0) 1 1 1 (0) 1 1 1 1 1 (1) 1 1 1 1 1 (1) 1 1 1 1 1 (1) 1 1 1 1 1 (1) 1 1 1 1 1 (0) 1 1 1 1 (1) 1 1 1 1 (1) 1 1
Déjà Vu
transitions(:
local :t {}
while /= ) dup:
set-to t swap & rot & rot rot &
t drop
take-from tape:
if tape:
pop-from tape
else:
:B
paste-together a h b:
push-to b h
while a:
push-to b pop-from a
b
universal-turing-machine transitions initial final tape:
local :tape-left []
local :state initial
local :head take-from tape
local :move { :stay @pass }
move!left:
push-to tape head
set :head take-from tape-left
move!right:
push-to tape-left head
set :head take-from tape
while /= state final:
if opt-get transitions & state head:
set :state &<>
set :head &<>
move!
else:
return paste-together tape-left head tape
paste-together tape-left head tapeSimple incrementer
:q0 :qf [ 1 1 1 ]
)
:q0 1 1 :right :q0
:q0 :B 1 :stay :qf
!. universal-turing-machine transitions(- Output:
[ 1 1 1 1 ]
Three-state busy beaver
:a :halt []
)
:a :B 1 :right :b
:a 1 1 :left :c
:b :B 1 :left :a
:b 1 1 :right :b
:c :B 1 :left :b
:c 1 1 :stay :halt
!. universal-turing-machine transitions(- Output:
[ 1 1 1 1 1 1 ]
5-state, 2-symbol probable Busy Beaver machine
:A :H []
)
:A :B 1 :right :B
:A 1 1 :left :C
:B :B 1 :right :C
:B 1 1 :right :B
:C :B 1 :right :D
:C 1 :B :left :E
:D :B 1 :left :A
:D 1 1 :left :D
:E :B 1 :stay :H
:E 1 :B :left :A
!. universal-turing-machine transitions((Output omitted because of length.)
EchoLisp
We define a Turing machine as an instance of TM struct, which stores the definition values (states,symbols,rules) and the current state values (state, tape, position). It can be stopped, restarted, called as a sub-program, or transformed into a sequence or stream.'Huge' TM are run in the background. Rules are compiled into a vector indexed by state * symbol.
Turing Machines
(require 'struct)
(struct TM (read-only: name states symbs final rules mem state-values: tape pos state))
(define-syntax-rule (rule-idx state symb numstates)
(+ state (* symb numstates)))
(define-syntax-rule (make-TM name states symbs rules)
(_make-TM name 'states 'symbs 'rules))
;; a rule is (state symbol --> write move new-state)
;; index for rule = state-num + (number of states) * symbol-num
;; convert states/symbol into vector indices
(define (compile-rule T rule into: rules)
(define numstates (vector-length (TM-states T)))
(define state (vector-index [rule 0](TM-states T) )) ; index
(define symb (vector-index [rule 1](TM-symbs T) ))
(define write-symb (vector-index [rule 2] (TM-symbs T) ))
(define move (1- (vector-index [rule 3] #(left stay right) )))
(define new-state (vector-index [rule 4](TM-states T)))
(define rulenum (rule-idx state symb numstates))
(vector-set! rules rulenum (vector write-symb move new-state))
; (writeln 'rule rulenum [rules rulenum])
)
(define (_make-TM name states symbs rules)
(define T (TM name (list->vector states) (list->vector symbs) null null))
(set-TM-final! T (1- (length states))) ;; assume one final state
(set-TM-rules! T (make-vector (* (length states) (length symbs))))
(for ((rule rules)) (compile-rule T (list->vector rule) into: (TM-rules T)))
T ) ; returns a TM
;;------------------
;; TM-trace
;;-------------------
(string-delimiter "")
(define (TM-print T symb-index: symb (hilite #f))
(cond
((= 0 symb) (if hilite "🔲" "◽️" ))
((= 1 symb) (if hilite "🔳 " "◾️" ))
(else "X")))
(define (TM-trace T tape pos state step)
(if (= (TM-final T) state)
(write "🔴")
(write "🔵"))
(for [(p (in-range (- (TM-mem T) 7) (+ (TM-mem T) 8)))]
(write (TM-print T [tape p] (= p pos))))
(write step)
(writeln))
;;---------------
;; TM-init : alloc and init tape
;;---------------
(define (TM-init T input-symbs (mem 20))
;; init state variables
(set-TM-tape! T (make-vector (* 2 mem)))
(set-TM-pos! T mem)
(set-TM-state! T 0)
(set-TM-mem! T mem)
(for [(symb input-symbs) (i (in-naturals))]
(vector-set! (TM-tape T) [+ i (TM-pos T)] (vector-index symb (TM-symbs T))))
(TM-trace T (TM-tape T) mem 0 0)
mem )
;;---------------
;; TM-run : run at most maxsteps
;;---------------
(define (TM-run T (verbose #f) (maxsteps 1_000_000))
(define count 0)
(define final (TM-final T))
(define rules (TM-rules T))
(define rule 0)
(define numstates (vector-length (TM-states T)))
;; set current state vars
(define pos (TM-pos T))
(define state (TM-state T))
(define tape (TM-tape T))
(when (and (zero? state) (= pos (TM-mem T)))
(writeln 'Starting (TM-name T))
(TM-trace T tape pos 0 count))
(while (and (!= state final) (< count maxsteps))
(++ count)
;; The machine
(set! rule [rules (rule-idx state [tape pos] numstates)])
(when (= rule 0) (error "missing rule" (list state [tape pos])))
(vector-set! tape pos [rule 0])
(set! state [rule 2])
(+= pos [rule 1])
;; end machine
(when verbose (TM-trace T tape pos state count )))
;; save TM state
(set-TM-pos! T pos)
(set-TM-state! T state)
(when (= final state) (writeln 'Stopping (TM-name T) 'at-pos (- pos (TM-mem T))))
count)
- Output:
(define T (make-TM "TM: incrementer"
(q0 qf)
(B 1)
((q0 1 1 right q0)
(q0 B 1 stay qf))))
(TM-init T '(1 1 1) 20)
(TM-run T #t)
(TM-run T #t)
🔵 ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ 🔳 ◾️ ◾️ ◽️ ◽️ ◽️ ◽️ ◽️ 0
Starting TM: incrementer
🔵 ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ 🔳 ◾️ ◾️ ◽️ ◽️ ◽️ ◽️ ◽️ 0
🔵 ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ ◾️ 🔳 ◾️ ◽️ ◽️ ◽️ ◽️ ◽️ 1
🔵 ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ ◾️ ◾️ 🔳 ◽️ ◽️ ◽️ ◽️ ◽️ 2
🔵 ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ ◾️ ◾️ ◾️ 🔲 ◽️ ◽️ ◽️ ◽️ 3
🔴 ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ ◾️ ◾️ ◾️ 🔳 ◽️ ◽️ ◽️ ◽️ 4
Stopping TM: incrementer at-pos 3
;; three-states busy beaver
(define T (make-TM "TM: three-states busy beaver"
(a b c halt)
(0 1)
((a 0 1 right b)
(a 1 1 left c)
(b 0 1 left a)
(b 1 1 right b)
(c 0 1 left b)
(c 1 1 stay halt))))
(TM-init T null 100)
Starting TM: three-states busy beaver
🔵 ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ 🔲 ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ 0
🔵 ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ ◾️ 🔲 ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ 1
🔵 ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ 🔳 ◾️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ 2
🔵 ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ 🔲 ◾️ ◾️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ 3
🔵 ◽️ ◽️ ◽️ ◽️ ◽️ 🔲 ◾️ ◾️ ◾️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ 4
🔵 ◽️ ◽️ ◽️ ◽️ 🔲 ◾️ ◾️ ◾️ ◾️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ 5
🔵 ◽️ ◽️ ◽️ ◽️ ◾️ 🔳 ◾️ ◾️ ◾️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ 6
🔵 ◽️ ◽️ ◽️ ◽️ ◾️ ◾️ 🔳 ◾️ ◾️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ 7
🔵 ◽️ ◽️ ◽️ ◽️ ◾️ ◾️ ◾️ 🔳 ◾️ ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ 8
🔵 ◽️ ◽️ ◽️ ◽️ ◾️ ◾️ ◾️ ◾️ 🔳 ◽️ ◽️ ◽️ ◽️ ◽️ ◽️ 9
🔵 ◽️ ◽️ ◽️ ◽️ ◾️ ◾️ ◾️ ◾️ ◾️ 🔲 ◽️ ◽️ ◽️ ◽️ ◽️ 10
🔵 ◽️ ◽️ ◽️ ◽️ ◾️ ◾️ ◾️ ◾️ 🔳 ◾️ ◽️ ◽️ ◽️ ◽️ ◽️ 11
🔵 ◽️ ◽️ ◽️ ◽️ ◾️ ◾️ ◾️ 🔳 ◾️ ◾️ ◽️ ◽️ ◽️ ◽️ ◽️ 12
🔴 ◽️ ◽️ ◽️ ◽️ ◾️ ◾️ ◾️ 🔳 ◾️ ◾️ ◽️ ◽️ ◽️ ◽️ ◽️ 13
Stopping TM: three-states busy beaver at-pos 0
Possible best busy beaver
We create a task to run it in the background.
- Output:
(define steps 0)
(define (TM-task T)
(define count (TM-run T #f 1000000))
(when (zero? steps) (writeln 'START (date)))
(+= steps count)
(writeln 'TM-steps steps (date))
(when (zero? count) (writeln 'END steps (date)))
(if (zero? count) #f T)) ;; return #f to signal end of task
;; 5-states 2-symbols busy beaver
;; Result: 4098 "1"s with 8191 "0"s interspersed in 47,176,870 steps.
(lib 'tasks)
(define T (make-TM "TM: 5-states 2-symbols busy beaver"
(A B C D E H)
(0 1)
((A 0 1 right B)
(A 1 1 left C)
(B 0 1 right C)
(B 1 1 right B)
(C 0 1 right D)
(C 1 0 left E)
(D 0 1 left A)
(D 1 1 left D)
(E 0 1 stay H)
(E 1 0 left A))))
(TM-init T null 20000)
(task-run (make-task TM-task T) 1000)
;; Firefox : 192 sec Chrome:342 sec Safari: 232 sec
START Mon Feb 08 2016 18:34:15 GMT+0100 (CET)
TM-steps 47176870 Mon Feb 08 2016 18:38:23 GMT+0100 (CET)
END 47176870 Mon Feb 08 2016 18:38:23 GMT+0100 (CET)
;; check result : count written "1"
(for/sum ((s (TM-tape T))) s)
→ 4098
EDSAC order code
This program was designed with the aim of running the 5-state busy beaver on an EDSAC simulator. It can be done, although the run time in real life would have been impossibly long (see below).
We know from other solutions that the Turing tape for the 5-state busy beaver requires 12289 cells. Since EDSAC memory was 1024 17-bit words, we are limited to one bit per cell, and hence to two symbols, say 0 and 1. This is enough for the Rosetta Code tasks, though not for the extra sorting task.
The tape is implemented as a circular buffer. Positive positions ascend from the low end of the tape area, while negative positions descend from the high end. Thus we don't need to know in advance where the zero position of the tape should be. Each EDSAC location in the tape area corresponds to 16 cells of the tape (17 would be awkward to code). If the positive and negative pointers collide, the program prints "OV" to indicate overflow.
When running the 5-state busy beaver, the simulator executed 2.7 billion EDSAC orders, which would have taken about 48 days on the original EDSAC. (Cf the estimate of "months" for the same task in ZX81 Basic.)
[Attempt at Turing machine for Rosetta Code.]
[EDSAC program, Initial Orders 2.]
[Library subroutine M3 prints header and is then overwritten.]
PFGKIFAFRDLFUFOFE@A6FG@E8FEZPF
*!!NR!STEPS@&#..PZ [..PZ marks end of header]
T48K [& (delta) parameter: Turing machine tape.]
P8F [Overwrites most of initial orders.]
T50K [X parameter: once-only code.]
P100F [Gets overwritten by the Turing machine tape.]
[Put the following as high in memory as possible,
to make room for the Turing machine tape.]
T52K [A parameter: rules and initial pattern. Also marks end]
P781F [of Turing tape, so must go immediately after tape area.]
T55K [V parameter: program-wide variables.]
P810F [Even address, 9 locations]
T46K [N parameter: constants.]
P820F [Even address]
T47K [M parameter: main routine.]
P859F
T51K [G parameter: library subroutine P7]
P988F [Even address, 35 locations.]
[============================= A parameter ===============================]
E25K TA GK
[0] [End of Turing tape area]
[Comment-in the desired task, or add another (2 symbols only).]
[Counts are stored in the address field.]
[Each rule is defined by an EDSAC pseudo-order, as follows:
Function letter: L = left, R = right, S = stay
Address field = new state number
Code letter: F if new symbol = 0, D if new symbol = 1.
No rule is needed for the halt state.]
[0]
[Simple incrementer: states are q0 = 0, qf = halt = 1
P1F [1 state, excluding the halt state
S1D RD [2 rules for each state (symbols 0 and 1)
P1F [1 word in tape area to be initialized
PF P3D [location 0 relative to tape, init to 7]
[3-state busy beaver: states are a = 0, b = 1, c = 2, halt = 3]
P3F [3 states, excluding the halt state]
R1D L2D [2 rules for each state (symbols 0 and 1)]
L0D R1D
L1D S3D
PF [0 words to be initialized (start with empty tape)]
[5-state busy beaver: states are A = 0, ..., E = 4, halt = 5
P5F 5 states, excluding the halt state
R1D L2D 2 rules for each state (symbols 0 and 1)
R2D R1D
R3D L4F
L0D L3D
S5D L0F
PF 0 words to be initialized (start with empty tape)]
[============================= X parameter ===============================]
E25K TX GK
[The following once-only code is loaded into the Turing machine tape area.]
[It runs at start-up, then gets overwritten when the tape is cleared.]
[Enter with acc = 0.]
[0] T2V [initial state assumed to be state 0]
T3V [tape head starts at position 0 on Turing tape]
T#V [reset count of steps]
T4V [initialize maximum position]
T5V [initialize minimum position]
[Calculate number of available tape positions; store in address field]
A22N [T order for exclusive end of tape]
S21N [T order for start of tape]
L4F [times 16, since each location holds 16 positions]
T25N [store for later use]
[Set up the loop in the main program that writes the initial pattern.
The main program has a list of position-value pairs.
This follows the list of rules, 2 rules per Turing machine state.]
[9] AA [number of states]
LD [times 2, because 2 rules per state]
A2F [plus 1 for the count of states]
A9@ [make A order to load number of position-value pairs]
T14@ [plant order]
[14] AM [load number of pairs (in address field)]
LD [times 2 for length of table]
TF [temp store in 0F]
A14@ [load order that was planted above]
A2F [make order to load first position]
U13M [plant in main routine]
AF [make A order for exclusive end]
T28M [plant in main routine]
[Set up order to load rules]
A26@
A2F
T18N
[Here with acc = 0. Jump to main routine.]
EM
[26] AA
[============================= V parameter ===============================]
E25K TV GK
[0] PFPF [number of steps (35-bit, must be at even address)]
[2] PF [current state of Turing machine]
[3] PF [current tape position, stored in address field]
[4] PF [maximum position on the tape so far]
[5] PF [minimum position on the tape so far]
[6] PF [rule for current state and symbol]
[7] PF [working group of 16 cells (1 EDSAC location)]
[8] PF [mask to select bit for current cell]
[============================= N parameter ===============================]
E25K TN GK
[17-bit masks: 11111111111111110, 11111111111111101, ..., 10111111111111111]
[0] V2047F V2046D V2045D V2043D V2039D V2031D V2015D V1983D
V1919D V1791D V1535D V1023D C2047D B2047D G2047D M2047D
[16] OF [add to A order to make T order with same address]
[17] AN [A order to load first mask in table]
[18] AF [A order to load first rule]
[19] A& [A order for start of tape]
[20] AA [A order for end of tape]
[21] T& [T order for start of tape]
[22] TA [T order for exclusive end of tape]
[23] P2047F [mask to pick out state from a Turing machine rule]
[24] P15F [mask to extract bit number from position]
[25] PF [number of tape positions available (calculated)]
[26] @F [carriage return]
[27] &F [line feed]
[28] K4096F [null]
[29] K2048F [set letters on teleprinter]
[============================= M parameter ===============================]
E25K TM GK
[Once-only code jumps to here with acc = 0]
[Clear the tape; this overwrites the once-only code]
[0] A21N [load T order for start of tape]
E3@ [always jump (since T > 0)]
[2] A22N [loop here after testing for end]
[3] T4@ [plant order to clear 1 location]
[4] TF [execute order]
A4@ [load order just executed]
A2F [inc address]
S22N [test for end]
G2@ [if not end, loop back]
[Here with acc = 0]
[Set up the starting pattern, i.e write 1's at zero or more positions on the tape.]
[To save space, the orders marked (*) are set up by the once-only code.]
[9] A13@ [load A order for next relative addess]
S28@ [compare with A order for exclusive end]
E29@ [if all done, jump out with acc = 0]
TF [clear acc]
[13] AF [(*) load relative address from table]
G17@ [jump if < 0]
A21N [make T order, addr counted from low end of tape]
E18@ [join commoon code (always jumps since T > 0)]
[17] A22N [make T order, addr counted from high end of tape]
[18] T23@ [plant T order in code]
A13@ [make order to load value from table]
A2F
T22@ [plant in code]
[22] AF [load value from table]
[23] TF [store in tape]
A22@ [make A order for next address]
A2F
T13@ [plant in code]
E9@ [always loop back]
[28] AF [(*) A order for exclusive end of list]
[29]
[Next step, i.e. set up new symbol, state and tape position.]
[Acc must be 0 here.]
[Get tape position and deduce corresponding EDSAC location and bit number.]
[29] H24N [mask for bit number]
C3V [acc := bit number]
UF [save bit number in 0F address field]
A17N [make order to load from mask table]
T44@ [plant order in code]
A3V [position]
SF [remove bit number part]
R4F [divide by 16 for relative address]
[If it's a non-negative address, add it to the start of the tape in EDSAC memory.]
[If it's a negative address, add it to the end of the tape.]
G40@ [jump if negative address]
A19N [make A order to load from tape]
G41@ [always jump to common code, since A < 0]
[40] A20N [here if negative address]
[41] U46@ [store order to load current group of 16 bits]
A16N [convert to T order at same address]
T69@ [store T order (a fair way down the code)]
[44] AF [load mask]
T8V
[46] AF [load group]
T7V
[Get rule for this state and symbol (where symbol = 0 or 1)]
H8V
C7V [acc := bit group with current bit cleared]
S7V [acc := 0 if bit is 0, -1 if bit is 1]
E54@
TF [clear acc]
A2F [to inc rule address if symbol is 1]
[54] A2V [add state twice (because each state has 2 rules)]
A2V
A18N [manufacture A order to load rule]
T58@
[58] AF [load rule]
T6V [to work space]
[Write new symbol (0 or 1) to tape. New symbol is in low bit of rule.]
HN [H register := 111...1110]
C6V
S6V [result = 0 if new symbol is 0; -1 if it's 1]
H8V [H register = mask 1...101...1 for current bit]
G67@ [jump to set the bit]
C7V [clear the bit]
E69@ [always jump (because top bit in tape store is always 0)]
[Set bit, assuming acc = -1 here (reason why it works is a bit complicated)]
[67] C7V
S8V
[69] TF [manufactured order]
[Update position of tape head, i.e. inc by 1, dec by 1, or no change.]
[Move is in top 2 bits of rule, thus]
[1x = move left, i.e. dec position (function letter can be L)]
[00 = move right, i.e. inc position (function letter can be R)]
[01 = stay, i.e. don't change position (function letter can be S)]
A6V
G83@ [left if top bit is 1]
[72] LD [else test next bit]
G95@ [skip move if next bit is 1]
[74] TF [here to move right]
A3V [inc position]
A2F
U3V
[Here we update the maximum position if latest >= maximum.]
[This is unnecessary if latest = maximum, but code is simpler this way.]
S4V [test against maximum position]
G95@ [skip if latest < maximum]
A4V [restore after test]
T4V [update maximum]
E91@ [always jump, to check for overflow]
[83] TF [here to move left]
A3V [dec position]
S2F
[86] U3V
S5V [test against current minimum position]
E95@ [jump if >= minimum]
A5V [restore acc after test]
T5V [update minimum]
[After updating maximum or minimum position, check that
available memory hasn't been exceeded.]
[91] A4V [maximum position]
S5V [subtract minimum position]
S25N [compare against number available]
E107@ [jump out if overflow]
[The next order also serves as a constant]
[95] TF [clear acc for next part]
[Increment the number of steps]
A#V
YFYF
T#V
[Finally set the new state.]
[100] H23N [mask for state bits in rule]
C6V [acc := new state]
SA [is it the last state?]
E111@ [if yes, halt the Turing machine]
AA [restore acc after test]
T2V [update state]
E29@ [loop back for next step]
[Overflow, i.e. non-negative tape positions (ascending in EDSAC memory)
collide with negative tape positions (descending).]
[107] O29N [set teleprinter to letters]
O107@ ON [print 'OV' to indicate overflow]
E116@ [jump to exit]
[Print number of steps]
[111] TF A#V [clear accc, load number of steps]
TD [number of steps to 0D for print subroutine
[114] A114 @GG [call print subroutine]
[116] O26N O27N [print CR, LF]
O28N [print null to flush teleprinter buffer]
ZF [stop]
[============================= G parameter ===============================]
E25K TG
[Library subroutine P7. 35 locations, even address. WWG page 18.]
[Prints non-negative integer, up to 10 digits, right-justified.]
GKA3FT26@H28#@NDYFLDT4DS27@TFH8@S8@T1FV4DAFG31@SFLDUFOFFFSFL4F
T4DA1FA27@G11@XFT28#ZPFT27ZP1024FP610D@524D!FO30@SFL8FE22@
[========================== X parameter again ===============================]
E25K TX GK
EZ [define entry point]
PF [enter with acc = 0]
[end]- Output:
[Simple incrementer]
NR STEPS
4
[3-state busy beaver]
NR STEPS
13
[5-state busy beaver; requires fast EDSAC simulator]
NR STEPS
47176870
Erlang
The following code is an Escript which can be placed into a file and run as escript filename or simply marked as executable and run directly via the provided shebang header. -type and -spec declarations have not been used; using the typer utility can get a head start on this process should a more robust solution be desired.
In this universal Turing machine simulator, a machine is defined by giving it a configuration function that returns the initial state, the halting states and the blank symbol, as well as a function for the rules. These are passed in to the public interface turing/3 as funs, together with the initial tape setup.
#!/usr/bin/env escript
-module(turing).
-mode(compile).
-export([main/1]).
% Incrementer definition:
% States: a | halt
% Initial state: a
% Halting states: halt
% Symbols: b | '1'
% Blank symbol: b
incrementer_config() -> {a, [halt], b}.
incrementer(a, '1') -> {'1', right, a};
incrementer(a, b) -> {'1', stay, halt}.
% Busy beaver definition:
% States: a | b | c | halt
% Initial state: a
% Halting states: halt
% Symbols: '0' | '1'
% Blank symbol: '0'
busy_beaver_config() -> {a, [halt], '0'}.
busy_beaver(a, '0') -> {'1', right, b};
busy_beaver(a, '1') -> {'1', left, c};
busy_beaver(b, '0') -> {'1', left, a};
busy_beaver(b, '1') -> {'1', right, b};
busy_beaver(c, '0') -> {'1', left, b};
busy_beaver(c, '1') -> {'1', stay, halt}.
% Mainline code.
main([]) ->
io:format("==============================~n"),
io:format("Turing machine simulator test.~n"),
io:format("==============================~n"),
Tape1 = turing(fun incrementer_config/0, fun incrementer/2, ['1','1','1']),
io:format("~w~n", [Tape1]),
Tape2 = turing(fun busy_beaver_config/0, fun busy_beaver/2, []),
io:format("~w~n", [Tape2]).
% Universal Turing machine simulator.
turing(Config, Rules, Input) ->
{Start, _, _} = Config(),
{Left, Right} = perform(Config, Rules, Start, {[], Input}),
lists:reverse(Left) ++ Right.
perform(Config, Rules, State, Input = {LeftInput, RightInput}) ->
{_, Halts, Blank} = Config(),
case lists:member(State, Halts) of
true -> Input;
false ->
{NewRight, Symbol} = symbol(RightInput, Blank),
{NewSymbol, Action, NewState} = Rules(State, Symbol),
NewInput = action(Action, Blank, {LeftInput, [NewSymbol| NewRight]}),
perform(Config, Rules, NewState, NewInput)
end.
symbol([], Blank) -> {[], Blank};
symbol([S|R], _) -> {R, S}.
action(left, Blank, {[], Right}) -> {[], [Blank|Right]};
action(left, _, {[L|Ls], Right}) -> {Ls, [L|Right]};
action(stay, _, Tape) -> Tape;
action(right, Blank, {Left, []}) -> {[Blank|Left], []};
action(right, _, {Left, [R|Rs]}) -> {[R|Left], Rs}.
Fortran
The plan
A Turing machine is the apotheosis of spaghetti code, because every action is followed by a GO TO another state. This is rather like SNOBOL, where every statement succeeds or fails, and is (optionally) followed by a GO TO for each condition. This in turn means that there is no special ordering of the states, aside from indicating which is to be the starting point. One can reorder them to taste. Accordingly, the plan is to number the states 1, 2, etc. with zero reserved to mean HALT and the starting state to be number one, no matter what names happen to be associated with a description of the states and their transitions. This has the advantage that the test for stopping is not STATE = ENDSTATE where STATE and ENDSTATE are some cute texts involving "HALT" or somesuch, but just STATE = 0, and not some arbitrary integer in a variable but a constant and a special constant at that. In short, the computer should be able to make this test without any subtraction, just inspecting the value of STATE as by loading into the accumulator whereby indicators are set. Alas, intel cpus employ cmp dword ptr [STATE (00476620)],0 so they do perform a subtraction anyway, and I wonder why I bother. But, principle is to be upheld!
Similarly, various symbols are described as being on the tape but they can be deemed to be symbol number 0, symbol number 1, and so on, as numerical values and not character codes (be they ASCII or whatever), in whatever order is convenient.
With this in mind, after some cogitation it was clear that the transition tables could be specified in a basic way involving only numerical values, and that each state would have NSYMBOL entries. Thus, the table would be indexed as a two-dimensional array, (STATE,SYMBOL) where SYMBOL was the (number of) the symbol under the tape's read/write head. Since this project involves a variety of Turing machines that manipulate a varying collection of symbols, the transition table would have to be dimensioned according to the problem. But this is distasteful, because it means that the array indexing involves variables rather than constants, slowing the calculation. Similarly, one could define a compound data aggregate such as TRANSITION(state,symbol) with sub entries MARK, MOVE, and NEXT at the cost of more complex indexing again - plus requiring the facilities of F90 and later.
Instead the plan was to use one dimensional arrays, and to perform the indexing via IT = STATE*NSYMBOL - S for symbol number S after which the transition entries would be MARK(IT), MOVE(IT), and NEXT(IT). This scheme has no CASE-statements nor compound IF-statements to identify the individual states and the particular symbol under the read/write head so as to select the correct transition. Hopefully, the compiler will recognise the similar shape of each array and reduce its code accordingly. The subtraction of the symbol code rather than addition is due to the more normal IT = (STATE - 1)*NSYMBOL + S requiring an extra subtraction, which is to be avoided. Otherwise the transition table would have unused entries at the start (because state zero requires no transitions) and this waste causes an itch.
The input scheme initially required a numbers-only style of description (so free-format reading could be used), and translating to the reduced scheme from the different presentation styles of the example Turing machine descriptions was tricky. The initial format required that the transitions for each state be entered in reverse numerical order of the number of the symbol being recognised, with no mention of the symbol because it was deduced by the order of entry, just the MARK, MOVE, and NEXT values, as follows:
1 State 1 1,-1, 3 1,+1, 2 2 State 2 1,+1, 2 1,-1, 1
Since only one number was required on the STATE lines, a comment can follow the value. Although this avoided redundant data (there being no need to give the state number on every line as well) it proved to be troublesome to prepare, so for the larger descriptions (involving a conversion of state names to numbers, and symbol glyphs to symbol numbers), a retreat: present the symbol number, and allow the entries to be in any order - such as increasing. When the transition table as ingested is printed, this means that the entry numbers are in an odd order, but no matter. Should there be an error in the input, a mess will result but preparing a more flexible input arrangement would require a lot more code.
Specifying the initial state of the tape proved a bit tricky in the case when it was to be entirely blank, as end-of-file problems arose. So for that case a single zero was supplied and the problem went away... The tape's initial state is all zero anyway. Remember that in the scheme here this is symbol number zero, not "0".
After the first version was working, annoyance with the numerical symbol specification scheme prompted an escalation to the ability to specify an alphabet of arbitrary symbols, rather than symbol number 0, number 1, etc. but this could cause difficulty with the free-format reading, especially if one symbol were to be a blank. It transpired that character data could be read into character variables in free-format with or without quotes around them, provided that outside quotes a desired symbol is neither a space nor a comma. Thus the initial specification of two symbols might be "01" or "10" or "AB" or AB for the content of SYMBOLS, and when the transition table is read, the supplied symbol is sought within SYMBOLS. The internal working continues with the number of the symbol, not the symbol's numerical value.
The next escalation concerned attempts to keep track of the bounds of the read/write head's position. The original display was a bit disappointing as the great expanse of zero values was oppressive to the eye. By introducing variables HMIN and HMAX a track could be kept of what span of the tape had been explicitly initialised and seen by the read/write head. Tape cells outside those bounds still have symbol number zero (sometimes named as B) but blanks were shown. However, maintaining those bounds added to the effort of the simple interpretation loop, and after some thought it became clear that a special additional symbol could be introduced so that symbol number zero would have a space as its symbol, and the declared symbols would be numbered one to N rather than 0 to N - 1. By making its action entries the same as for the first symbol, all would be well. At the end of the run that tape (alas, a finite tape) could be scanned to ascertain the bounds of action.
In short, state one is special, it is the starting state. Symbol one (that would previously have been counted as symbol number zero) is special, it is synonymous with blank tape - which holds symbol(0), which when printed is a blank...
Source
This is in F77 style, and just one mainline. The Turing action is effected by just five statements:
200 I = STATE*NSYMBOL - ICHAR(TAPE(HEAD)) !Index the transition.
TAPE(HEAD) = MARK(I) !Do it. Possibly not changing the symbol.
HEAD = HEAD + MOVE(I) !Possibly not moving the head.
STATE = ICHAR(NEXT(I)) !Hopefully, something has changed!
IF (STATE.GT.0) GO TO 200 !Otherwise, we might loop forever...
Which manifests the ability to read and write memory, simple arithmetic, and a conditional branch. This is all it takes. And thanks to Kurt Gödel, it is known that simple arithmetic is enough to construct statements that are true, but unprovable.
The use of ICHAR() and CHAR() hopefully involves no machine code effort, they are merely a genuflection to the rules of type checking to calm the compiler. Most of the blather is concerned with acquiring the description (with minimal checking), and preparing some trace output.
Showing the head movement with a plus sign for positive movement but not zero proved a bit tricky, given that an uninitialised MOVE entry has the obtrusive value of -60 so that function SIGN doesn't help: SIGN(1,-60) gives -1, but SIGN(1,0) gives 1. SIGN(MAX(-1,i),i) would do and now this is becoming messy, so array indexing trickery is out unless a special function is to be supplied - but this is a mainline-only attempt. The SP format code causes the appearance of a plus for subsequent fields but alas, zero has a plus sign also. So, write to a text variable, place the plus sign if wanted, and write the resulting text.
PROGRAM U !Reads a specification of a Turing machine, and executes it.
Careful! Reserves a symbol #0 to represent blank tape as a blank.
INTEGER MANY,FIRST,LAST !Some sizes must be decided upon.
PARAMETER (MANY = 66, FIRST = 1, LAST = 666) !These should do.
INTEGER HERE(MANY)
CHARACTER*1 MARK(MANY) !The transition table.
INTEGER*1 MOVE(MANY) !Three related arrays.
CHARACTER*1 NEXT(MANY) !All with the same indexing.
CHARACTER*1 TAPE(FIRST:LAST)!Notionally, no final bound, in both directions - a potential infinity..
INTEGER STATE !Execution starts with state 1.
INTEGER HEAD !And the tape read/write head at position 1.
INTEGER STEP !And we might as well keep count.
INTEGER OFFSET !An affine shift.
INTEGER NSTATE !Counts can be helpful.
INTEGER NSYMBOL !The count of recognised symbols.
INTEGER S,S1 !Symbol numbers.
CHARACTER*1 RS,WS !Input scanning: read symbol, write symbol.
CHARACTER*1 SYMBOL(0:MANY) !I reserve SYMBOL(0).
CHARACTER*(MANY) SYMBOLS !Up to 255, for single character variables.
EQUIVALENCE (SYMBOL(1),SYMBOLS) !Individually or collectively.
INTEGER I,J,K,L,IT !Assistants.
INTEGER LONG !Now for some text scanning.
PARAMETER (LONG = 80) !This should suffice.
CHARACTER*(LONG) ALINE !A scratchpad.
REAL T0,T1 !Some CPU time attempts.
INTEGER KBD,MSG,INF !Some I/O unit numbers.
KBD = 5 !Standard input.
MSG = 6 !Standard output
INF = 10 !Suitable for a disc file.
OPEN (INF,FILE = "TestAdd1.dat",ACTION="READ") !Go for one.
READ (INF,1) ALINE !The first line is to be a heding.
1 FORMAT (A) !Just plain text.
WRITE (MSG,2) ALINE !Reveal it.
2 FORMAT ("Turing machine simulation for... ",A) !Announce the plan.
READ (INF,*) SYMBOLS !Allows a quoted string.
NSYMBOL = LEN_TRIM(SYMBOLS) !How many symbols? (Trailing spaces will be lost)
WRITE (MSG,3) NSYMBOL,SYMBOLS(1:NSYMBOL) !They will be symbol number 0, 1, ..., NSYMBOL - 1.
3 FORMAT (I0," symbols: >",A,"<") !And this is their count.
IF (NSYMBOL.LE.1) STOP "Expect at least two symbols!"
SYMBOL(0) = " " !My special state meaning "never before seen".
NSYMBOL = NSYMBOL + 1 !So, one more is in actual use.
NSTATE = 0 !As for states, I haven't seen any.
MOVE = -66 !This should cause trouble and be noticed!
MARK = CHAR(0) !In case a state is omitted.
NEXT = CHAR(0) !Like, mention state seven, but omit mention of state six.
HERE = 0 !Clear the counts.
Collate the transition table.
10 READ (INF,*) STATE !Read this once, rather than for every transition.
IF (STATE.LE.0) GO TO 20 !Ah, finished.
WRITE (MSG,11) STATE !But they can come in any order.
NSTATE = MAX(STATE,NSTATE)!And I'd like to know how many.
11 FORMAT ("Entry: Read Write Move Next. For state ",I0) !Prepare a nice heading.
IF (STATE.LE.0) STOP "Positive STATE numbers only!" !It may not be followed.
IF (STATE*NSYMBOL.GT.MANY) STOP"My transition table is too small!" !But the value of STATE is shown.
DO S = 0,NSYMBOL - 1 !Initialise the transitions for STATE.
IT = STATE*NSYMBOL - S !Finger the one for S.
MARK(IT) = CHAR(S) !No change to what's under the head.
NEXT(IT) = CHAR(0) !And this stops the run.
END DO !Just in case a symbol's number is omitted.
DO S = 1,NSYMBOL - 1 !A transition for every symbol must be given or the read process will get out of step.
READ(INF,*) RS,WS,K,L !Read symbol, write symbol, move, next.
I = INDEX(SYMBOLS(1:NSYMBOL - 1),RS) !Convert the character to a symbol number.
J = INDEX(SYMBOLS(1:NSYMBOL - 1),WS) !To enable decorative glyphs, not just digits.
IF (I.LE.0) STOP "Unrecognised read symbol!" !This really should be more helpful.
IF (J.LE.0) STOP "Unrecognised write symbol!" !By reading into ALINE and showing it, etc.
IT = STATE*NSYMBOL - I !Locate the entry for the state x symbol pair.
MARK(IT) = CHAR(J) !The value to be written.
MOVE(IT) = K !The movement of the tape head.
NEXT(IT) = CHAR(L) !The next state.
IF (I.EQ.1) S1 = IT !This transition will be duplicated. SYMBOL(1) is for blank tape.
END DO !On to the next symbol's transition.
Copy SYMBOL(1)'s transition to the transition for the secret extra, SYMBOL(0).
IT = STATE*NSYMBOL !Finger the interpolated entry for SYMBOL(0).
MARK(IT) = MARK(S1) !Thus will SYMBOL(0), shown as a space, be overwritten.
MOVE(IT) = MOVE(S1) !And SYMBOL(0) treated
NEXT(IT) = NEXT(S1) !Exactly as if it were SYMBOL(1).
Cast forth the transition table for STATE, not mentioning SYMBOL(0) - but see label 911.
DO S = 1,NSYMBOL - 1 !Roll them out in the order as given in SYMBOL.
IT = STATE*NSYMBOL - S !But the entry number will be odd.
WRITE (ALINE,12) IT,SYMBOL(S), !The character's code value is irrelevant.
1 SYMBOL(ICHAR(MARK(IT))),MOVE(IT),ICHAR(NEXT(IT)) !Append the details just read.
12 FORMAT (I5,":",2X,'"',A1,'"',3X'"',A1,'"',I5,I5,I13) !Revealing the symbols, not their number.
IF (MOVE(IT).GT.0) ALINE(21:21) = "+" !I want a leading + for positive, not zero.
WRITE (MSG,1) ALINE(1:27) !The SP format code is unhelpful for zero.
END DO !Hopefully, I'm still in sync with the input.
GO TO 10 !Perhaps another state follows.
Chew tape. The initial state is some sequence of symbols, starting at TAPE(1).
20 TAPE = CHAR(0) !Set every cell to zero. Not blank.
OFFSET = 12 !Affine shift. The numerical value of HEAD is not seen.
READ (INF,1) ALINE !Get text, for the tape's initial state.
L = LEN_TRIM(ALINE) !Last non-blank. Flexible format this isn't.
DO I = 1,L !Step through cells 1 to L.
TAPE(I + OFFSET - 1) = CHAR(INDEX(SYMBOLS,ALINE(I:I))) !Character code to symbol number.
END DO !Rather than reading as I1.
CLOSE (INF) !Finished with the input, and not much checking either.
WRITE (MSG,*) !Take a breath.
Cast forth a heading..
WRITE (MSG,99) !Announce.
99 FORMAT ("Starts with State 1 and the tape head at 1.") !Positioned for OFFSET = 12.
ALINE = " Step: Head State|Tape..." !Prepare a heading for the trace.
L = 18 + OFFSET*2 !Locate the start position.
ALINE(L - 1:L + 1) = "<H>"!No underlining, no overprinting, no colour (neither background nor foreground). Sigh.
WRITE (MSG,1) ALINE !Take that!
CALL CPU_TIME(T0) !Start the clock.
HEAD = OFFSET !This is counted as position one.
STATE = 1 !The initial state.
STEP = 0 !No steps yet.
Chase through the transitions. Could check that HEAD is within bounds FIRST:LAST.
100 IF (STEP.GE.200) GO TO 200 !Perhaps an extended campaign.
STEP = STEP + 1 !Otherwise, here we go.
DO I = 1,LONG/2 !Scan TAPE(1:LONG/2).
IT = 2*I - 1 !Allowing two positions each.
ALINE(IT:IT) = " " !So a leading space.
ALINE(IT + 1:IT + 1) = SYMBOL(ICHAR(TAPE(I))) !And the indicated symbol.
END DO !On to the enxt.
I = HEAD*2 !The head's location in the display span.
IF (I.GT.1 .AND. I.LT.LONG) THEN !Within range?
IF (ALINE(I:I).EQ.SYMBOL(0)) ALINE(I:I) = SYMBOL(1) !Yes. Am I looking at a new cell?
ALINE(I - 1:I - 1) = "<" !Bracket the head's cell.
ALINE(I + 1:I + 1) = ">" !In ALINE.
END IF !So much for showing the head's position.
WRITE (MSG,102) STEP,HEAD - OFFSET + 1,STATE,ALINE !Splot the state.
102 FORMAT (I5,":",I5,I6,"|",A) !Aligns with FORMAT 99.
I = STATE*NSYMBOL - ICHAR(TAPE(HEAD)) !For this STATE and the symbol under TAPE(HEAD)
HERE(I) = HERE(I) + 1 !Count my visits.
TAPE(HEAD) = MARK(I) !Place the new symbol.
HEAD = HEAD + MOVE(I) !Move the head.
IF (HEAD.LT.FIRST .OR. HEAD.GT.LAST) GO TO 110 !Check the bounds.
STATE = ICHAR(NEXT(I)) !The new state.
IF (STATE.GT.0) GO TO 100 !Go to it.
Cease.
I = HEAD*2 !Locate HEAD within ALINE.
IF (I.GT.1 .AND. I.LT.LONG) ALINE(I:I) = SYMBOL(ICHAR(TAPE(HEAD))) !The only change.
WRITE (MSG,103) HEAD - OFFSET + 1,STATE,ALINE !Show.
103 FORMAT ("HALT!",I6,I6,"|",A) !But, no step count to start with. See FORMAT 102.
GO TO 900 !Done.
Can't continue! Insufficient tape, alas.
110 WRITE (MSG,*) "Insufficient tape!" !Oh dear.
GO TO 900 !Give in.
Change into high gear: no trace and no test thereof neither.
200 STEP = STEP + 1 !So, advance.
IF (MOD(STEP,10000000).EQ.0) WRITE (MSG,201) STEP !Ah, still some timewasting.
201 FORMAT ("Step ",I0) !No screen action is rather discouraging.
I = STATE*NSYMBOL - ICHAR(TAPE(HEAD)) !Index the transition.
HERE(I) = HERE(I) + 1 !Another visit.
TAPE(HEAD) = MARK(I) !Do it. Possibly not changing the symbol.
HEAD = HEAD + MOVE(I) !Possibly not moving the head.
IF (HEAD.LT.FIRST .OR. HEAD.GT.LAST) GO TO 110 !But checking the bounds just in case.
STATE = ICHAR(NEXT(I)) !Hopefully, something has changed!
IF (STATE.GT.0) GO TO 200 !Otherwise, we might loop forever...
Closedown.
900 CALL CPU_TIME(T1) !Where did it all go?
WRITE (MSG,901) STEP,STATE !Announce the ending.
901 FORMAT ("After step ",I0,", state = ",I0,".") !Thus.
DO I = FIRST,LAST !Scan the tape.
IF (ICHAR(TAPE(I)).NE.0) EXIT !This is the whole point of SYMBOL(0).
END DO !So that the bounds
DO J = LAST,FIRST,-1 !Of tape access
IF (ICHAR(TAPE(J)).NE.0) EXIT !(and placement of the initial state)
END DO !Can be found without tedious ongoing MIN and MAX.
WRITE (MSG,902) HEAD - OFFSET + 1, !Tediously,
1 I - OFFSET + 1, !Reverse the offset
2 J - OFFSET + 1 !So as to seem that HEAD = 1, to start with.
902 FORMAT ("The head is at position ",I0, !Now announce the results.
1 " and wandered over ",I0," to ",I0) !This will affect the dimension chosen for TAPE.
T1 = T1 - T0 !Some time may have been accurately measured.
IF (T1.GT.0.1) WRITE (MSG,903) T1 !And this may be sort of correct.
903 FORMAT ("CPU time",F9.3) !Though distinct from elapsed time.
Curious about the usage of the transition table?
910 WRITE (MSG,911) !Possibly not,
911 FORMAT (/,35X,"Usage.") !But here it comes.
DO STATE = 1,NSTATE !For every state
WRITE (MSG,11) STATE !Name the state, as before.
DO S = 0,NSYMBOL - 1 !But this time, roll every symbol.
IT = STATE*NSYMBOL - S !Including my "secret" symbol.
WRITE (ALINE,12) IT,SYMBOL(S), !The same sequence,
1 SYMBOL(ICHAR(MARK(IT))),MOVE(IT),ICHAR(NEXT(IT)),HERE(IT) !But with an addendum here.
IF (MOVE(IT).GT.0) ALINE(21:21) = "+" !SIGN(i,i) gives -1, 0, +1 but -60 for -60.
WRITE (MSG,1) ALINE(1:40) !When what I want is -1. SIGN(1,i) doesn't give zero.
END DO !On to the next symbol in the order as supplied.
END DO !And the next state, in numbers order.
END !That was fun.
Results
Rather than show the input file and its possibly oddly-ordered entries, its presentation in the output is more regular if with oddly-ordered entry numbering... The symbols on the tape are presented in a two-character field so that the one under the tape read/write head can be shown <bracketed>.
Simple base one incrementer
Turing machine simulation for... A simple base 1 incrementer.
2 symbols: >01<
Entry: Read Write Move Next. For state 1
2: "0" "1" 0 0
1: "1" "1" +1 1
Starts with State 1 and the tape head at 1.
Step: Head State|Tape... <H>
1: 1 1| <1>1 1
2: 2 1| 1<1>1
3: 3 1| 1 1<1>
4: 4 1| 1 1 1<0>
HALT! 4 0| 1 1 1<1>
After step 4, state = 0.
The head is at position 4 and wandered over 1 to 4
Usage.
Entry: Read Write Move Next. For state 1
3: " " "1" 0 0 1
2: "0" "1" 0 0 0
1: "1" "1" +1 1 3
After the run the transition table is shown again, augmented by counts for each entry, and with the extra symbol shown as well as those in the supplied table. Here is the same specification but with 01 replaced by BA:
A simple base 1 incrementer. BA 1 State 1 A, A,+1, 1 B, A, 0, 0 0 No more states. AAA
With the equivalent output. No significance is ascribed to the glyphs, other than that the first symbol is taken as being synonymous with blank tape. For each state, the entry for that first symbol is duplicated for symbol zero, which is reserved for blank tape that will be depicted as blank not whatever the first symbol is - which can't be a space, given free-format input without quoting...
Turing machine simulation for... A simple base 1 incrementer.
2 symbols: >BA<
Entry: Read Write Move Next. For state 1
2: "B" "A" 0 0
1: "A" "A" +1 1
Starts with State 1 and the tape head at 1.
Step: Head State|Tape... <H>
1: 1 1| <A>A A
2: 2 1| A<A>A
3: 3 1| A A<A>
4: 4 1| A A A<B>
HALT! 4 0| A A A<A>
After step 4, state = 0.
The head is at position 4 and wandered over 1 to 4
Usage.
Entry: Read Write Move Next. For state 1
3: " " "A" 0 0 1
2: "B" "A" 0 0 0
1: "A" "A" +1 1 3
In other words, the glyphs chosen for symbol number one, symbol number two, etc. do not matter so long as they are used consistently.
Three-state Busy Beaver
Turing machine simulation for... A three-state Busy Beaver.
2 symbols: >01<
Entry: Read Write Move Next. For state 1
2: "0" "1" +1 2
1: "1" "1" -1 3
Entry: Read Write Move Next. For state 2
5: "0" "1" -1 1
4: "1" "1" +1 2
Entry: Read Write Move Next. For state 3
8: "0" "1" -1 2
7: "1" "1" 0 0
Starts with State 1 and the tape head at 1.
Step: Head State|Tape... <H>
1: 1 1| <0>
2: 2 2| 1<0>
3: 1 1| <1>1
4: 0 3| <0>1 1
5: -1 2| <0>1 1 1
6: -2 1| <0>1 1 1 1
7: -1 2| 1<1>1 1 1
8: 0 2| 1 1<1>1 1
9: 1 2| 1 1 1<1>1
10: 2 2| 1 1 1 1<1>
11: 3 2| 1 1 1 1 1<0>
12: 2 1| 1 1 1 1<1>1
13: 1 3| 1 1 1<1>1 1
HALT! 1 0| 1 1 1<1>1 1
After step 13, state = 0.
The head is at position 1 and wandered over -2 to 3
Usage.
Entry: Read Write Move Next. For state 1
3: " " "1" +1 2 1
2: "0" "1" +1 2 1
1: "1" "1" -1 3 2
Entry: Read Write Move Next. For state 2
6: " " "1" -1 1 3
5: "0" "1" -1 1 0
4: "1" "1" +1 2 4
Entry: Read Write Move Next. For state 3
9: " " "1" -1 2 1
8: "0" "1" -1 2 0
7: "1" "1" 0 0 1
Sorting
The TestSort.dat file is the last prepared in the reverse symbol order. Translating from the verbose description to the reduced style required care, especially because it did not specify some combinations, that presumably would never arise.
Turing machine simulation for... A sorting test.
4 symbols: >0123<
Entry: Read Write Move Next. For state 1
4: "0" "0" -1 5
3: "1" "1" +1 1
2: "2" "3" +1 2
1: "3" "3" 0 0
Entry: Read Write Move Next. For state 2
9: "0" "0" -1 3
8: "1" "1" +1 2
7: "2" "2" +1 2
6: "3" "3" 0 0
Entry: Read Write Move Next. For state 3
14: "0" "0" 0 0
13: "1" "2" -1 4
12: "2" "2" -1 3
11: "3" "2" -1 5
Entry: Read Write Move Next. For state 4
19: "0" "0" 0 0
18: "1" "1" -1 4
17: "2" "2" -1 4
16: "3" "1" +1 1
Entry: Read Write Move Next. For state 5
24: "0" "0" +1 0
23: "1" "1" -1 5
22: "2" "2" 0 0
21: "3" "3" 0 0
Starts with State 1 and the tape head at 1.
Step: Head State|Tape... <H>
1: 1 1| <2>2 2 1 2 2 1 2 1 2 1 2 1 2
2: 2 2| 3<2>2 1 2 2 1 2 1 2 1 2 1 2
3: 3 2| 3 2<2>1 2 2 1 2 1 2 1 2 1 2
4: 4 2| 3 2 2<1>2 2 1 2 1 2 1 2 1 2
5: 5 2| 3 2 2 1<2>2 1 2 1 2 1 2 1 2
6: 6 2| 3 2 2 1 2<2>1 2 1 2 1 2 1 2
7: 7 2| 3 2 2 1 2 2<1>2 1 2 1 2 1 2
8: 8 2| 3 2 2 1 2 2 1<2>1 2 1 2 1 2
9: 9 2| 3 2 2 1 2 2 1 2<1>2 1 2 1 2
10: 10 2| 3 2 2 1 2 2 1 2 1<2>1 2 1 2
11: 11 2| 3 2 2 1 2 2 1 2 1 2<1>2 1 2
12: 12 2| 3 2 2 1 2 2 1 2 1 2 1<2>1 2
13: 13 2| 3 2 2 1 2 2 1 2 1 2 1 2<1>2
14: 14 2| 3 2 2 1 2 2 1 2 1 2 1 2 1<2>
15: 15 2| 3 2 2 1 2 2 1 2 1 2 1 2 1 2<0>
16: 14 3| 3 2 2 1 2 2 1 2 1 2 1 2 1<2>0
17: 13 3| 3 2 2 1 2 2 1 2 1 2 1 2<1>2 0
18: 12 4| 3 2 2 1 2 2 1 2 1 2 1<2>2 2 0
19: 11 4| 3 2 2 1 2 2 1 2 1 2<1>2 2 2 0
20: 10 4| 3 2 2 1 2 2 1 2 1<2>1 2 2 2 0
21: 9 4| 3 2 2 1 2 2 1 2<1>2 1 2 2 2 0
22: 8 4| 3 2 2 1 2 2 1<2>1 2 1 2 2 2 0
23: 7 4| 3 2 2 1 2 2<1>2 1 2 1 2 2 2 0
24: 6 4| 3 2 2 1 2<2>1 2 1 2 1 2 2 2 0
25: 5 4| 3 2 2 1<2>2 1 2 1 2 1 2 2 2 0
26: 4 4| 3 2 2<1>2 2 1 2 1 2 1 2 2 2 0
27: 3 4| 3 2<2>1 2 2 1 2 1 2 1 2 2 2 0
28: 2 4| 3<2>2 1 2 2 1 2 1 2 1 2 2 2 0
29: 1 4| <3>2 2 1 2 2 1 2 1 2 1 2 2 2 0
30: 2 1| 1<2>2 1 2 2 1 2 1 2 1 2 2 2 0
31: 3 2| 1 3<2>1 2 2 1 2 1 2 1 2 2 2 0
32: 4 2| 1 3 2<1>2 2 1 2 1 2 1 2 2 2 0
33: 5 2| 1 3 2 1<2>2 1 2 1 2 1 2 2 2 0
34: 6 2| 1 3 2 1 2<2>1 2 1 2 1 2 2 2 0
35: 7 2| 1 3 2 1 2 2<1>2 1 2 1 2 2 2 0
36: 8 2| 1 3 2 1 2 2 1<2>1 2 1 2 2 2 0
37: 9 2| 1 3 2 1 2 2 1 2<1>2 1 2 2 2 0
38: 10 2| 1 3 2 1 2 2 1 2 1<2>1 2 2 2 0
39: 11 2| 1 3 2 1 2 2 1 2 1 2<1>2 2 2 0
40: 12 2| 1 3 2 1 2 2 1 2 1 2 1<2>2 2 0
41: 13 2| 1 3 2 1 2 2 1 2 1 2 1 2<2>2 0
42: 14 2| 1 3 2 1 2 2 1 2 1 2 1 2 2<2>0
43: 15 2| 1 3 2 1 2 2 1 2 1 2 1 2 2 2<0>
44: 14 3| 1 3 2 1 2 2 1 2 1 2 1 2 2<2>0
45: 13 3| 1 3 2 1 2 2 1 2 1 2 1 2<2>2 0
46: 12 3| 1 3 2 1 2 2 1 2 1 2 1<2>2 2 0
47: 11 3| 1 3 2 1 2 2 1 2 1 2<1>2 2 2 0
48: 10 4| 1 3 2 1 2 2 1 2 1<2>2 2 2 2 0
49: 9 4| 1 3 2 1 2 2 1 2<1>2 2 2 2 2 0
50: 8 4| 1 3 2 1 2 2 1<2>1 2 2 2 2 2 0
51: 7 4| 1 3 2 1 2 2<1>2 1 2 2 2 2 2 0
52: 6 4| 1 3 2 1 2<2>1 2 1 2 2 2 2 2 0
53: 5 4| 1 3 2 1<2>2 1 2 1 2 2 2 2 2 0
54: 4 4| 1 3 2<1>2 2 1 2 1 2 2 2 2 2 0
55: 3 4| 1 3<2>1 2 2 1 2 1 2 2 2 2 2 0
56: 2 4| 1<3>2 1 2 2 1 2 1 2 2 2 2 2 0
57: 3 1| 1 1<2>1 2 2 1 2 1 2 2 2 2 2 0
58: 4 2| 1 1 3<1>2 2 1 2 1 2 2 2 2 2 0
59: 5 2| 1 1 3 1<2>2 1 2 1 2 2 2 2 2 0
60: 6 2| 1 1 3 1 2<2>1 2 1 2 2 2 2 2 0
61: 7 2| 1 1 3 1 2 2<1>2 1 2 2 2 2 2 0
62: 8 2| 1 1 3 1 2 2 1<2>1 2 2 2 2 2 0
63: 9 2| 1 1 3 1 2 2 1 2<1>2 2 2 2 2 0
64: 10 2| 1 1 3 1 2 2 1 2 1<2>2 2 2 2 0
65: 11 2| 1 1 3 1 2 2 1 2 1 2<2>2 2 2 0
66: 12 2| 1 1 3 1 2 2 1 2 1 2 2<2>2 2 0
67: 13 2| 1 1 3 1 2 2 1 2 1 2 2 2<2>2 0
68: 14 2| 1 1 3 1 2 2 1 2 1 2 2 2 2<2>0
69: 15 2| 1 1 3 1 2 2 1 2 1 2 2 2 2 2<0>
70: 14 3| 1 1 3 1 2 2 1 2 1 2 2 2 2<2>0
71: 13 3| 1 1 3 1 2 2 1 2 1 2 2 2<2>2 0
72: 12 3| 1 1 3 1 2 2 1 2 1 2 2<2>2 2 0
73: 11 3| 1 1 3 1 2 2 1 2 1 2<2>2 2 2 0
74: 10 3| 1 1 3 1 2 2 1 2 1<2>2 2 2 2 0
75: 9 3| 1 1 3 1 2 2 1 2<1>2 2 2 2 2 0
76: 8 4| 1 1 3 1 2 2 1<2>2 2 2 2 2 2 0
77: 7 4| 1 1 3 1 2 2<1>2 2 2 2 2 2 2 0
78: 6 4| 1 1 3 1 2<2>1 2 2 2 2 2 2 2 0
79: 5 4| 1 1 3 1<2>2 1 2 2 2 2 2 2 2 0
80: 4 4| 1 1 3<1>2 2 1 2 2 2 2 2 2 2 0
81: 3 4| 1 1<3>1 2 2 1 2 2 2 2 2 2 2 0
82: 4 1| 1 1 1<1>2 2 1 2 2 2 2 2 2 2 0
83: 5 1| 1 1 1 1<2>2 1 2 2 2 2 2 2 2 0
84: 6 2| 1 1 1 1 3<2>1 2 2 2 2 2 2 2 0
85: 7 2| 1 1 1 1 3 2<1>2 2 2 2 2 2 2 0
86: 8 2| 1 1 1 1 3 2 1<2>2 2 2 2 2 2 0
87: 9 2| 1 1 1 1 3 2 1 2<2>2 2 2 2 2 0
88: 10 2| 1 1 1 1 3 2 1 2 2<2>2 2 2 2 0
89: 11 2| 1 1 1 1 3 2 1 2 2 2<2>2 2 2 0
90: 12 2| 1 1 1 1 3 2 1 2 2 2 2<2>2 2 0
91: 13 2| 1 1 1 1 3 2 1 2 2 2 2 2<2>2 0
92: 14 2| 1 1 1 1 3 2 1 2 2 2 2 2 2<2>0
93: 15 2| 1 1 1 1 3 2 1 2 2 2 2 2 2 2<0>
94: 14 3| 1 1 1 1 3 2 1 2 2 2 2 2 2<2>0
95: 13 3| 1 1 1 1 3 2 1 2 2 2 2 2<2>2 0
96: 12 3| 1 1 1 1 3 2 1 2 2 2 2<2>2 2 0
97: 11 3| 1 1 1 1 3 2 1 2 2 2<2>2 2 2 0
98: 10 3| 1 1 1 1 3 2 1 2 2<2>2 2 2 2 0
99: 9 3| 1 1 1 1 3 2 1 2<2>2 2 2 2 2 0
100: 8 3| 1 1 1 1 3 2 1<2>2 2 2 2 2 2 0
101: 7 3| 1 1 1 1 3 2<1>2 2 2 2 2 2 2 0
102: 6 4| 1 1 1 1 3<2>2 2 2 2 2 2 2 2 0
103: 5 4| 1 1 1 1<3>2 2 2 2 2 2 2 2 2 0
104: 6 1| 1 1 1 1 1<2>2 2 2 2 2 2 2 2 0
105: 7 2| 1 1 1 1 1 3<2>2 2 2 2 2 2 2 0
106: 8 2| 1 1 1 1 1 3 2<2>2 2 2 2 2 2 0
107: 9 2| 1 1 1 1 1 3 2 2<2>2 2 2 2 2 0
108: 10 2| 1 1 1 1 1 3 2 2 2<2>2 2 2 2 0
109: 11 2| 1 1 1 1 1 3 2 2 2 2<2>2 2 2 0
110: 12 2| 1 1 1 1 1 3 2 2 2 2 2<2>2 2 0
111: 13 2| 1 1 1 1 1 3 2 2 2 2 2 2<2>2 0
112: 14 2| 1 1 1 1 1 3 2 2 2 2 2 2 2<2>0
113: 15 2| 1 1 1 1 1 3 2 2 2 2 2 2 2 2<0>
114: 14 3| 1 1 1 1 1 3 2 2 2 2 2 2 2<2>0
115: 13 3| 1 1 1 1 1 3 2 2 2 2 2 2<2>2 0
116: 12 3| 1 1 1 1 1 3 2 2 2 2 2<2>2 2 0
117: 11 3| 1 1 1 1 1 3 2 2 2 2<2>2 2 2 0
118: 10 3| 1 1 1 1 1 3 2 2 2<2>2 2 2 2 0
119: 9 3| 1 1 1 1 1 3 2 2<2>2 2 2 2 2 0
120: 8 3| 1 1 1 1 1 3 2<2>2 2 2 2 2 2 0
121: 7 3| 1 1 1 1 1 3<2>2 2 2 2 2 2 2 0
122: 6 3| 1 1 1 1 1<3>2 2 2 2 2 2 2 2 0
123: 5 5| 1 1 1 1<1>2 2 2 2 2 2 2 2 2 0
124: 4 5| 1 1 1<1>1 2 2 2 2 2 2 2 2 2 0
125: 3 5| 1 1<1>1 1 2 2 2 2 2 2 2 2 2 0
126: 2 5| 1<1>1 1 1 2 2 2 2 2 2 2 2 2 0
127: 1 5| <1>1 1 1 1 2 2 2 2 2 2 2 2 2 0
128: 0 5| <0>1 1 1 1 1 2 2 2 2 2 2 2 2 2 0
HALT! 1 0| <0>1 1 1 1 1 2 2 2 2 2 2 2 2 2 0
After step 128, state = 0.
The head is at position 1 and wandered over 0 to 15
Usage.
Entry: Read Write Move Next. For state 1
5: " " "0" -1 5 0
4: "0" "0" -1 5 0
3: "1" "1" +1 1 1
2: "2" "3" +1 2 5
1: "3" "3" 0 0 0
Entry: Read Write Move Next. For state 2
10: " " "0" -1 3 1
9: "0" "0" -1 3 4
8: "1" "1" +1 2 13
7: "2" "2" +1 2 40
6: "3" "3" 0 0 0
Entry: Read Write Move Next. For state 3
15: " " "0" 0 0 0
14: "0" "0" 0 0 0
13: "1" "2" -1 4 4
12: "2" "2" -1 3 24
11: "3" "2" -1 5 1
Entry: Read Write Move Next. For state 4
20: " " "0" 0 0 0
19: "0" "0" 0 0 0
18: "1" "1" -1 4 9
17: "2" "2" -1 4 16
16: "3" "1" +1 1 4
Entry: Read Write Move Next. For state 5
25: " " "0" +1 0 1
24: "0" "0" +1 0 0
23: "1" "1" -1 5 5
22: "2" "2" 0 0 0
21: "3" "3" 0 0 0
Five-state Busy Beaver
The TestBB5.dat run required modifying the source with FIRST = -12345 and OFFSET = 20 so that the trace didn't hit the bounds.
Turing machine simulation for... A five-state Busy Beaver.
2 symbols: >01<
Entry: Read Write Move Next. For state 1
2: "0" "1" +1 2
1: "1" "1" -1 3
Entry: Read Write Move Next. For state 2
5: "0" "1" +1 3
4: "1" "1" +1 2
Entry: Read Write Move Next. For state 3
8: "0" "1" +1 4
7: "1" "0" -1 5
Entry: Read Write Move Next. For state 4
11: "0" "1" -1 1
10: "1" "1" -1 4
Entry: Read Write Move Next. For state 5
14: "0" "1" 0 0
13: "1" "0" -1 1
Starts with State 1 and the tape head at 1.
Step: Head State|Tape... <H>
1: 1 1| <0>
2: 2 2| 1<0>
3: 3 3| 1 1<0>
4: 4 4| 1 1 1<0>
5: 3 1| 1 1<1>1
6: 2 3| 1<1>1 1
7: 1 5| <1>0 1 1
8: 0 1| <0>0 0 1 1
9: 1 2| 1<0>0 1 1
10: 2 3| 1 1<0>1 1
11: 3 4| 1 1 1<1>1
12: 2 4| 1 1<1>1 1
13: 1 4| 1<1>1 1 1
14: 0 4| <1>1 1 1 1
15: -1 4| <0>1 1 1 1 1
16: -2 1| <0>1 1 1 1 1 1
17: -1 2| 1<1>1 1 1 1 1
18: 0 2| 1 1<1>1 1 1 1
19: 1 2| 1 1 1<1>1 1 1
20: 2 2| 1 1 1 1<1>1 1
21: 3 2| 1 1 1 1 1<1>1
22: 4 2| 1 1 1 1 1 1<1>
23: 5 2| 1 1 1 1 1 1 1<0>
24: 6 3| 1 1 1 1 1 1 1 1<0>
25: 7 4| 1 1 1 1 1 1 1 1 1<0>
26: 6 1| 1 1 1 1 1 1 1 1<1>1
27: 5 3| 1 1 1 1 1 1 1<1>1 1
28: 4 5| 1 1 1 1 1 1<1>0 1 1
29: 3 1| 1 1 1 1 1<1>0 0 1 1
30: 2 3| 1 1 1 1<1>1 0 0 1 1
31: 1 5| 1 1 1<1>0 1 0 0 1 1
32: 0 1| 1 1<1>0 0 1 0 0 1 1
33: -1 3| 1<1>1 0 0 1 0 0 1 1
34: -2 5| <1>0 1 0 0 1 0 0 1 1
35: -3 1| <0>0 0 1 0 0 1 0 0 1 1
36: -2 2| 1<0>0 1 0 0 1 0 0 1 1
37: -1 3| 1 1<0>1 0 0 1 0 0 1 1
38: 0 4| 1 1 1<1>0 0 1 0 0 1 1
39: -1 4| 1 1<1>1 0 0 1 0 0 1 1
40: -2 4| 1<1>1 1 0 0 1 0 0 1 1
41: -3 4| <1>1 1 1 0 0 1 0 0 1 1
42: -4 4| <0>1 1 1 1 0 0 1 0 0 1 1
43: -5 1| <0>1 1 1 1 1 0 0 1 0 0 1 1
44: -4 2| 1<1>1 1 1 1 0 0 1 0 0 1 1
45: -3 2| 1 1<1>1 1 1 0 0 1 0 0 1 1
46: -2 2| 1 1 1<1>1 1 0 0 1 0 0 1 1
47: -1 2| 1 1 1 1<1>1 0 0 1 0 0 1 1
48: 0 2| 1 1 1 1 1<1>0 0 1 0 0 1 1
49: 1 2| 1 1 1 1 1 1<0>0 1 0 0 1 1
50: 2 3| 1 1 1 1 1 1 1<0>1 0 0 1 1
51: 3 4| 1 1 1 1 1 1 1 1<1>0 0 1 1
52: 2 4| 1 1 1 1 1 1 1<1>1 0 0 1 1
53: 1 4| 1 1 1 1 1 1<1>1 1 0 0 1 1
54: 0 4| 1 1 1 1 1<1>1 1 1 0 0 1 1
55: -1 4| 1 1 1 1<1>1 1 1 1 0 0 1 1
56: -2 4| 1 1 1<1>1 1 1 1 1 0 0 1 1
57: -3 4| 1 1<1>1 1 1 1 1 1 0 0 1 1
58: -4 4| 1<1>1 1 1 1 1 1 1 0 0 1 1
59: -5 4| <1>1 1 1 1 1 1 1 1 0 0 1 1
60: -6 4| <0>1 1 1 1 1 1 1 1 1 0 0 1 1
61: -7 1| <0>1 1 1 1 1 1 1 1 1 1 0 0 1 1
62: -6 2| 1<1>1 1 1 1 1 1 1 1 1 0 0 1 1
63: -5 2| 1 1<1>1 1 1 1 1 1 1 1 0 0 1 1
64: -4 2| 1 1 1<1>1 1 1 1 1 1 1 0 0 1 1
65: -3 2| 1 1 1 1<1>1 1 1 1 1 1 0 0 1 1
66: -2 2| 1 1 1 1 1<1>1 1 1 1 1 0 0 1 1
67: -1 2| 1 1 1 1 1 1<1>1 1 1 1 0 0 1 1
68: 0 2| 1 1 1 1 1 1 1<1>1 1 1 0 0 1 1
69: 1 2| 1 1 1 1 1 1 1 1<1>1 1 0 0 1 1
70: 2 2| 1 1 1 1 1 1 1 1 1<1>1 0 0 1 1
71: 3 2| 1 1 1 1 1 1 1 1 1 1<1>0 0 1 1
72: 4 2| 1 1 1 1 1 1 1 1 1 1 1<0>0 1 1
73: 5 3| 1 1 1 1 1 1 1 1 1 1 1 1<0>1 1
74: 6 4| 1 1 1 1 1 1 1 1 1 1 1 1 1<1>1
75: 5 4| 1 1 1 1 1 1 1 1 1 1 1 1<1>1 1
76: 4 4| 1 1 1 1 1 1 1 1 1 1 1<1>1 1 1
77: 3 4| 1 1 1 1 1 1 1 1 1 1<1>1 1 1 1
78: 2 4| 1 1 1 1 1 1 1 1 1<1>1 1 1 1 1
79: 1 4| 1 1 1 1 1 1 1 1<1>1 1 1 1 1 1
80: 0 4| 1 1 1 1 1 1 1<1>1 1 1 1 1 1 1
81: -1 4| 1 1 1 1 1 1<1>1 1 1 1 1 1 1 1
82: -2 4| 1 1 1 1 1<1>1 1 1 1 1 1 1 1 1
83: -3 4| 1 1 1 1<1>1 1 1 1 1 1 1 1 1 1
84: -4 4| 1 1 1<1>1 1 1 1 1 1 1 1 1 1 1
85: -5 4| 1 1<1>1 1 1 1 1 1 1 1 1 1 1 1
86: -6 4| 1<1>1 1 1 1 1 1 1 1 1 1 1 1 1
87: -7 4| <1>1 1 1 1 1 1 1 1 1 1 1 1 1 1
88: -8 4| <0>1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
89: -9 1| <0>1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
90: -8 2| 1<1>1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
91: -7 2| 1 1<1>1 1 1 1 1 1 1 1 1 1 1 1 1 1
92: -6 2| 1 1 1<1>1 1 1 1 1 1 1 1 1 1 1 1 1
93: -5 2| 1 1 1 1<1>1 1 1 1 1 1 1 1 1 1 1 1
94: -4 2| 1 1 1 1 1<1>1 1 1 1 1 1 1 1 1 1 1
95: -3 2| 1 1 1 1 1 1<1>1 1 1 1 1 1 1 1 1 1
96: -2 2| 1 1 1 1 1 1 1<1>1 1 1 1 1 1 1 1 1
97: -1 2| 1 1 1 1 1 1 1 1<1>1 1 1 1 1 1 1 1
98: 0 2| 1 1 1 1 1 1 1 1 1<1>1 1 1 1 1 1 1
99: 1 2| 1 1 1 1 1 1 1 1 1 1<1>1 1 1 1 1 1
100: 2 2| 1 1 1 1 1 1 1 1 1 1 1<1>1 1 1 1 1
101: 3 2| 1 1 1 1 1 1 1 1 1 1 1 1<1>1 1 1 1
102: 4 2| 1 1 1 1 1 1 1 1 1 1 1 1 1<1>1 1 1
103: 5 2| 1 1 1 1 1 1 1 1 1 1 1 1 1 1<1>1 1
104: 6 2| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1<1>1
105: 7 2| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1<1>
106: 8 2| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1<0>
107: 9 3| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1<0>
108: 10 4| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1<0>
109: 9 1| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1<1>1
110: 8 3| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1<1>1 1
111: 7 5| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1<1>0 1 1
112: 6 1| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1<1>0 0 1 1
113: 5 3| 1 1 1 1 1 1 1 1 1 1 1 1 1 1<1>1 0 0 1 1
114: 4 5| 1 1 1 1 1 1 1 1 1 1 1 1 1<1>0 1 0 0 1 1
115: 3 1| 1 1 1 1 1 1 1 1 1 1 1 1<1>0 0 1 0 0 1 1
116: 2 3| 1 1 1 1 1 1 1 1 1 1 1<1>1 0 0 1 0 0 1 1
117: 1 5| 1 1 1 1 1 1 1 1 1 1<1>0 1 0 0 1 0 0 1 1
118: 0 1| 1 1 1 1 1 1 1 1 1<1>0 0 1 0 0 1 0 0 1 1
119: -1 3| 1 1 1 1 1 1 1 1<1>1 0 0 1 0 0 1 0 0 1 1
120: -2 5| 1 1 1 1 1 1 1<1>0 1 0 0 1 0 0 1 0 0 1 1
121: -3 1| 1 1 1 1 1 1<1>0 0 1 0 0 1 0 0 1 0 0 1 1
122: -4 3| 1 1 1 1 1<1>1 0 0 1 0 0 1 0 0 1 0 0 1 1
123: -5 5| 1 1 1 1<1>0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
124: -6 1| 1 1 1<1>0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
125: -7 3| 1 1<1>1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
126: -8 5| 1<1>0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
127: -9 1| <1>0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
128: -10 3| <0>1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
129: -9 4| 1<1>0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
130: -10 4| <1>1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
131: -11 4| <0>1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
132: -12 1| <0>1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
133: -11 2| 1<1>1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
134: -10 2| 1 1<1>1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
135: -9 2| 1 1 1<1>0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
136: -8 2| 1 1 1 1<0>0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
137: -7 3| 1 1 1 1 1<0>1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
138: -6 4| 1 1 1 1 1 1<1>0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
139: -7 4| 1 1 1 1 1<1>1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
140: -8 4| 1 1 1 1<1>1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
141: -9 4| 1 1 1<1>1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
142: -10 4| 1 1<1>1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
143: -11 4| 1<1>1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
144: -12 4| <1>1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
145: -13 4| <0>1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
146: -14 1| <0>1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
147: -13 2| 1<1>1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
148: -12 2| 1 1<1>1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
149: -11 2| 1 1 1<1>1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
150: -10 2| 1 1 1 1<1>1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
151: -9 2| 1 1 1 1 1<1>1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
152: -8 2| 1 1 1 1 1 1<1>1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
153: -7 2| 1 1 1 1 1 1 1<1>1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
154: -6 2| 1 1 1 1 1 1 1 1<1>0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
155: -5 2| 1 1 1 1 1 1 1 1 1<0>0 1 0 0 1 0 0 1 0 0 1 0 0 1 1
156: -4 3| 1 1 1 1 1 1 1 1 1 1<0>1 0 0 1 0 0 1 0 0 1 0 0 1 1
157: -3 4| 1 1 1 1 1 1 1 1 1 1 1<1>0 0 1 0 0 1 0 0 1 0 0 1 1
158: -4 4| 1 1 1 1 1 1 1 1 1 1<1>1 0 0 1 0 0 1 0 0 1 0 0 1 1
159: -5 4| 1 1 1 1 1 1 1 1 1<1>1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
160: -6 4| 1 1 1 1 1 1 1 1<1>1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
161: -7 4| 1 1 1 1 1 1 1<1>1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
162: -8 4| 1 1 1 1 1 1<1>1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
163: -9 4| 1 1 1 1 1<1>1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
164: -10 4| 1 1 1 1<1>1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
165: -11 4| 1 1 1<1>1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
166: -12 4| 1 1<1>1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
167: -13 4| 1<1>1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
168: -14 4| <1>1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
169: -15 4| <0>1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
170: -16 1| <0>1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
171: -15 2| 1<1>1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
172: -14 2| 1 1<1>1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
173: -13 2| 1 1 1<1>1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
174: -12 2| 1 1 1 1<1>1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
175: -11 2| 1 1 1 1 1<1>1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
176: -10 2| 1 1 1 1 1 1<1>1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
177: -9 2| 1 1 1 1 1 1 1<1>1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
178: -8 2| 1 1 1 1 1 1 1 1<1>1 1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
179: -7 2| 1 1 1 1 1 1 1 1 1<1>1 1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
180: -6 2| 1 1 1 1 1 1 1 1 1 1<1>1 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
181: -5 2| 1 1 1 1 1 1 1 1 1 1 1<1>1 1 0 0 1 0 0 1 0 0 1 0 0 1 1
182: -4 2| 1 1 1 1 1 1 1 1 1 1 1 1<1>1 0 0 1 0 0 1 0 0 1 0 0 1 1
183: -3 2| 1 1 1 1 1 1 1 1 1 1 1 1 1<1>0 0 1 0 0 1 0 0 1 0 0 1 1
184: -2 2| 1 1 1 1 1 1 1 1 1 1 1 1 1 1<0>0 1 0 0 1 0 0 1 0 0 1 1
185: -1 3| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1<0>1 0 0 1 0 0 1 0 0 1 1
186: 0 4| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1<1>0 0 1 0 0 1 0 0 1 1
187: -1 4| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1<1>1 0 0 1 0 0 1 0 0 1 1
188: -2 4| 1 1 1 1 1 1 1 1 1 1 1 1 1 1<1>1 1 0 0 1 0 0 1 0 0 1 1
189: -3 4| 1 1 1 1 1 1 1 1 1 1 1 1 1<1>1 1 1 0 0 1 0 0 1 0 0 1 1
190: -4 4| 1 1 1 1 1 1 1 1 1 1 1 1<1>1 1 1 1 0 0 1 0 0 1 0 0 1 1
191: -5 4| 1 1 1 1 1 1 1 1 1 1 1<1>1 1 1 1 1 0 0 1 0 0 1 0 0 1 1
192: -6 4| 1 1 1 1 1 1 1 1 1 1<1>1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1
193: -7 4| 1 1 1 1 1 1 1 1 1<1>1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1
194: -8 4| 1 1 1 1 1 1 1 1<1>1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1
195: -9 4| 1 1 1 1 1 1 1<1>1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1
196: -10 4| 1 1 1 1 1 1<1>1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1
197: -11 4| 1 1 1 1 1<1>1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1
198: -12 4| 1 1 1 1<1>1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1
199: -13 4| 1 1 1<1>1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1
200: -14 4| 1 1<1>1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1
Step 10000000
Step 20000000
Step 30000000
Step 40000000
After step 47176870, state = 0.
The head is at position -12242 and wandered over -12242 to 46
CPU time 0.641
Usage.
Entry: Read Write Move Next. For state 1
3: " " "1" +1 2 6121
2: "0" "1" +1 2 1
1: "1" "1" -1 3 10210
Entry: Read Write Move Next. For state 2
6: " " "1" +1 3 15
5: "0" "1" +1 3 6107
4: "1" "1" +1 2 23563940
Entry: Read Write Move Next. For state 3
9: " " "1" +1 4 22
8: "0" "1" +1 4 6107
7: "1" "0" -1 5 10203
Entry: Read Write Move Next. For state 4
12: " " "1" -1 1 6129
11: "0" "1" -1 1 0
10: "1" "1" -1 4 23557812
Entry: Read Write Move Next. For state 5
15: " " "1" 0 0 1
14: "0" "1" 0 0 0
13: "1" "0" -1 1 10202
Removing the code maintaining the bounds HMIN and HMAX as HEAD wandered about reduced the cpu time from 0·578 to 0·562, and reducing the progress report to every ten-millionth step reduced that to 0·516. But adding the code to count the usage of each transition raised it to 0·641. Removing the usage count and the progress report got it down to 0·484. Removing the compiler's array bound checking and activating "maximum optimisations" achieved 0·312. Reverting to normal runs and introducing the check IF (HEAD.LT.FIRST .OR. HEAD.GT.LAST) changed the run time from 0·703 to 0·75. Thus, 47 million tests in 0·05 seconds - not much of a cost for peace of mind.
Fōrmulæ
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.
Programs in Fōrmulæ are created/edited online in its website, However they run on execution servers. By default remote servers are used, but they are limited in memory and processing power, since they are intended for demonstration and casual use. A local server can be downloaded and installed, it has no limitations (it runs in your own computer). Because of that, example programs can be fully visualized and edited, but some of them will not run if they require a moderate or heavy computation/memory resources, and no local server is being used.
In this page you can see the program(s) related to this task and their results.
Go
package turing
type Symbol byte
type Motion byte
const (
Left Motion = 'L'
Right Motion = 'R'
Stay Motion = 'N'
)
type Tape struct {
data []Symbol
pos, left int
blank Symbol
}
// NewTape returns a new tape filled with 'data' and position set to 'start'.
// 'start' does not need to be range, the tape will be extended if required.
func NewTape(blank Symbol, start int, data []Symbol) *Tape {
t := &Tape{
data: data,
blank: blank,
}
if start < 0 {
t.Left(-start)
}
t.Right(start)
return t
}
func (t *Tape) Stay() {}
func (t *Tape) Data() []Symbol { return t.data[t.left:] }
func (t *Tape) Read() Symbol { return t.data[t.pos] }
func (t *Tape) Write(s Symbol) { t.data[t.pos] = s }
func (t *Tape) Dup() *Tape {
t2 := &Tape{
data: make([]Symbol, len(t.Data())),
blank: t.blank,
}
copy(t2.data, t.Data())
t2.pos = t.pos - t.left
return t2
}
func (t *Tape) String() string {
s := ""
for i := t.left; i < len(t.data); i++ {
b := t.data[i]
if i == t.pos {
s += "[" + string(b) + "]"
} else {
s += " " + string(b) + " "
}
}
return s
}
func (t *Tape) Move(a Motion) {
switch a {
case Left:
t.Left(1)
case Right:
t.Right(1)
case Stay:
t.Stay()
}
}
const minSz = 16
func (t *Tape) Left(n int) {
t.pos -= n
if t.pos < 0 {
// Extend left
var sz int
for sz = minSz; cap(t.data[t.left:])-t.pos >= sz; sz <<= 1 {
}
newd := make([]Symbol, sz)
newl := len(newd) - cap(t.data[t.left:])
n := copy(newd[newl:], t.data[t.left:])
t.data = newd[:newl+n]
t.pos += newl - t.left
t.left = newl
}
if t.pos < t.left {
if t.blank != 0 {
for i := t.pos; i < t.left; i++ {
t.data[i] = t.blank
}
}
t.left = t.pos
}
}
func (t *Tape) Right(n int) {
t.pos += n
if t.pos >= cap(t.data) {
// Extend right
var sz int
for sz = minSz; t.pos >= sz; sz <<= 1 {
}
newd := make([]Symbol, sz)
n := copy(newd[t.left:], t.data[t.left:])
t.data = newd[:t.left+n]
}
if i := len(t.data); t.pos >= i {
t.data = t.data[:t.pos+1]
if t.blank != 0 {
for ; i < len(t.data); i++ {
t.data[i] = t.blank
}
}
}
}
type State string
type Rule struct {
State
Symbol
Write Symbol
Motion
Next State
}
func (i *Rule) key() key { return key{i.State, i.Symbol} }
func (i *Rule) action() action { return action{i.Write, i.Motion, i.Next} }
type key struct {
State
Symbol
}
type action struct {
write Symbol
Motion
next State
}
type Machine struct {
tape *Tape
start, state State
transition map[key]action
l func(string, ...interface{}) // XXX
}
func NewMachine(rules []Rule) *Machine {
m := &Machine{transition: make(map[key]action, len(rules))}
if len(rules) > 0 {
m.start = rules[0].State
}
for _, r := range rules {
m.transition[r.key()] = r.action()
}
return m
}
func (m *Machine) Run(input *Tape) (int, *Tape) {
m.tape = input.Dup()
m.state = m.start
for cnt := 0; ; cnt++ {
if m.l != nil {
m.l("%3d %4s: %v\n", cnt, m.state, m.tape)
}
sym := m.tape.Read()
act, ok := m.transition[key{m.state, sym}]
if !ok {
return cnt, m.tape
}
m.tape.Write(act.write)
m.tape.Move(act.Motion)
m.state = act.next
}
}
An example program using the above package:
package main
import (
".." // XXX path to above turing package
"fmt"
)
func main() {
var incrementer = turing.NewMachine([]turing.Rule{
{"q0", '1', '1', turing.Right, "q0"},
{"q0", 'B', '1', turing.Stay, "qf"},
})
input := turing.NewTape('B', 0, []turing.Symbol{'1', '1', '1'})
cnt, output := incrementer.Run(input)
fmt.Println("Turing machine halts after", cnt, "operations")
fmt.Println("Resulting tape:", output)
var beaver = turing.NewMachine([]turing.Rule{
{"a", '0', '1', turing.Right, "b"},
{"a", '1', '1', turing.Left, "c"},
{"b", '0', '1', turing.Left, "a"},
{"b", '1', '1', turing.Right, "b"},
{"c", '0', '1', turing.Left, "b"},
{"c", '1', '1', turing.Stay, "halt"},
})
cnt, output = beaver.Run(turing.NewTape('0', 0, nil))
fmt.Println("Turing machine halts after", cnt, "operations")
fmt.Println("Resulting tape:", output)
beaver = turing.NewMachine([]turing.Rule{
{"A", '0', '1', turing.Right, "B"},
{"A", '1', '1', turing.Left, "C"},
{"B", '0', '1', turing.Right, "C"},
{"B", '1', '1', turing.Right, "B"},
{"C", '0', '1', turing.Right, "D"},
{"C", '1', '0', turing.Left, "E"},
{"D", '0', '1', turing.Left, "A"},
{"D", '1', '1', turing.Left, "D"},
{"E", '0', '1', turing.Stay, "H"},
{"E", '1', '0', turing.Left, "A"},
})
cnt, output = beaver.Run(turing.NewTape('0', 0, nil))
fmt.Println("Turing machine halts after", cnt, "operations")
fmt.Println("Resulting tape has", len(output.Data()), "cells")
var sort = turing.NewMachine([]turing.Rule{
// Moving right, first b→B;s1
{"s0", 'a', 'a', turing.Right, "s0"},
{"s0", 'b', 'B', turing.Right, "s1"},
{"s0", ' ', ' ', turing.Left, "se"},
// Conintue right to end of tape → s2
{"s1", 'a', 'a', turing.Right, "s1"},
{"s1", 'b', 'b', turing.Right, "s1"},
{"s1", ' ', ' ', turing.Left, "s2"},
// Continue left over b. a→b;s3, B→b;se
{"s2", 'a', 'b', turing.Left, "s3"},
{"s2", 'b', 'b', turing.Left, "s2"},
{"s2", 'B', 'b', turing.Left, "se"},
// Continue left until B→a;s0
{"s3", 'a', 'a', turing.Left, "s3"},
{"s3", 'b', 'b', turing.Left, "s3"},
{"s3", 'B', 'a', turing.Right, "s0"},
// Move to tape start → halt
{"se", 'a', 'a', turing.Left, "se"},
{"se", ' ', ' ', turing.Right, "see"},
})
input = turing.NewTape(' ', 0, []turing.Symbol("abbabbabababab"))
cnt, output = sort.Run(input)
fmt.Println("Turing machine halts after", cnt, "operations")
fmt.Println("Resulting tape:", output)
}
- Output:
Turing machine halts after 4 operations Resulting tape: 1 1 1 [1] Turing machine halts after 13 operations Resulting tape: 1 1 1 [1] 1 1 Turing machine halts after 47176870 operations Resulting tape has 12289 cells Turing machine halts after 118 operations Resulting tape: [a] a a a a a b b b b b b b b
Haskell
Simple Universal Turing Machine
In this program the tape is infinite, and the machines rules are coded in Haskell as a function from state and value to action, using Haskell as a DSL.
-- Some elementary types for Turing Machine
data Move = MLeft | MRight | Stay deriving (Show, Eq)
data Tape a = Tape a [a] [a]
data Action state val = Action val Move state deriving (Show)
instance (Show a) => Show (Tape a) where
show (Tape x lts rts) = concat $ left ++ [hd] ++ right
where hd = "[" ++ show x ++ "]"
left = map show $ reverse $ take 10 lts
right = map show $ take 10 rts
-- new tape
tape blank lts rts | null rts = Tape blank left blanks
| otherwise = Tape (head rts) left right
where blanks = repeat blank
left = reverse lts ++ blanks
right = tail rts ++ blanks
-- Turing Machine
step rules (state, Tape x (lh:lts) (rh:rts)) = (state', tape')
where Action x' dir state' = rules state x
tape' = move dir
move Stay = Tape x' (lh:lts) (rh:rts)
move MLeft = Tape lh lts (x':rh:rts)
move MRight = Tape rh (x':lh:lts) rts
runUTM rules stop start tape = steps ++ [final]
where (steps, final:_) = break ((== stop) . fst) $ iterate (step rules) (start, tape)
Increment machine
incr "q0" 1 = Action 1 MRight "q0"
incr "q0" 0 = Action 1 Stay "qf"
tape1 = tape 0 [] [1,1, 1]
machine1 = runUTM incr "qf" "q0" tape1
The output of the increment machine :
*Main> mapM_ print machine1
("q0",0000000000[1]1100000000)
("q0",0000000001[1]1000000000)
("q0",0000000011[1]0000000000)
("q0",0000000111[0]0000000000)
("qf",0000000111[1]0000000000)
Beaver machine
beaver "a" 0 = Action 1 MRight "b"
beaver "a" 1 = Action 1 MLeft "c"
beaver "b" 0 = Action 1 MLeft "a"
beaver "b" 1 = Action 1 MRight "b"
beaver "c" 0 = Action 1 MLeft "b"
beaver "c" 1 = Action 1 Stay "halt"
tape2 = tape 0 [] []
machine2 = runUTM beaver "halt" "a" tape2
Sorting test
sorting "A" 1 = Action 1 MRight "A"
sorting "A" 2 = Action 3 MRight "B"
sorting "A" 0 = Action 0 MLeft "E"
sorting "B" 1 = Action 1 MRight "B"
sorting "B" 2 = Action 2 MRight "B"
sorting "B" 0 = Action 0 MLeft "C"
sorting "C" 1 = Action 2 MLeft "D"
sorting "C" 2 = Action 2 MLeft "C"
sorting "C" 3 = Action 2 MLeft "E"
sorting "D" 1 = Action 1 MLeft "D"
sorting "D" 2 = Action 2 MLeft "D"
sorting "D" 3 = Action 1 MRight "A"
sorting "E" 1 = Action 1 MLeft "E"
sorting "E" 0 = Action 0 MRight "STOP"
tape3 = tape 0 [] [2,2,2,1,2,2,1,2,1,2,1,2,1,2]
machine3 = runUTM sorting "STOP" "A" tape3
Using State Monad
A state monad represents the machine. It works with an arbitrary number of symbols and states, but all of them must be of the same type (integer, string...) Intermediate states can be logged during execution, or they can be discarded. The initial and final states as well as errors are always logged. Three functions are added so that machines can be written to a file and parsed/run from there. Examples are provided.
import Control.Monad.State
import Data.List (intersperse, nub, find)
data TapeMovement = MoveLeft | MoveRight | Stay deriving (Show, Eq)
-- Rule = (state 1, input, output, movement, state 2)
type Rule a = (a, a, a, TapeMovement, a)
-- Execution = (tape position, current machine state, tape)
type Execution a = (Int, a, [a])
type Log a = [Execution a]
type UTM a b = State (Machine a) b
-- can work with data of any type
data Machine a = Machine
{ allStates :: [a] -- not used actually
, initialState :: a -- not used actually, initial state in "current"
, finalStates :: [a]
, symbols :: [a] -- not used actually
, blank :: a
, noOpSymbol :: a -- means: don't change input / don't shift tape
, rules :: [Rule a]
, current :: Execution a
, machineLog :: Log a -- stores state changes from last to first
, machineLogActive :: Bool -- if true, intermediate steps are stored
, noRuleMsg :: a -- error symbol if no rule matches
, stopMsg :: a } -- symbol to append to the end result
deriving (Show)
-- it is not checked whether the input and output symbols are valid
apply :: Eq a => Rule a -> UTM a a
apply (_, _, output, direction, stateUpdate) = do
m <- get
let (pos, currentState, tape) = current m
tapeUpdate = if output == noOpSymbol m
then tape
else take pos tape ++ [output] ++ drop (pos + 1) tape
newTape
| pos == 0 && direction == MoveLeft = blank m : tapeUpdate
| succ pos == length tape && direction == MoveRight = tapeUpdate ++ [blank m]
| otherwise = tapeUpdate
newPosition = case direction of
MoveLeft -> if pos == 0 then 0 else pred pos
MoveRight -> succ pos
Stay -> pos
newState = if stateUpdate == noOpSymbol m
then currentState
else stateUpdate
put $! m { current = (newPosition, newState, newTape) }
return newState
-- rules with no-operation symbols and states must be underneath
-- rules with defined symbols and states
lookupRule :: Eq a => UTM a (Maybe (Rule a))
lookupRule = do
m <- get
let (pos, currentState, tape) = current m
item = tape !! pos
isValid (e, i, _, _, _) = e == currentState &&
(i == item || i == noOpSymbol m)
return $! find isValid (rules m)
msgToLog :: a -> UTM a ()
msgToLog e = do
m <- get
let (pos, currentState, tape) = current m
put $! m { machineLog = (pos, currentState, tape ++ [e]) : machineLog m }
toLog :: UTM a ()
toLog = do
m <- get
put $! m { machineLog = current m : machineLog m }
-- execute the machine's program
execute :: Eq a => UTM a ()
execute = do
toLog -- log the initial state
loop
where
loop = do
m <- get
r <- lookupRule -- look for a matching rule
case r of
Nothing -> msgToLog (noRuleMsg m)
Just rule -> do
stateUpdate <- apply rule
if stateUpdate `elem` finalStates m
then msgToLog (stopMsg m)
else do
when (machineLogActive m) toLog
loop
---------------------------
-- convenient functions
---------------------------
-- run execute, format and print the output
runMachine :: Machine String -> IO ()
runMachine m@(Machine { current = (_, _, tape) }) =
if null tape
then putStrLn "NO TAPE"
else case machineLog $ execState execute m of
[] -> putStrLn "NO OUTPUT"
xs -> do
mapM_ (\(pos, _, output) -> do
let formatOutput = concat output
putStrLn formatOutput
putStrLn (replicate pos ' ' ++ "^")) $ reverse xs
putStrLn $ show (length xs) ++ " STEPS. FINAL STATE: " ++
let (_, finalState, _) = head xs in show finalState
-- convert a string with format state+space+input+space+output+space+
-- direction+space+new state to a rule
toRule :: String -> Rule String
toRule xs =
let [a, b, c, d, e] = take 5 $ words xs
dir = case d of
"l" -> MoveLeft
"r" -> MoveRight
"*" -> Stay
in (a, b, c, dir, e)
-- load a text file and parse it to a machine.
-- see comments and examples
-- lines in the file starting with ';' are header lines or comments
-- header and input lines must contain a ':' and after that the content to be parsed
-- so there can be comments between ';' and ':' in those lines
loadMachine :: FilePath -> IO (Machine String)
loadMachine n = do
f <- readFile n
let ls = lines f
-- header: first 4 lines
([e1, e2, e3, e4], rest) = splitAt 4 ls
-- rules and input: rest of the file
re = map toRule . filter (not . null) $ map (takeWhile (/= ';')) rest
ei = head . words . tail . snd $ break (== ':') e1
va = head . words . tail . snd $ break (== ':') e3
ci = words . intersperse ' ' . tail . snd $ break (== ':') $ last rest
return Machine
{ rules = re
, initialState = ei
, finalStates = words . tail . snd $ break (== ':') e2
, blank = va
, noOpSymbol = head . words . tail . snd $ break (== ':') e4
, allStates = nub $ concatMap (\(a, _, _, _, e) -> [a, e]) re
, symbols = nub $ concatMap (\(_, b, c, _, _) -> [b, c]) re
, current = (0, ei, if null ci then [va] else ci)
-- we assume
, noRuleMsg = "\tNO RULE." -- error: no matching rule found
, stopMsg = "\tHALT." -- message: machine reached a final state
, machineLog = []
, machineLogActive = True }
Examples for machine files:
; Initial state: q0 ; Final states: qf ; Blank symbol: B ; No-op symbol: * ; Simple incrementer q0 1 1 r q0 q0 B 1 * qf ; Initial tape: 111
; Initial state: a ; Final states: halt ; Blank symbol: 0 ; No-op symbol: * ; Three-state busy beaver a 0 1 r b a 1 1 l c b 0 1 l a b 1 1 r b c 0 1 l b c 1 1 * halt ; Initial tape:
To run a machine:
loadMachine "machine1" >>= runMachine
Output (simple incrementer):
111 ^ 111 ^ 111 ^ 111B ^ 1111 HALT. ^ 5 STEPS. FINAL STATE: "qf"
Output (three-state busy beaver):
0
^
10
^
11
^
011
^
0111
^
01111
^
11111
^
11111
^
11111
^
11111
^
111110
^
111111
^
111111
^
111111 HALT.
^
14 STEPS. FINAL STATE: "halt"
Icon and Unicon
The following works in both languages. The state machine input format differs slightly from the example given above. Various options exist for tracing the actions of the machine. This particular UTM halts when entering a final state or when a motion of 'halt' is acted on.
record TM(start,final,delta,tape,blank)
record delta(old_state, input_symbol, new_state, output_symbol, direction)
global start_tape
global show_count, full_display, trace_list # trace flags
procedure main(args)
init(args)
runTuringMachine(get_tm())
end
procedure init(args)
trace_list := ":"
while arg := get(args) do {
if arg == "-f" then full_display := "yes"
else if match("-t",arg) then trace_list ||:= arg[3:0]||":"
else show_count := integer(arg)
}
end
procedure get_tm()
D := table()
writes("What is the start state? ")
start := !&input
writes("What are the final states (colon separated)? ")
finals := !&input
(finals||":") ? every insert(fStates := set(), 1(tab(upto(':')),move(1)))
writes("What is the tape blank symbol?")
blank := !&input
write("Enter the delta mappings, using the following format:")
write("\tenter delta(curState,tapeSymbol) = (newState,newSymbol,direct) as")
write("\t curState:tapeSymbol:newState:newSymbol:direct");
write("\t\twhere direct is left, right, stay, or halt")
write("End with a blank line.")
write("")
every line := !&input do {
if *line = 0 then break
line ?
if (os := tab(upto(':')), move(1), ic := tab(upto(':')), move(1),
ns := tab(upto(':')), move(1), oc := tab(upto(':')), move(1),
d := map(tab(0))) then D[os||":"||ic] := delta(os,ic,ns,oc,d)
else write(line, " is in bad form, correct it")
}
if /start_tape then {
write("Enter the input tape")
start_tape := !&input
}
return TM(start,fStates,D,start_tape,blank)
end
procedure runTuringMachine(tm)
trans := tm.delta
rightside := tm.tape
if /rightside | (*rightside = 0) then rightside := tm.blank
leftside := ""
cur_state := tm.start
write("Machine starts in ",cur_state," with tape:")
show_tape(tm,leftside,rightside)
while mapping := \trans[cur_state||":"||rightside[1]] do {
rightside[1] := mapping.output_symbol
case mapping.direction of {
"left" : {
if *leftside = 0 then leftside := tm.blank
rightside := leftside[-1] || rightside
leftside[-1] := ""
}
"right" : {
leftside ||:= rightside[1]
rightside[1] := ""
if *rightside = 0 then rightside := tm.blank
}
"halt" : break
}
cur_state := mapping.new_state
if member(tm.final,cur_state) then break
trace(tm,cur_state,leftside,rightside)
}
write()
write("Machine halts in ",cur_state," with tape:")
show_tape(tm,leftside,rightside)
end
procedure trace(tm,cs,ls,rs)
static count, last_state
initial {
count := 0
last_state := ""
}
count +:= 1
if \show_count & (count % show_count = 0) then show_tape(tm,ls,rs)
if find(":"||cs||":",trace_list) & (last_state ~== cs) then {
writes("\tnow in state: ",cs," ")
if \full_display then show_delta(tm.delta[cs||":"||rs[1]])
else write()
}
last_state := cs
return
end
procedure show_delta(m)
if /m then write("NO MOVE!")
else {
writes("\tnext move is ")
writes("delta(",m.old_state,",",m.input_symbol,") ::= ")
write("(",m.new_state,",",m.output_symbol,",",m.direction,")")
}
end
procedure show_tape(tm,l,r)
l := reverse(trim(reverse(l),tm.blank))
r := trim(r,tm.blank)
write(l,r)
write(repl(" ",*l),"^")
end
First sample machine, with tape changes on each transition traced:
->turing 1
What is the start state? q0
What are the final states (colon separated)? qf
What is the tape blank symbol?B
Enter the delta mappings, using the following format:
enter delta(curState,tapeSymbol) = (newState,newSymbol,direct) as
curState:tapeSymbol:newState:newSymbol:direct
where direct is left, right, stay, or halt
End with a blank line.
q0:1:q0:1:right
q0:B:qf:1:stay
Enter the input tape
111
Machine starts in q0 with tape:
111
^
111
^
111
^
111
^
Machine halts in qf with tape:
1111
^
->
Second sample machine, with all tracing off (only first and last tapes are displayed):
->turing
What is the start state? a
What are the final states (colon separated)? halt
What is the tape blank symbol?0
Enter the delta mappings, using the following format:
enter delta(curState,tapeSymbol) = (newState,newSymbol,direct) as
curState:tapeSymbol:newState:newSymbol:direct
where direct is left, right, stay, or halt
End with a blank line.
a:0:b:1:right
a:1:c:1:left
b:0:a:1:left
b:1:b:1:right
c:0:b:1:left
c:1:halt:1:stay
Enter the input tape
Machine starts in a with tape:
^
Machine halts in halt with tape:
111111
^
->
J
Source for this task was slightly adapted from http://www.2bestsystems.com/j/J_Conference_2012. All the information for the Turing machines is represented by integers, the halting state is set as _1 (minus one), and head movements are mapped as (left, stay, right) ➜ (_1, 0, 1). A Turing machine is executed until a halt state is issued or a trivial infinite regress in the form of a single changeless cycle is detected. The transition table entry format is similar to the one in http://drb9.drb.insel.de/~heiner/BB/simAB3Y_SB.html.
The universal (stateless point-free) Turing machine
The universal Turing machine is defined in terms of fixed tacit (stateless point-free) code, showing that this dialect of J is Turing complete.
". noun define -. CRLF NB. Fixed tacit universal Turing machine code...
utm=.
(((":@:(]&:>)@:(6&({::)) ,: (":@] 9&({::))) ,. ':'"_) ,. 2&({::) >@:(((48 + ]
) { a."_)@[ ; (] $ ' '"_) , '^'"_) 3&({::))@:([ (0 0 $ 1!:2&2)@:('A changeles
s cycle was detected!'"_)^:(-.@:(_1"_ = 1&({::))))@:((((3&({::) + 8&({::)) ;
1 + 9&({::)) 3 9} ])@:(<@:((0 (0 {:: ])`(<@:(1 {:: ]))`(2 {:: ])} ])@:(7 3 2&
{)) 2} ])@:(<"0@:(6&({::) (<@[ { ]) 0&({::)) 7 8 1} ])@:([ (0 0 $ 1!:2&2)@:((
(":@:(]&:>)@:(6&({::)) ,: (":@] 9&({::))) ,. ':'"_) ,. 2&({::) >@:(((48 + ])
{ a."_)@[ ; (] $ ' '"_) , '^'"_) 3&({::))^:(0 = 4&({::) | 9&({::)))@:(<@:(1&(
{::) ; 3&({::) { 2&({::)) 6} ])@:(<@:(3&({::) + _1 = 3&({::)) 3} ])@:(<@:(((_
1 = 3&({::)) {:: 5&({::)) , 2&({::) , (3&({::) = #@:(2&({::))) {:: 5&({::)) 2
} ])^:(-.@:(_1"_ = 1&({::)))^:_)@:((0 ; (({. , ({: % 3:) , 3:)@:$ $ ,)@:(}."1
)@:(".;._2)@:(0&({::))) 9 0} ])@:(<@:('' ; 0"_) 5} ])@:(,&(;:',,,,,'))@:(,~)
)
The incrementer machine
Noun=. ".@('(0 : 0)'"_)
NB. Simple Incrementer...
NB. 0 1 Tape Symbol Scan
NB. S p m g p m g (p,m,g) → (print,move,goto)
QS=. (Noun _) ; 0 NB. Reading the transition table and setting the initial state
0 1 0 _1 1 1 0
)
TPF=. 1 1 1 ; 0 ; 1 NB. Setting the tape, its pointer and the display frequency
TPF utm QS NB. Running the Turing machine...
0 1:111
0 :^
0 1:111
1 : ^
0 1:111
2 : ^
0 0:1110
3 : ^
0 0:1111
4 : ^
The three-state busy beaver machine
NB. Three-state busy beaver..
NB. 0 1 Tape Symbol Scan
NB. S p m g p m g (p,m,g) → (print,move,goto)
QS=. (Noun _) ; 0 NB. Reading the transition table and setting the initial state
0 1 1 1 1 _1 2
1 1 _1 0 1 1 1
2 1 _1 1 1 0 _1
)
TPF=. 0 ; 0 ; 1 NB. Setting the tape, its pointer and the display frequency
TPF utm QS NB. Running the Turing machine...
0 0:0
0 :^
1 0:10
1 : ^
0 1:11
2 :^
2 0:011
3 :^
1 0:0111
4 :^
0 0:01111
5 :^
1 1:11111
6 : ^
1 1:11111
7 : ^
1 1:11111
8 : ^
1 1:11111
9 : ^
1 0:111110
10 : ^
0 1:111111
11 : ^
2 1:111111
12 : ^
2 1:111111
13 : ^
The probable 5-state, 2-symbol busy beaver machine
NB. Probable 5-state, 2-symbol busy beaver...
NB. 0 1 Tape Symbol Scan
NB. S p m g p m g (p,m,g) → (print,move,goto)
QS=. (Noun _) ; 0 NB. Reading the transition table and setting the state
0 1 1 1 1 _1 2
1 1 1 2 1 1 1
2 1 1 3 0 _1 4
3 1 _1 0 1 _1 3
4 1 1 _1 0 _1 0
)
TPF=. 0 ; 0 ; _ NB. Setting the tape, its pointer and the display frequency
TPF utm QS NB. Running the Turing machine...
0 0:0
0 :^
4 0 :101001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001...
47176870: ^
The sorting stress test machine
NB. Sorting stress test...
NB. 0 1 2 3 Tape Symbol Scan
NB. S p m g p m g p m g p m g (p,m,g) ➜ (print,move,goto)
QS=. (Noun _) ; 0 NB. Reading the transition table and setting the initial state
0 0 _1 4 1 1 0 3 1 1 _ _ _
1 0 _1 2 1 1 1 2 1 1 _ _ _
2 _ _ _ 2 _1 3 2 _1 2 2 _1 4
3 _ _ _ 1 _1 3 2 _1 3 1 1 0
4 0 1 _1 1 _1 4 _ _ _ _ _ _
)
TPF=. 1 2 2 1 2 2 1 2 1 2 1 2 1 2 ; 0 ; 50 NB. Setting the tape, its pointer and the display frequency
TPF utm QS NB. Running the Turing machine...
0 1:12212212121212
0 :^
3 2:113122121222220
50 : ^
1 2:111111322222220
100: ^
4 0:0111111222222220
118: ^
The structured derivation of the universal Turing machine
The fixed tacit code was produced by means of an unorthodox tacit toolkit; however, the verb produced is orthodox (i.e., compliant with the language specifications):
NB. Structured derivation of the universal Turing machine...
NB.--------------------------------------------------------------------------------------
NB. Quick and dirty tacit toolkit...
o=. @:
c=."_
ver=. (0:`)([:^:)
d=. (fix=. (;:'f.')ver) (train=.(;:'`:')ver&6) (an=. <@:((,'0') (,&<) ]))
ver=. (an f. o fix'ver')ver o an f.
z=. ((an'')`($ ,)`) (`:6)
d=. (a0=. `'') (a1=. (@:[) ((<'&')`) (`:6)) (a2=. (`(<(":0);_)) (`:6))
av=. ((an o fix'a0')`) (`(an o fix'a1')) (`(an o fix'a2') ) (`:6)
Fetch=. (ver o train ;:'&{::')&.> o i. f.av
tie=. ver o train ;:'`'
indices=. (, $~ 1 -.~ $) o (train"0 o ((1 -: L.)S:1 # <S:1) o (tie&'') o fix :: ] @:[)
f=. ((ver o train ;:'&{')) o indices o train f.av
'A B'=. 2 Fetch
head=. (;:'<@:') {.~ 2 * 1 = # o [
h=. train o (indices o train o (A f) (head , (B f) o ] , < o an o [ , (;:'}]')c) ]) f.av
DropIfNB=. < o ('('"_ , ] , ')'"_) o ((}: ^: ('NB.' -: 3&{. o > o {:)) &. ;:)
pipe=. ([ , ' o ' , ])&:>/ o |.
is=. ". o (, o ": o > , '=. ' , pipe o (DropIfNB;._2) o ". o ('0 ( : 0)'c)) f.av
NB.--------------------------------------------------------------------------------------
NB. Producing the main (dyadic) verb utm...
Note 0
NB. X (boxed list)...
Q - Instruction table
S - Turing machine initial state
NB. Y (boxed list)...
T - Data tape
P - Head position pointer
F - Display frequency
NB. Local...
B - Blank defaults
M - State and tape symbol read
PRINT - Printing symbol
MOVE - Tape head moving instruction
C - Step Counter
)
'Q S T P F B M PRINT MOVE C'=. 10 Fetch NB. Fetching 10 Boxes
DisplayTape=. > o (((48 + ]) { a.c)@[ ; ((] $ ' 'c) , '^'c))
display=. ((((": o (]&:>) o M) ,: (":@] C)) ,. ':'c ) ,. (T DisplayTape P))
NB. Displaying state, symbol, tape / step and pointer
amend=. 0 (0 {:: ])`(<@:(1 {:: ]))`(2 {:: ])} ]
NB. execute (monadic verb)...
FillLeft=. (_1 = P ) {:: B NB. Expanding and filling the tape
FillRight=. ( P = # o T) {:: B NB. with 0's (if necessary)
ia=. <@[ { ] NB. Selecting by the indices of an array
execute is
T`(FillLeft , T , FillRight)h NB. Adjusting the tape
P`(P + _1 = P) h NB. and the pointer (if necessary)
M`(S ; P { T) h NB. Updating the state and reading the tape symbol
[ (smoutput o display)^:(0 = F | C) NB. Displaying intermediate cycles
(PRINT MOVE S)`(<"0 o (M ia Q))h NB. Performing the printing, moving and state actions
T`(amend o ((PRINT P T)f)) h NB. Printing symbol on tape at the pointer position
(P C)`((P + MOVE) ; 1 + C) h NB. Updating the pointer and the counter
)
cc=. 'A changeless cycle was detected!'c
halt=. _1 c = S NB. Halting when the current state is _1
rt=. ((({. , ({: % 3:) , 3:) o $) $ ,) o (}."1) o (". ;. _2)
NB. Reshaping the transition table as a 3D array (state,symbol,action)
utm is NB. Universal Turing Machine (dyadic verb)
,~ NB. Appending the arguments in reverse order
,&(;:5$',') NB. Appending 5 local boxes (B M PRINT MOVE C)
B`('' ; 0 c) h NB. Setting empty blank defaults as 0
(C Q)`(0 ; rt o Q)h NB. Setting the counter and the transition table
execute^:(-. o halt)^:_ NB. Executing until a halt instruction is issued
[ smoutput o cc ^: (-. o halt) NB. or a changeless single cycle is detected
display NB. Displaying (returning) the final status
)
utm=. utm f. NB. Fixing the universal Turing machine code
NB. The simulation code is produced by 77 (-@:[ ]\ 5!:5@<@:]) 'utm'
Java
This is an implementation of the universal Turing machine in plain Java using standard libraries only. As generics are used, Java 5 is required. The examples (incrementer and busy beaver) are implemented directly in the main method and executed sequentially; as an additional third example, a sorting algorithm is implemented and executed in the end of the main method. During execution the complete tape and the current active transition are printed out in every step. The state names and tape symbols may contain several characters, so arbitrary strings such as "q1", "q2", ... can be valid state names or tape symbols. The machine is deterministic as the transitions are stored in a HashMap which uses state / tape symbol pairs as keys. This is self-coded, not a standard implementation, so there is no guarantee of correctness.
import java.util.HashMap;
import java.util.HashSet;
import java.util.LinkedList;
import java.util.ListIterator;
import java.util.List;
import java.util.Set;
import java.util.Map;
public class UTM {
private List<String> tape;
private String blankSymbol;
private ListIterator<String> head;
private Map<StateTapeSymbolPair, Transition> transitions = new HashMap<StateTapeSymbolPair, Transition>();
private Set<String> terminalStates;
private String initialState;
public UTM(Set<Transition> transitions, Set<String> terminalStates, String initialState, String blankSymbol) {
this.blankSymbol = blankSymbol;
for (Transition t : transitions) {
this.transitions.put(t.from, t);
}
this.terminalStates = terminalStates;
this.initialState = initialState;
}
public static class StateTapeSymbolPair {
private String state;
private String tapeSymbol;
public StateTapeSymbolPair(String state, String tapeSymbol) {
this.state = state;
this.tapeSymbol = tapeSymbol;
}
// These methods can be auto-generated by Eclipse.
@Override
public int hashCode() {
final int prime = 31;
int result = 1;
result = prime * result
+ ((state == null) ? 0 : state.hashCode());
result = prime
* result
+ ((tapeSymbol == null) ? 0 : tapeSymbol
.hashCode());
return result;
}
// These methods can be auto-generated by Eclipse.
@Override
public boolean equals(Object obj) {
if (this == obj)
return true;
if (obj == null)
return false;
if (getClass() != obj.getClass())
return false;
StateTapeSymbolPair other = (StateTapeSymbolPair) obj;
if (state == null) {
if (other.state != null)
return false;
} else if (!state.equals(other.state))
return false;
if (tapeSymbol == null) {
if (other.tapeSymbol != null)
return false;
} else if (!tapeSymbol.equals(other.tapeSymbol))
return false;
return true;
}
@Override
public String toString() {
return "(" + state + "," + tapeSymbol + ")";
}
}
public static class Transition {
private StateTapeSymbolPair from;
private StateTapeSymbolPair to;
private int direction; // -1 left, 0 neutral, 1 right.
public Transition(StateTapeSymbolPair from, StateTapeSymbolPair to, int direction) {
this.from = from;
this.to = to;
this.direction = direction;
}
@Override
public String toString() {
return from + "=>" + to + "/" + direction;
}
}
public void initializeTape(List<String> input) { // Arbitrary Strings as symbols.
tape = input;
}
public void initializeTape(String input) { // Uses single characters as symbols.
tape = new LinkedList<String>();
for (int i = 0; i < input.length(); i++) {
tape.add(input.charAt(i) + "");
}
}
public List<String> runTM() { // Returns null if not in terminal state.
if (tape.size() == 0) {
tape.add(blankSymbol);
}
head = tape.listIterator();
head.next();
head.previous();
StateTapeSymbolPair tsp = new StateTapeSymbolPair(initialState, tape.get(0));
while (transitions.containsKey(tsp)) { // While a matching transition exists.
System.out.println(this + " --- " + transitions.get(tsp));
Transition trans = transitions.get(tsp);
head.set(trans.to.tapeSymbol); // Write tape symbol.
tsp.state = trans.to.state; // Change state.
if (trans.direction == -1) { // Go left.
if (!head.hasPrevious()) {
head.add(blankSymbol); // Extend tape.
}
tsp.tapeSymbol = head.previous(); // Memorize tape symbol.
} else if (trans.direction == 1) { // Go right.
head.next();
if (!head.hasNext()) {
head.add(blankSymbol); // Extend tape.
head.previous();
}
tsp.tapeSymbol = head.next(); // Memorize tape symbol.
head.previous();
} else {
tsp.tapeSymbol = trans.to.tapeSymbol;
}
}
System.out.println(this + " --- " + tsp);
if (terminalStates.contains(tsp.state)) {
return tape;
} else {
return null;
}
}
@Override
public String toString() {
try {
int headPos = head.previousIndex();
String s = "[ ";
for (int i = 0; i <= headPos; i++) {
s += tape.get(i) + " ";
}
s += "[H] ";
for (int i = headPos + 1; i < tape.size(); i++) {
s += tape.get(i) + " ";
}
return s + "]";
} catch (Exception e) {
return "";
}
}
public static void main(String[] args) {
// Simple incrementer.
String init = "q0";
String blank = "b";
Set<String> term = new HashSet<String>();
term.add("qf");
Set<Transition> trans = new HashSet<Transition>();
trans.add(new Transition(new StateTapeSymbolPair("q0", "1"), new StateTapeSymbolPair("q0", "1"), 1));
trans.add(new Transition(new StateTapeSymbolPair("q0", "b"), new StateTapeSymbolPair("qf", "1"), 0));
UTM machine = new UTM(trans, term, init, blank);
machine.initializeTape("111");
System.out.println("Output (si): " + machine.runTM() + "\n");
// Busy Beaver (overwrite variables from above).
init = "a";
term.clear();
term.add("halt");
blank = "0";
trans.clear();
// Change state from "a" to "b" if "0" is read on tape, write "1" and go to the right. (-1 left, 0 nothing, 1 right.)
trans.add(new Transition(new StateTapeSymbolPair("a", "0"), new StateTapeSymbolPair("b", "1"), 1));
trans.add(new Transition(new StateTapeSymbolPair("a", "1"), new StateTapeSymbolPair("c", "1"), -1));
trans.add(new Transition(new StateTapeSymbolPair("b", "0"), new StateTapeSymbolPair("a", "1"), -1));
trans.add(new Transition(new StateTapeSymbolPair("b", "1"), new StateTapeSymbolPair("b", "1"), 1));
trans.add(new Transition(new StateTapeSymbolPair("c", "0"), new StateTapeSymbolPair("b", "1"), -1));
trans.add(new Transition(new StateTapeSymbolPair("c", "1"), new StateTapeSymbolPair("halt", "1"), 0));
machine = new UTM(trans, term, init, blank);
machine.initializeTape("");
System.out.println("Output (bb): " + machine.runTM());
// Sorting test (overwrite variables from above).
init = "s0";
blank = "*";
term = new HashSet<String>();
term.add("see");
trans = new HashSet<Transition>();
trans.add(new Transition(new StateTapeSymbolPair("s0", "a"), new StateTapeSymbolPair("s0", "a"), 1));
trans.add(new Transition(new StateTapeSymbolPair("s0", "b"), new StateTapeSymbolPair("s1", "B"), 1));
trans.add(new Transition(new StateTapeSymbolPair("s0", "*"), new StateTapeSymbolPair("se", "*"), -1));
trans.add(new Transition(new StateTapeSymbolPair("s1", "a"), new StateTapeSymbolPair("s1", "a"), 1));
trans.add(new Transition(new StateTapeSymbolPair("s1", "b"), new StateTapeSymbolPair("s1", "b"), 1));
trans.add(new Transition(new StateTapeSymbolPair("s1", "*"), new StateTapeSymbolPair("s2", "*"), -1));
trans.add(new Transition(new StateTapeSymbolPair("s2", "a"), new StateTapeSymbolPair("s3", "b"), -1));
trans.add(new Transition(new StateTapeSymbolPair("s2", "b"), new StateTapeSymbolPair("s2", "b"), -1));
trans.add(new Transition(new StateTapeSymbolPair("s2", "B"), new StateTapeSymbolPair("se", "b"), -1));
trans.add(new Transition(new StateTapeSymbolPair("s3", "a"), new StateTapeSymbolPair("s3", "a"), -1));
trans.add(new Transition(new StateTapeSymbolPair("s3", "b"), new StateTapeSymbolPair("s3", "b"), -1));
trans.add(new Transition(new StateTapeSymbolPair("s3", "B"), new StateTapeSymbolPair("s0", "a"), 1));
trans.add(new Transition(new StateTapeSymbolPair("se", "a"), new StateTapeSymbolPair("se", "a"), -1));
trans.add(new Transition(new StateTapeSymbolPair("se", "*"), new StateTapeSymbolPair("see", "*"), 1));
machine = new UTM(trans, term, init, blank);
machine.initializeTape("babbababaa");
System.out.println("Output (sort): " + machine.runTM() + "\n");
}
}- Output:
-- [H] denotes the head; its position on the tape is over the symbol printed right from it.
[ [H] 1 1 1 ] --- (q0,1)=>(q0,1)/1 [ 1 [H] 1 1 ] --- (q0,1)=>(q0,1)/1 [ 1 1 [H] 1 ] --- (q0,1)=>(q0,1)/1 [ 1 1 1 [H] b ] --- (q0,b)=>(qf,1)/0 [ 1 1 1 [H] 1 ] --- (qf,1) Output (si): [1, 1, 1, 1] [ [H] 0 ] --- (a,0)=>(b,1)/1 [ 1 [H] 0 ] --- (b,0)=>(a,1)/-1 [ [H] 1 1 ] --- (a,1)=>(c,1)/-1 [ [H] 0 1 1 ] --- (c,0)=>(b,1)/-1 [ [H] 0 1 1 1 ] --- (b,0)=>(a,1)/-1 [ [H] 0 1 1 1 1 ] --- (a,0)=>(b,1)/1 [ 1 [H] 1 1 1 1 ] --- (b,1)=>(b,1)/1 [ 1 1 [H] 1 1 1 ] --- (b,1)=>(b,1)/1 [ 1 1 1 [H] 1 1 ] --- (b,1)=>(b,1)/1 [ 1 1 1 1 [H] 1 ] --- (b,1)=>(b,1)/1 [ 1 1 1 1 1 [H] 0 ] --- (b,0)=>(a,1)/-1 [ 1 1 1 1 [H] 1 1 ] --- (a,1)=>(c,1)/-1 [ 1 1 1 [H] 1 1 1 ] --- (c,1)=>(halt,1)/0 [ 1 1 1 [H] 1 1 1 ] --- (halt,1) Output (bb): [1, 1, 1, 1, 1, 1] [ [H] b a b b a b a b a a ] --- (s0,b)=>(s1,B)/1 [ B [H] a b b a b a b a a ] --- (s1,a)=>(s1,a)/1 [ B a [H] b b a b a b a a ] --- (s1,b)=>(s1,b)/1 [ B a b [H] b a b a b a a ] --- (s1,b)=>(s1,b)/1 [ B a b b [H] a b a b a a ] --- (s1,a)=>(s1,a)/1 [ B a b b a [H] b a b a a ] --- (s1,b)=>(s1,b)/1 [ B a b b a b [H] a b a a ] --- (s1,a)=>(s1,a)/1 [ B a b b a b a [H] b a a ] --- (s1,b)=>(s1,b)/1 [ B a b b a b a b [H] a a ] --- (s1,a)=>(s1,a)/1 [ B a b b a b a b a [H] a ] --- (s1,a)=>(s1,a)/1 [ B a b b a b a b a a [H] * ] --- (s1,*)=>(s2,*)/-1 [ B a b b a b a b a [H] a * ] --- (s2,a)=>(s3,b)/-1 [ B a b b a b a b [H] a b * ] --- (s3,a)=>(s3,a)/-1 [ B a b b a b a [H] b a b * ] --- (s3,b)=>(s3,b)/-1 [ B a b b a b [H] a b a b * ] --- (s3,a)=>(s3,a)/-1 [ B a b b a [H] b a b a b * ] --- (s3,b)=>(s3,b)/-1 [ B a b b [H] a b a b a b * ] --- (s3,a)=>(s3,a)/-1 [ B a b [H] b a b a b a b * ] --- (s3,b)=>(s3,b)/-1 [ B a [H] b b a b a b a b * ] --- (s3,b)=>(s3,b)/-1 [ B [H] a b b a b a b a b * ] --- (s3,a)=>(s3,a)/-1 [ [H] B a b b a b a b a b * ] --- (s3,B)=>(s0,a)/1 [ a [H] a b b a b a b a b * ] --- (s0,a)=>(s0,a)/1 [ a a [H] b b a b a b a b * ] --- (s0,b)=>(s1,B)/1 [ a a B [H] b a b a b a b * ] --- (s1,b)=>(s1,b)/1 [ a a B b [H] a b a b a b * ] --- (s1,a)=>(s1,a)/1 [ a a B b a [H] b a b a b * ] --- (s1,b)=>(s1,b)/1 [ a a B b a b [H] a b a b * ] --- (s1,a)=>(s1,a)/1 [ a a B b a b a [H] b a b * ] --- (s1,b)=>(s1,b)/1 [ a a B b a b a b [H] a b * ] --- (s1,a)=>(s1,a)/1 [ a a B b a b a b a [H] b * ] --- (s1,b)=>(s1,b)/1 [ a a B b a b a b a b [H] * ] --- (s1,*)=>(s2,*)/-1 [ a a B b a b a b a [H] b * ] --- (s2,b)=>(s2,b)/-1 [ a a B b a b a b [H] a b * ] --- (s2,a)=>(s3,b)/-1 [ a a B b a b a [H] b b b * ] --- (s3,b)=>(s3,b)/-1 [ a a B b a b [H] a b b b * ] --- (s3,a)=>(s3,a)/-1 [ a a B b a [H] b a b b b * ] --- (s3,b)=>(s3,b)/-1 [ a a B b [H] a b a b b b * ] --- (s3,a)=>(s3,a)/-1 [ a a B [H] b a b a b b b * ] --- (s3,b)=>(s3,b)/-1 [ a a [H] B b a b a b b b * ] --- (s3,B)=>(s0,a)/1 [ a a a [H] b a b a b b b * ] --- (s0,b)=>(s1,B)/1 [ a a a B [H] a b a b b b * ] --- (s1,a)=>(s1,a)/1 [ a a a B a [H] b a b b b * ] --- (s1,b)=>(s1,b)/1 [ a a a B a b [H] a b b b * ] --- (s1,a)=>(s1,a)/1 [ a a a B a b a [H] b b b * ] --- (s1,b)=>(s1,b)/1 [ a a a B a b a b [H] b b * ] --- (s1,b)=>(s1,b)/1 [ a a a B a b a b b [H] b * ] --- (s1,b)=>(s1,b)/1 [ a a a B a b a b b b [H] * ] --- (s1,*)=>(s2,*)/-1 [ a a a B a b a b b [H] b * ] --- (s2,b)=>(s2,b)/-1 [ a a a B a b a b [H] b b * ] --- (s2,b)=>(s2,b)/-1 [ a a a B a b a [H] b b b * ] --- (s2,b)=>(s2,b)/-1 [ a a a B a b [H] a b b b * ] --- (s2,a)=>(s3,b)/-1 [ a a a B a [H] b b b b b * ] --- (s3,b)=>(s3,b)/-1 [ a a a B [H] a b b b b b * ] --- (s3,a)=>(s3,a)/-1 [ a a a [H] B a b b b b b * ] --- (s3,B)=>(s0,a)/1 [ a a a a [H] a b b b b b * ] --- (s0,a)=>(s0,a)/1 [ a a a a a [H] b b b b b * ] --- (s0,b)=>(s1,B)/1 [ a a a a a B [H] b b b b * ] --- (s1,b)=>(s1,b)/1 [ a a a a a B b [H] b b b * ] --- (s1,b)=>(s1,b)/1 [ a a a a a B b b [H] b b * ] --- (s1,b)=>(s1,b)/1 [ a a a a a B b b b [H] b * ] --- (s1,b)=>(s1,b)/1 [ a a a a a B b b b b [H] * ] --- (s1,*)=>(s2,*)/-1 [ a a a a a B b b b [H] b * ] --- (s2,b)=>(s2,b)/-1 [ a a a a a B b b [H] b b * ] --- (s2,b)=>(s2,b)/-1 [ a a a a a B b [H] b b b * ] --- (s2,b)=>(s2,b)/-1 [ a a a a a B [H] b b b b * ] --- (s2,b)=>(s2,b)/-1 [ a a a a a [H] B b b b b * ] --- (s2,B)=>(se,b)/-1 [ a a a a [H] a b b b b b * ] --- (se,a)=>(se,a)/-1 [ a a a [H] a a b b b b b * ] --- (se,a)=>(se,a)/-1 [ a a [H] a a a b b b b b * ] --- (se,a)=>(se,a)/-1 [ a [H] a a a a b b b b b * ] --- (se,a)=>(se,a)/-1 [ [H] a a a a a b b b b b * ] --- (se,a)=>(se,a)/-1 [ [H] * a a a a a b b b b b * ] --- (se,*)=>(see,*)/1 [ * [H] a a a a a b b b b b * ] --- (see,a) Output (sort): [*, a, a, a, a, a, b, b, b, b, b, *]
JavaScript
function tm(d,s,e,i,b,t,... r) {
document.write(d, '<br>')
if (i<0||i>=t.length) return
var re=new RegExp(b,'g')
write('*',s,i,t=t.split(''))
var p={}; r.forEach(e=>((s,r,w,m,n)=>{p[s+'.'+r]={w,n,m:[0,1,-1][1+'RL'.indexOf(m)]}})(... e.split(/[ .:,]+/)))
for (var n=1; s!=e; n+=1) {
with (p[s+'.'+t[i]]) t[i]=w,s=n,i+=m
if (i==-1) i=0,t.unshift(b)
else if (i==t.length) t[i]=b
write(n,s,i,t)
}
document.write('<br>')
function write(n, s, i, t) {
t = t.join('')
t = t.substring(0,i) + '<u>' + t.charAt(i) + '</u>' + t.substr(i+1)
document.write((' '+n).slice(-3).replace(/ /g,' '), ': ', s, ' [', t.replace(re,' '), ']', '<br>')
}
}
tm( 'Unary incrementer',
// s e i b t
'a', 'h', 0, 'B', '111',
// s.r: w, m, n
'a.1: 1, L, a',
'a.B: 1, S, h'
)
tm( 'Unary adder',
1, 0, 0, '0', '1110111',
'1.1: 0, R, 2', // write 0 rigth goto 2
'2.1: 1, R, 2', // while (1) rigth
'2.0: 1, S, 0' // write 1 stay halt
)
tm( 'Three-state busy beaver',
1, 0, 0, '0', '0',
'1.0: 1, R, 2',
'1.1: 1, R, 0',
'2.0: 0, R, 3',
'2.1: 1, R, 2',
'3.0: 1, L, 3',
'3.1: 1, L, 1'
)
- Output:
Unary incrementer
*: a [111]
1: a [ 111]
2: h [1111]
Unary adder
*: 1 [111 111]
1: 2 [ 11 111]
2: 3 [ 11 111]
3: 3 [ 11 111]
4: 0 [ 111111]
Three-state busy beaver
*: 1 [ ]
1: 2 [1 ]
2: 3 [1 ]
3: 3 [1 1]
4: 3 [111]
5: 1 [ 111]
6: 2 [1111]
7: 2 [1111]
8: 2 [1111]
9: 2 [1111 ]
10: 3 [1111 ]
11: 3 [1111 1]
12: 3 [111111]
13: 1 [111111]
14: 0 [111111]
Julia
import Base.show
@enum Move Left=1 Stay Right
mutable struct MachineState
state::String
tape::Dict{Int, String}
headpos::Int
end
struct Rule
instate::String
s1::String
s2::String
move::Move
outstate::String
end
struct Program
title::String
initial::String
final::String
blank::String
rules::Vector{Rule}
end
const testprograms = [
(Program("Simple incrementer", "q0", "qf", "B",
[Rule("q0", "1", "1", Right, "q0"), Rule("q0", "B", "1", Stay, "qf")]),
Dict(1 =>"1", 2 => "1", 3 => "1"), true),
(Program("Three-state busy beaver", "a", "halt", "0",
[Rule("a", "0", "1", Right, "b"), Rule("a", "1", "1", Left, "c"),
Rule("b", "0", "1", Left, "a"), Rule("b", "1", "1", Right, "b"),
Rule("c", "0", "1", Left, "b"), Rule("c", "1", "1", Stay, "halt")]),
Dict(), true),
(Program("Five-state busy beaver", "A", "H", "0",
[Rule("A", "0", "1", Right, "B"), Rule("A", "1", "1", Left, "C"),
Rule("B", "0", "1", Right, "C"), Rule("B", "1", "1", Right, "B"),
Rule("C", "0", "1", Right, "D"), Rule("C", "1", "0", Left, "E"),
Rule("D", "0", "1", Left, "A"), Rule("D", "1", "1", Left, "D"),
Rule("E", "0", "1", Stay, "H"), Rule("E", "1", "0", Left, "A")]),
Dict(), false)]
function show(io::IO, mstate::MachineState)
ibracket(i, curpos, val) = (i == curpos) ? "[$val]" : " $val "
print(io, rpad("($(mstate.state))", 12))
for i in sort(collect(keys(mstate.tape)))
print(io, " $(ibracket(i, mstate.headpos, mstate.tape[i]))")
end
end
function turing(program, tape, verbose)
println("\n$(program.title)")
verbose && println(" State \tTape [head]\n--------------------------------------------------")
mstate = MachineState(program.initial, tape, 1)
stepcount = 0
while true
if !haskey(mstate.tape, mstate.headpos)
mstate.tape[mstate.headpos] = program.blank
end
verbose && println(mstate)
for rule in program.rules
if rule.instate == mstate.state && rule.s1 == mstate.tape[mstate.headpos]
mstate.tape[mstate.headpos] = rule.s2
if rule.move == Left
mstate.headpos -= 1
elseif rule.move == Right
mstate.headpos += 1
end
mstate.state = rule.outstate
break
end
end
stepcount += 1
if mstate.state == program.final
break
end
end
verbose && println(mstate)
println("Total steps: $stepcount")
end
for (prog, tape, verbose) in testprograms
turing(prog, tape, verbose)
end
- Output:
Simple incrementer State Tape [head] -------------------------------------------------- (q0) [1] 1 1 (q0) 1 [1] 1 (q0) 1 1 [1] (q0) 1 1 1 [B] (qf) 1 1 1 [1] Total steps: 4 Three-state busy beaver State Tape [head] -------------------------------------------------- (a) [0] (b) 1 [0] (a) [1] 1 (c) [0] 1 1 (b) [0] 1 1 1 (a) [0] 1 1 1 1 (b) 1 [1] 1 1 1 (b) 1 1 [1] 1 1 (b) 1 1 1 [1] 1 (b) 1 1 1 1 [1] (b) 1 1 1 1 1 [0] (a) 1 1 1 1 [1] 1 (c) 1 1 1 [1] 1 1 (halt) 1 1 1 [1] 1 1 Total steps: 13 Five-state busy beaver Total steps: 47176870
Kotlin
// version 1.2.10
enum class Dir { LEFT, RIGHT, STAY }
class Rule(
val state1: String,
val symbol1: Char,
val symbol2: Char,
val dir: Dir,
val state2: String
)
class Tape(
var symbol: Char,
var left: Tape? = null,
var right: Tape? = null
)
class Turing(
val states: List<String>,
val finalStates: List<String>,
val symbols: CharArray,
val blank: Char,
var state: String,
tapeInput: CharArray,
rules: List<Rule>
) {
var tape: Tape? = null
val transitions = Array(states.size) { arrayOfNulls<Rule>(symbols.size) }
init {
for (i in 0 until tapeInput.size) {
move(Dir.RIGHT)
tape!!.symbol = tapeInput[i]
}
if (tapeInput.size == 0) move(Dir.RIGHT)
while (tape!!.left != null) tape = tape!!.left
for (i in 0 until rules.size) {
val rule = rules[i]
transitions[stateIndex(rule.state1)][symbolIndex(rule.symbol1)] = rule
}
}
private fun stateIndex(state: String): Int {
val i = states.indexOf(state)
return if (i >= 0) i else 0
}
private fun symbolIndex(symbol: Char): Int {
val i = symbols.indexOf(symbol)
return if (i >= 0) i else 0
}
private fun move(dir: Dir) {
val orig = tape
when (dir) {
Dir.RIGHT -> {
if (orig != null && orig.right != null) {
tape = orig.right
}
else {
tape = Tape(blank)
if (orig != null) {
tape!!.left = orig
orig.right = tape
}
}
}
Dir.LEFT -> {
if (orig != null && orig.left != null) {
tape = orig.left
}
else {
tape = Tape(blank)
if (orig != null) {
tape!!.right = orig
orig.left = tape
}
}
}
Dir.STAY -> {}
}
}
fun printState() {
print("%-10s ".format(state))
var t = tape
while (t!!.left != null ) t = t.left
while (t != null) {
if (t == tape) print("[${t.symbol}]")
else print(" ${t.symbol} ")
t = t.right
}
println()
}
fun run(maxLines: Int = 20) {
var lines = 0
while (true) {
printState()
for (finalState in finalStates) {
if (finalState == state) return
}
if (++lines == maxLines) {
println("(Only the first $maxLines lines displayed)")
return
}
val rule = transitions[stateIndex(state)][symbolIndex(tape!!.symbol)]
tape!!.symbol = rule!!.symbol2
move(rule.dir)
state = rule.state2
}
}
}
fun main(args: Array<String>) {
println("Simple incrementer")
Turing(
states = listOf("q0", "qf"),
finalStates = listOf("qf"),
symbols = charArrayOf('B', '1'),
blank = 'B',
state = "q0",
tapeInput = charArrayOf('1', '1', '1'),
rules = listOf(
Rule("q0", '1', '1', Dir.RIGHT, "q0"),
Rule("q0", 'B', '1', Dir.STAY, "qf")
)
).run()
println("\nThree-state busy beaver")
Turing(
states = listOf("a", "b", "c", "halt"),
finalStates = listOf("halt"),
symbols = charArrayOf('0', '1'),
blank = '0',
state = "a",
tapeInput = charArrayOf(),
rules = listOf(
Rule("a", '0', '1', Dir.RIGHT, "b"),
Rule("a", '1', '1', Dir.LEFT, "c"),
Rule("b", '0', '1', Dir.LEFT, "a"),
Rule("b", '1', '1', Dir.RIGHT, "b"),
Rule("c", '0', '1', Dir.LEFT, "b"),
Rule("c", '1', '1', Dir.STAY, "halt")
)
).run()
println("\nFive-state two-symbol probable busy beaver")
Turing(
states = listOf("A", "B", "C", "D", "E", "H"),
finalStates = listOf("H"),
symbols = charArrayOf('0', '1'),
blank = '0',
state = "A",
tapeInput = charArrayOf(),
rules = listOf(
Rule("A", '0', '1', Dir.RIGHT, "B"),
Rule("A", '1', '1', Dir.LEFT, "C"),
Rule("B", '0', '1', Dir.RIGHT, "C"),
Rule("B", '1', '1', Dir.RIGHT, "B"),
Rule("C", '0', '1', Dir.RIGHT, "D"),
Rule("C", '1', '0', Dir.LEFT, "E"),
Rule("D", '0', '1', Dir.LEFT, "A"),
Rule("D", '1', '1', Dir.LEFT, "D"),
Rule("E", '0', '1', Dir.STAY, "H"),
Rule("E", '1', '0', Dir.LEFT, "A")
)
).run()
}
- Output:
Simple incrementer q0 [1] 1 1 q0 1 [1] 1 q0 1 1 [1] q0 1 1 1 [B] qf 1 1 1 [1] Three-state busy beaver a [0] b 1 [0] a [1] 1 c [0] 1 1 b [0] 1 1 1 a [0] 1 1 1 1 b 1 [1] 1 1 1 b 1 1 [1] 1 1 b 1 1 1 [1] 1 b 1 1 1 1 [1] b 1 1 1 1 1 [0] a 1 1 1 1 [1] 1 c 1 1 1 [1] 1 1 halt 1 1 1 [1] 1 1 Five-state two-symbol probable busy beaver A [0] B 1 [0] C 1 1 [0] D 1 1 1 [0] A 1 1 [1] 1 C 1 [1] 1 1 E [1] 0 1 1 A [0] 0 0 1 1 B 1 [0] 0 1 1 C 1 1 [0] 1 1 D 1 1 1 [1] 1 D 1 1 [1] 1 1 D 1 [1] 1 1 1 D [1] 1 1 1 1 D [0] 1 1 1 1 1 A [0] 1 1 1 1 1 1 B 1 [1] 1 1 1 1 1 B 1 1 [1] 1 1 1 1 B 1 1 1 [1] 1 1 1 B 1 1 1 1 [1] 1 1 (Only the first 20 lines displayed)
Lambdatalk
{require lib_H} // associative arrays library
{def tm
{def tm.r
{lambda {:data :rules :state :end :blank :i :N}
{if {or {W.equal? :state :end} {> :N 400}} // recursion limited to 400
then :data
else {let { {:data :data} {:rules :rules}
{:state {H.get :state :rules}}
{:end :end} {:blank :blank}
{:i :i} {:N :N}
{:cell {if {W.equal? {A.get :i :data} undefined}
then :blank
else {A.get :i :data}}}
} {tm.r {A.set! :i {H.get write {H.get :cell :state}} :data}
:rules
{H.get next {H.get :cell :state}}
:end
:blank
{+ :i {H.get move {H.get :cell :state}} }
{+ :N 1} } }}}}
{lambda {:name :data :rules :state :end :blank}
:name: {A.duplicate :data} ->
{tm.r :data :rules :state :end :blank 0 0}}}
-> tm
{tm zero2one
{A.new 0 0 0}
{H.new A {H.new 0 {H.new write 1 | move 1 | next A} |
B {H.new write . | move 1 | next H} }
} A H B}
output: zero2one: [0,0,0] -> [1,1,1,.]
{tm add_one
{A.new 1 1 1}
{H.new A {H.new 1 {H.new write 1 | move 1 | next A} |
B {H.new write 1 | move 0 | next B} }
} A B B}
output: add_one: [1,1,1] -> [1,1,1,1]
{tm unary_adder
{A.new 1 1 1 0 1 1 1}
{H.new A {H.new 1 {H.new write 0 | move 1 | next B} } |
B {H.new 1 {H.new write 1 | move 1 | next B} |
0 {H.new write 1 | move 0 | next H} }
} A H B}
output: unary_adder: [1,1,1,0,1,1,1] -> [0,1,1,1,1,1,1]
{tm duplicate
{A.new 1 1 1}
{H.new q0 {H.new 1 {H.new write B | move 1 | next q1} } |
q1 {H.new 1 {H.new write 1 | move 1 | next q1} |
0 {H.new write 0 | move 1 | next q2} |
B {H.new write 0 | move 1 | next q2}} |
q2 {H.new 1 {H.new write 1 | move 1 | next q2} |
B {H.new write 1 | move -1 | next q3}} |
q3 {H.new 1 {H.new write 1 | move -1 | next q3} |
0 {H.new write 0 | move -1 | next q3} |
B {H.new write 1 | move 1 | next q4}} |
q4 {H.new 1 {H.new write B | move 1 | next q1} |
0 {H.new write 0 | move 1 | next qf}}
} q0 qf B}
output: duplicate: [1,1,1] -> [1,1,1,0,1,1,1]
{tm sort
{A.new 2 1 2 2 2 1 1}
{H.new A {H.new 1 {H.new write 1 | move 1 | next A} |
2 {H.new write 3 | move 1 | next B} |
0 {H.new write 0 | move -1 | next E}} |
B {H.new 1 {H.new write 1 | move 1 | next B} |
2 {H.new write 2 | move 1 | next B} |
0 {H.new write 0 | move -1 | next C}} |
C {H.new 1 {H.new write 2 | move -1 | next D} |
2 {H.new write 2 | move -1 | next C} |
3 {H.new write 2 | move -1 | next E}} |
D {H.new 1 {H.new write 1 | move -1 | next D} |
2 {H.new write 2 | move -1 | next D} |
3 {H.new write 1 | move 1 | next A}} |
E {H.new 1 {H.new write 1 | move -1 | next E} |
0 {H.new write 0 | move 1 | next H}}
} A H 0}
output: sort: [2,1,2,2,2,1,1] -> [1,1,1,2,2,2,2,0]
{tm busy_beaver
{A.new}
{H.new A {H.new 0 {H.new write 1 | move 1 | next B} |
1 {H.new write 1 | move -1 | next C}} |
B {H.new 0 {H.new write 1 | move -1 | next A} |
1 {H.new write 1 | move 1 | next B}} |
C {H.new 0 {H.new write 1 | move -1 | next B} |
1 {H.new write 1 | move 0 | next H}}
} A H 0}
output: busy_beaver: [] -> [1,1,1]
{tm busy_beaver2
{A.new}
{H.new A {H.new 0 {H.new write 1 | move 1 | next B} |
1 {H.new write 1 | move -1 | next C}} |
B {H.new 0 {H.new write 1 | move 1 | next C} |
1 {H.new write 1 | move 1 | next B}} |
C {H.new 0 {H.new write 1 | move 1 | next D} |
1 {H.new write 0 | move -1 | next E}} |
D {H.new 0 {H.new write 1 | move -1 | next A} |
1 {H.new write 1 | move -1 | next D}} |
E {H.new 0 {H.new write 1 | move 0 | next K} |
1 {H.new write 0 | move -1 | next A}}
} A K 0}
output: busy_beaver2: [] -> [1,1,1,1,1,1,1,1,1,1,1]
Lua
-- Machine definitions
local incrementer = {
name = "Simple incrementer",
initState = "q0",
endState = "qf",
blank = "B",
rules = {
{"q0", "1", "1", "right", "q0"},
{"q0", "B", "1", "stay", "qf"}
}
}
local threeStateBB = {
name = "Three-state busy beaver",
initState = "a",
endState = "halt",
blank = "0",
rules = {
{"a", "0", "1", "right", "b"},
{"a", "1", "1", "left", "c"},
{"b", "0", "1", "left", "a"},
{"b", "1", "1", "right", "b"},
{"c", "0", "1", "left", "b"},
{"c", "1", "1", "stay", "halt"}
}
}
local fiveStateBB = {
name = "Five-state busy beaver",
initState = "A",
endState = "H",
blank = "0",
rules = {
{"A", "0", "1", "right", "B"},
{"A", "1", "1", "left", "C"},
{"B", "0", "1", "right", "C"},
{"B", "1", "1", "right", "B"},
{"C", "0", "1", "right", "D"},
{"C", "1", "0", "left", "E"},
{"D", "0", "1", "left", "A"},
{"D", "1", "1", "left", "D"},
{"E", "0", "1", "stay", "H"},
{"E", "1", "0", "left", "A"}
}
}
-- Display a representation of the tape and machine state on the screen
function show (state, headPos, tape)
local leftEdge = 1
while tape[leftEdge - 1] do leftEdge = leftEdge - 1 end
io.write(" " .. state .. "\t| ")
for pos = leftEdge, #tape do
if pos == headPos then io.write("[" .. tape[pos] .. "] ") else io.write(" " .. tape[pos] .. " ") end
end
print()
end
-- Simulate a turing machine
function UTM (machine, tape, countOnly)
local state, headPos, counter = machine.initState, 1, 0
print("\n\n" .. machine.name)
print(string.rep("=", #machine.name) .. "\n")
if not countOnly then print(" State", "| Tape [head]\n---------------------") end
repeat
if not tape[headPos] then tape[headPos] = machine.blank end
if not countOnly then show(state, headPos, tape) end
for _, rule in ipairs(machine.rules) do
if rule[1] == state and rule[2] == tape[headPos] then
tape[headPos] = rule[3]
if rule[4] == "left" then headPos = headPos - 1 end
if rule[4] == "right" then headPos = headPos + 1 end
state = rule[5]
break
end
end
counter = counter + 1
until state == machine.endState
if countOnly then print("Steps taken: " .. counter) else show(state, headPos, tape) end
end
-- Main procedure
UTM(incrementer, {"1", "1", "1"})
UTM(threeStateBB, {})
UTM(fiveStateBB, {}, "countOnly")
- Output:
Simple incrementer ================== State | Tape [head] --------------------- q0 | [1] 1 1 q0 | 1 [1] 1 q0 | 1 1 [1] q0 | 1 1 1 [B] qf | 1 1 1 [1] Three-state busy beaver ======================= State | Tape [head] --------------------- a | [0] b | 1 [0] a | [1] 1 c | [0] 1 1 b | [0] 1 1 1 a | [0] 1 1 1 1 b | 1 [1] 1 1 1 b | 1 1 [1] 1 1 b | 1 1 1 [1] 1 b | 1 1 1 1 [1] b | 1 1 1 1 1 [0] a | 1 1 1 1 [1] 1 c | 1 1 1 [1] 1 1 halt | 1 1 1 [1] 1 1 Five-state busy beaver ====================== Steps taken: 47176870
M2000 Interpreter
Module CheckIt {
print "Universal Turing Machine"
print "------------------------"
class Machine {
private:
Head=1, Symbols=(,), States=(,)
Initial_State$, Terminating_state$, Blank_Symbol$
BS=0, Rules=list, caption$
tp$="{0:4} {1} {2} {3:5} {4:4}"
public:
Module States {
.States<=array([])
}
Module Symbols {
.Symbols<=array([])
}
Module Reset (.Initial_State$, .Terminating_state$, .Blank_Symbol$) {
if len(.States)=0 then error "No States defined"
if len(.Symbols)=0 then error "No Symbols defined"
if .States#nothave(.Initial_State$) then error "Initial State Not Exist"
if .States#nothave(.Terminating_state$) then error "Terminating State Not Exist"
it=.Symbols#pos(.Blank_Symbol$) : if it=-1 then error "Blank symbol not exist"
.BS<=it
.Rules<=List
}
Module Init (.caption$) {
flush // empty stack
print .caption$
}
Module AddRule (state$, read_symbol$, write_symbol$, action$, end_state$) {
if .States#nothave(state$) then Error "State not exist"
if .symbols#nothave(read_symbol$) then Error "Read Symbol not exist"
if .symbols#nothave(write_symbol$) then Error "Read Symbol not exist"
if ("right","left","stay")#nothave(action$) then Error "Action not exist"
if .States#nothave(end_state$) then Error "End state not exist"
try ok {
tuple=(.symbols#pos(write_symbol$), action$, end_state$)
Append .rules, state$+"_"+read_symbol$:=tuple
}
if not ok then error "rule "+ state$+"_"+read_symbol$+" already exist "
Pen 11 {
Print format$(.tp$, state$, read_symbol$, write_symbol$, action$, end_state$)
}
if stack.size>=5 then loop
}
Module Tape {
s=[]
m=each(s)
while m
it= .Symbols#pos(stackitem$(m))
if it=-1 then error "Tape symbol not exist at position ";m^
data it
end while
}
Module Run (steps as long, display as boolean) {
if len(.rules)=0 then error "No rules found"
if .Initial_State$="" or .Terminating_state$="" or .Blank_Symbol$="" then
error "Reset the machine please"
end if
if empty then push .BS
curState$=.Initial_State$
cont=true
.head<=1
dim inst$() : link inst$() to inst()
while curState$<>.Terminating_state$
if display then pen 15 {showstack()}
steps--
theRule$=curState$+"_"+.symbols#val$(stackitem(.head))
if not exist(.Rules, theRule$) then error "Undefined "+theRule$
inst$()=.Rules(theRule$)
shift .head : drop :push inst(0): shiftback .head
select case inst$(1)
case "right"
.head++ : if .head>stack.size then data .BS
case "left"
if .head<=1 then push .BS else .head--
else case
cont=false
end select
// change state
curState$=inst$(2)
// Show Stack
if steps=0 or not cont then exit
end while
if steps=0 then print over
Pen 12 {showstack()}
print "tape length: ";stack.size : flush
Refresh
sub showstack()
local d$=format$("{0:-5} {1::-5} ", curState$, .head)
local i: for i=1 to min.data(stack.size, 60): d$+=.symbols#val$(stackitem(i)):Next
print d$
end sub
}
}
Turing1=Machine()
For Turing1 {
.init "Simple incrementer"
.States "q0", "qf"
.Symbols "B", "1"
.Reset "q0", "qf", "B" // initial state, terminating state, blank symbol
.AddRule "q0", "1", "1", "right", "q0"
.AddRule "q0", "B", "1", "stay", "qf"
.tape "1", "1", "1"
.Run 100, true
}
Turing2=Machine()
For Turing2 {
.init "Three-state busy beaver"
.States "a", "b", "c", "halt"
.Symbols "0", "1"
.Reset "a", "halt", "0"
.AddRule "a", "0", "1", "right", "b", "a", "1", "1", "left", "c"
.AddRule "b", "0", "1", "left", "a", "b", "1", "1", "right", "b"
.AddRule "c", "0", "1", "left", "b", "c", "1", "1", "stay", "halt"
.Run 1000, true
}
For Turing1 {
.init "Sorter"
.States "A","B","C","D","E","X"
.Symbols "a","b","B","*"
.Reset "A", "X", "*"
.AddRule "A", "a", "a", "right", "A", "A", "b", "B", "right", "B"
.AddRule "A", "*", "*", "left", "E", "B", "a", "a", "right", "B"
.AddRule "B", "b", "b", "right", "B", "B", "*", "*", "left", "C"
.AddRule "C", "a", "b", "left", "D", "C", "b", "b", "left", "C"
.AddRule "C", "B", "b", "left", "E", "D", "a", "a", "left", "D"
.AddRule "D", "b", "b", "left", "D", "D", "B", "a", "right", "A"
.AddRule "E", "a", "a", "left", "E", "E", "*", "*", "right", "X"
.tape "b", "a", "b","b","b","a","a"
.Run 100, false
}
Turing1.tape "b","b","b","a","b","a","b","a","a","a","b","b","a"
Turing1.Run 1000, false
Turing3=Machine()
for Turing3 {
.init "5-state, 2-symbol probable Busy Beaver machine from Wikipedia"
.States "A","B","C","D", "E", "H"
.Symbols "0", "1"
.Reset "A", "H", "0"
.AddRule "A", "0", "1", "right", "B", "A", "1", "1", "left", "C"
.AddRule "B", "0", "1", "right", "C", "B", "1", "1", "right", "B"
.AddRule "C", "0", "1", "right", "D", "C", "1", "1", "left", "E"
.AddRule "D", "0", "1", "left", "A", "D", "1", "1", "left", "D"
.AddRule "E", "0", "1", "stay", "H", "E", "1", "0", "left", "A"
profiler
.Run 470, false //000000, false
Print round(timecount/1000,2);"s" // estimated 12.5 hours for 47000000 steps
}
}
CheckIt- Output:
(CUT VERSION)
Universal Turing Machine
------------------------
Simple incrementer
q0 1 111
q0 2 111
q0 3 111
q0 4 111B
qf 4 1111
tape length: 4
Three-state busy beaver
a 1 0
b 2 10
a 1 11
c 1 011
b 1 0111
a 1 01111
b 2 11111
b 3 11111
b 4 11111
b 5 11111
b 6 111110
a 5 111111
c 4 111111
halt 4 111111
tape length: 6
Mathematica/Wolfram Language
The universal machine
Updated to use dynamic definition of a function. Values computed for each input are saved. Functionally equivalent to computing a matrix for a set of inputs.
left = 1; right = -1; stay = 0;
cmp[s_] := ToExpression[StringSplit[s, ","]];
utm[rules_, initial_, head_] :=
Module[{tape = initial, rh = head, n = 1},
Clear[nxt];
nxt[state_, field_] :=
nxt[state, field] = Position[rules, {rules[[state, 5]], field, _, _, _}][[1, 1]];
n = Position[rules, {rules[[n, 1]], BitGet[tape, rh], _, _, _}][[1,1]];
While[rules[[n, 4]] != 0,
If[rules[[n, 3]] != BitGet[tape, rh],
If[rules[[n, 3]] == 1, tape = BitSet[tape, rh],
tape = BitClear[tape, rh]]];
rh = rh + rules[[n, 4]];
If[rh < 0, rh = 0; tape = 2*tape];
n = nxt[n, BitGet[tape, rh]];
]; {tape, rh}
];
];
A print routine and test drivers
printMachine[tape_,pos_]:=(mach=IntegerString[tape,2];
ptr=StringReplace[mach,{"0"-> " ","1"->" "}];
Print[mach];Print[StringInsert[ptr,"^",StringLength[ptr]-pos]];);
simpleIncr={"q0,1,1,right,q0","q0,B,1,stay,qf"};
simpleIncr=Map[cmp,simpleIncr]/.B->0;
fin=utm[simpleIncr,7,2];
printMachine[fin[[1]],fin[[2]]];
busyBeaver3S={
"a,0,1,right,b",
"a,1,1,left,c",
"b,0,1,left,a",
"b,1,1,right,b",
"c,0,1,left,b",
"c,1,1,stay,halt"};
fin=utm[Map[cmp,busyBeaver3S],0,0];
printMachine[fin[[1]],fin[[2]]];
Summary output from the 2 short machines
- Output:
1110 ^ 111111 ^
A machine with 47,176,870 steps
Runs in 4 minutes on an i5 desktop (with the dynamic function definiton). The resulting tape is very long, we'll print the result of treating the value as a binary encoded integer.
probable5S={
"A, 0, 1, right, B",
"A, 1, 1, left, C",
"B, 0, 1, right, C",
"B, 1, 1, right, B",
"C, 0, 1, right, D",
"C, 1, 0, left, E",
"D, 0, 1, left, A",
"D, 1, 1, left, D",
"E, 0, 1, stay, H",
"E, 1, 0, left, A"};
fin=utm[Map[cmp,probable5S],0,0];
]
fin[[1]]//N
3.254757786465838*10^3698
MATLAB
function tape=turing(rules,tape,initial,terminal)
%"rules" is cell array of cell arrays of the following form:
%First element is number representing initial state
%Second element is number representing input from the tape
%Third element is number representing output printed onto the tape
%Fourth element is 'l', 'r', or 's' representing whether to go right,
%left, or stay. Treats any input other than 'l' or 'r' as 's'.
%Final value is state we go to
%0 is always blank symbol
term=0;
ind=1;
while term==0
a=[];
for i=1:numel(rules)
if rules{i}{1}==initial
a=[a i];
end
end
possible=rules(a);
n=numel(possible);
while numel(tape)<ind
tape=[tape, 0]; %#ok<AGROW>
end
for i=1:n
if(tape(ind)==possible{i}{2})
break;
end
end
instruction=possible{i};
tape(ind)=instruction{3};
if instruction{4}=='r'
ind=ind+1;
elseif instruction{4}=='l'
if ind==1
tape=[0,tape]; %#ok<AGROW>
else
ind=ind-1;
end
end
if terminal==instruction{5}
term=1;
else
initial=instruction{5};
end
end
end
- Output:
function Rulesets()
C={{'q0',1,1,'r','q0'};
{'q0',0,1,'s','qf'}};
turing(C,[1,1,1],'q0','qf')
C={{'a',0,1,'r','b'};
{'a',1,1,'l','c'};
{'b',0,1,'l','a'};
{'b',1,1,'r','b'};
{'c',0,1,'l','b'};
{'c',1,1,'s','halt'}};
turing(C,[0],'a','halt')
C={{'A', 0, 1, 'r', 'B'};
{'A', 1, 1, 'l', 'C'};
{'B', 0, 1, 'r', 'C'};
{'B', 1, 1, 'r', 'B'};
{'C', 0, 1, 'r', 'D'};
{'C', 1, 0, 'l', 'E'};
{'D', 0, 1, 'l', 'A'};
{'D', 1, 1, 'l', 'D'};
{'E', 0, 1, 's', 'H'};
{'E', 1, 0, 'l', 'A'}};
a=(turing(C,[0],'A','H'));
numel(a)
sum(a)
end
>> Rulesets
ans =
1 1 1 1
ans =
1 1 1 1 1 1
ans =
12289
ans =
4098
%Therefore, the last one outputs a tape of length 12289 with 4098 ones, which we expect
Mercury
The universal machine
Source for this example was lightly adapted from https://bitbucket.org/ttmrichter/turing. Of particular interest in this implementation is that because of the type parameterisation of the config type, the machine being simulated cannot be compiled if there is any mistake in the states, symbols and actions. Also, because of Mercury's determinism detection and enforcement, it's impossible to pass in a non-deterministic set of rules. At most one answer can come back from the rules interface.
:- module turing.
:- interface.
:- import_module list.
:- import_module set.
:- type config(State, Symbol)
---> config(initial_state :: State,
halting_states :: set(State),
blank :: Symbol ).
:- type action ---> left ; stay ; right.
:- func turing(config(State, Symbol),
pred(State, Symbol, Symbol, action, State),
list(Symbol)) = list(Symbol).
:- mode turing(in,
pred(in, in, out, out, out) is semidet,
in) = out is det.
:- implementation.
:- import_module pair.
:- import_module require.
turing(Config@config(Start, _, _), Rules, Input) = Output :-
(Left-Right) = perform(Config, Rules, Start, ([]-Input)),
Output = append(reverse(Left), Right).
:- func perform(config(State, Symbol),
pred(State, Symbol, Symbol, action, State),
State, pair(list(Symbol))) = pair(list(Symbol)).
:- mode perform(in, pred(in, in, out, out, out) is semidet,
in, in) = out is det.
perform(Config@config(_, Halts, Blank), Rules, State,
Input@(LeftInput-RightInput)) = Output :-
symbol(RightInput, Blank, RightNew, Symbol),
( set.member(State, Halts) ->
Output = Input
; Rules(State, Symbol, NewSymbol, Action, NewState) ->
NewLeft = pair(LeftInput, [NewSymbol|RightNew]),
NewRight = action(Action, Blank, NewLeft),
Output = perform(Config, Rules, NewState, NewRight)
;
error("an impossible state has apparently become possible") ).
:- pred symbol(list(Symbol), Symbol, list(Symbol), Symbol).
:- mode symbol(in, in, out, out) is det.
symbol([], Blank, [], Blank).
symbol([Sym|Rem], _, Rem, Sym).
:- func action(action, State, pair(list(State))) = pair(list(State)).
action(left, Blank, ([]-Right)) = ([]-[Blank|Right]).
action(left, _, ([Left|Lefts]-Rights)) = (Lefts-[Left|Rights]).
action(stay, _, Tape) = Tape.
action(right, Blank, (Left-[])) = ([Blank|Left]-[]).
action(right, _, (Left-[Right|Rights])) = ([Right|Left]-Rights).The incrementer machine
This machine has been stripped of the Mercury ceremony around modules, imports, etc.
:- type incrementer_states ---> a ; halt.
:- type incrementer_symbols ---> b ; '1'.
:- func incrementer_config = config(incrementer_states, incrementer_symbols).
incrementer_config = config(a, % the initial state
set([halt]), % the set of halting states
b). % the blank symbol
:- pred incrementer(incrementer_states::in,
incrementer_symbols::in,
incrementer_symbols::out,
action::out,
incrementer_states::out) is semidet.
incrementer(a, '1', '1', right, a).
incrementer(a, b, '1', stay, halt).
TapeOut = turing(incrementer_config, incrementer, [1, 1, 1]).This will, on execution, fill TapeOut with [1, 1, 1, 1].
The busy beaver machine
This machine has been stripped of the Mercury ceremony around modules, imports, etc.
:- type busy_beaver_states ---> a ; b ; c ; halt.
:- type busy_beaver_symbols ---> '0' ; '1'.
:- func busy_beaver_config = config(busy_beaver_states, busy_beaver_symbols).
busy_beaver_config = config(a, % initial state
set([halt]), % set of terminating states
'0'). % blank symbol
:- pred busy_beaver(busy_beaver_states::in,
busy_beaver_symbols::in,
busy_beaver_symbols::out,
action::out,
busy_beaver_states::out) is semidet.
busy_beaver(a, '0', '1', right, b).
busy_beaver(a, '1', '1', left, c).
busy_beaver(b, '0', '1', left, a).
busy_beaver(b, '1', '1', right, b).
busy_beaver(c, '0', '1', left, b).
busy_beaver(c, '1', '1', stay, halt).
TapeOut = turing(busy_beaver_config, busy_beaver, []).This will, on execution, fill TapeOut with [1, 1, 1, 1, 1, 1].
NetLogo
The following is the Code section of the NetLogo file UTMachine_RSdan3dewey.nlogo which can be
downloaded from the page:
http://sites.google.com/site/dan3deweyscspaimsportfolio/extra-turing-machine
This page also has other information, screen shots, etc.
;; "A Turing Turtle": a Turing Machine implemented in NetLogo
;; by Dan Dewey 1/16/2016
;;
;; This NetLogo code implements a Turing Machine, see, e.g.,
;; http://en.wikipedia.org/wiki/Turing_machine
;; The Turing machine fits nicely into the NetLogo paradigm in which
;; there are agents (aka the turtles), that move around
;; in a world of "patches" (2D cells).
;; Here, a single agent represents the Turing machine read/write head
;; and the patches represent the Turing tape values via their colors.
;; The 2D array of patches is treated as a single long 1D tape in an
;; obvious way.
;; This program is presented as a NetLogo example on the page:
;; http://rosettacode.org/wiki/Universal_Turing_machine
;; This file may be larger than others on that page, note however
;; that I include many comments in the code and I have made no
;; effort to 'condense' the code, prefering clarity over compactness.
;; A demo and discussion of this program is on the web page:
;; http://sites.google.com/site/dan3deweyscspaimsportfolio/extra-turing-machine
;; The Copy example machine was taken from:
;; http://en.wikipedia.org/wiki/Turing_machine_examples
;; The "Busy Beaver" machines encoded below were taken from:
;; http://www.logique.jussieu.fr/~michel/ha.html
;; The implementation here allows 3 symbols (blank, 0, 1) on the tape
;; and 3 head motions (left, stay, right).
;; The 2D world is nominally set to be 29x29, going from (-14,-14) to
;; (14,14) from lower left to upper right and with (0,0) at the center.
;; This gives a total Turing tape length of 29^2 = 841 cells, sufficient for the
;; "Lazy" Beaver 5,2 example.
;; Since the max-pxcor variable is used in the code below (as opposed to
;; a hard-coded number), the effective tape size can be changed by
;; changing the size of the 2D world with the Settings... button on the interface.
;; The "Info" tab of the NetLogo interface contains some further comments.
;; - - - - - - -
;; - - - - - - - - - - - Global/Agent variables
;; These three 2D arrays (lists of lists) encode the Turing Machine rules:
;; WhatToWrite: -1 (Blank), 0, 1
;; HowToMove: -1 (left), 0(stay), 1 (right)
;; NextState: 0 to N-1, negative value goes to a halt state.
;; The above are a function of the current state and the current tape (patch) value.
;; MachineState is used by the turtle to pass the current state of the Turing machine
;; (or the halt code) to the observer.
globals [ WhatToWrite HowToMove NextState MachineState
;; some other golobals of secondary importance...
;; set different patch colors to record the Turing tape values
BlankColor ZeroColor OneColor
;; a delay constant to slow down the operation
RealTimePerTick ]
;; We'll have one turtle which is the Turing machine read/write head
;; it will keep track of the current Turing state in its own MyState value
turtles-own [ MyState ]
;; - - - - - - - - - - -
to Setup ;; sets up the world
clear-all ;; clears the world first
;; Try to not have (too many) ad hoc numbers in the code,
;; collect and set various values here especially if they might be used in multiple places:
;; The colors for Blank, Zero and One : (user can can change as desired)
set BlankColor 2 ;; dark gray
set OneColor green
set ZeroColor red
;; slow it down for the humans to watch
set RealTimePerTick 0.2 ;; have simulation go at nice realtime speed
create-turtles 1 ;; create the one Turing turtle
[ ;; set default parameters
set size 2 ;; set a nominal size
set color yellow ;; color of border
;; set the starting location, some Turing programs will adjust this if needed:
setxy 0 0 ;; -1 * max-pxcor -1 * max-pxcor
set shape "square2empty" ;; edited version of "square 2" to have clear in middle
;; set the starting state - always 0
set MyState 0
set MachineState 0 ;; the turtle will update this global value from now on
]
;; Define the Turing machine rules with 2D lists.
;; Based on the selection made on interface panel, setting the string Turing_Program_Selection.
;; This routine has all the Turing 'programs' in it - it's at the very bottom of this file.
LoadTuringProgram
;; the environment, e.g. the Turing tape
ask patches
[
;; all patches are set to the blank color
set pcolor BlankColor
]
;; keep track of time; each tick is a Turing step
reset-ticks
end
;; - - - - - - - - - - - - - - - -
to Go ;; this repeatedly does steps
;; The turtle does the main work
ask turtles
[
DoOneStep
wait RealTimePerTick
]
tick
;; The Turing turtle will die if it tries to go beyond the cells,
;; in that case (no turtles left) we'll stop.
;; Also stop if the MachineState has been set to a negative number (a halt state).
if ((count turtles = 0) or (MachineState < 0))
[ stop ]
end
to DoOneStep
;; have the turtle do one Turing step
;; First, 'read the tape', i.e., based on the patch color here:
let tapeValue GetTapeValue
;; using the tapeValue and MyState, get the desired actions here:
;; (the item commands extract the appropriate value from the list-of-lists)
let myWrite item (tapeValue + 1) (item MyState WhatToWrite)
let myMove item (tapeValue + 1) (item MyState HowToMove)
let myNextState item (tapeValue + 1) (item MyState NextState)
;; Write to the tape as appropriate
SetTapeValue myWrite
;; Move as appropriate
if (myMove = 1) [MoveForward]
if (myMove = -1) [MoveBackward]
;; Go to the next state; check if it is a halt state.
;; Update the global MachineState value
set MachineState myNextState
ifelse (myNextState < 0)
[
;; It's a halt state. The negative MachineState will signal the stop.
;; Go back to the starting state so it can be re-run if desired.
set MyState 0]
[
;; Not a halt state, so change to the desired next state
set MyState myNextState
]
end
to MoveForward
;; move the turtle forward one cell, including line wrapping.
set heading 90
ifelse (xcor = max-pxcor)
[set xcor -1 * max-pxcor
;; and go up a row if possible... otherwise die
ifelse ycor = max-pxcor
[ die ] ;; tape too short - a somewhat crude end of things ;-)
[set ycor ycor + 1]
]
[jump 1]
end
to MoveBackward
;; move the turtle backward one cell, including line-wrapping.
set heading -90
ifelse (xcor = -1 * max-pxcor)
[
set xcor max-pxcor
;; and go down a row... or die
ifelse ycor = -1 * max-pxcor
[ die ] ;; tape too short - a somewhat crude end of things ;-)
[set ycor ycor - 1]
]
[jump 1]
end
to-report GetTapeValue
;; report the tape color equivalent value
if (pcolor = ZeroColor) [report 0]
if (pcolor = OneColor) [report 1]
report -1
end
to SetTapeValue [ value ]
;; write the appropriate color on the tape
ifelse (value = 1)
[set pcolor OneColor]
[ ifelse (value = 0)
[set pcolor ZeroColor][set pcolor BlankColor]]
end
;; - - - - - OK, here are the data for the various Turing programs...
;; Note that besdes settting the rules (array values) these sections can also
;; include commands to clear the tape, position the r/w head, adjust wait time, etc.
to LoadTuringProgram
;; A template of the rules structure: a list of lists
;; E.g. values are given for States 0 to 4, when looking at Blank, Zero, One:
;; For 2-symbol machines use Blank(-1) and One(1) and ignore the middle values (never see zero).
;; Normal Halt will be state -1, the -9 default shows an unexpected halt.
;; state 0 state 1 state 2 state 3 state 4
set WhatToWrite (list (list -1 0 1) (list -1 0 1) (list -1 0 1) (list -1 0 1) (list -1 0 1) )
set HowToMove (list (list 0 0 0) (list 0 0 0) (list 0 0 0) (list 0 0 0) (list 0 0 0) )
set NextState(list (list -9 -9 -9) (list -9 -9 -9) (list -9 -9 -9) (list -9 -9 -9) (list -9 -9 -9) )
;; Fill the rules based on the selected case
if (Turing_Program_Selection = "Simple Incrementor")
[
;; simple Incrementor - this is from the RosettaCode Universal Turing Machine page - very simple!
set WhatToWrite (list (list 1 0 1) )
set HowToMove (list (list 0 0 1) )
set NextState (list (list -1 -9 0) )
]
;; Fill the rules based on the selected case
if (Turing_Program_Selection = "Incrementor w/Return")
[
;; modified Incrementor: it returns to the first 1 on the left.
;; This version allows the "Copy Ones to right" program to directly follow it.
;; move right append one back to beginning
set WhatToWrite (list (list -1 0 1) (list 1 0 1) (list -1 0 1) )
set HowToMove (list (list 1 0 1) (list 0 0 1) (list 1 0 -1) )
set NextState (list (list 1 -9 1) (list 2 -9 1) (list -1 -9 2) )
]
;; Fill the rules based on the selected case
if (Turing_Program_Selection = "Copy Ones to right")
[
;; "Copy" from Wiki "Turing machine examples" page; slight mod so that it ends on first 1
;; of the copy allowing Copy to be re-executed to create another copy.
;; Has 5 states and uses Blank and 1 to make a copy of a string of ones;
;; this can be run after runs of the "Incrementor w/Return".
;; state 0 state 1 state 2 state 3 state 4
set WhatToWrite (list (list -1 0 -1) (list -1 0 1) (list 1 0 1) (list -1 0 1) (list 1 0 1) )
set HowToMove (list (list 1 0 1) (list 1 0 1) (list -1 0 1) (list -1 0 -1) (list 1 0 -1) )
set NextState (list (list -1 -9 1) (list 2 -9 1) (list 3 -9 2) (list 4 -9 3) (list 0 -9 4) )
]
;; Fill the rules based on the selected case
if (Turing_Program_Selection = "Binary Counter")
[
;; Count in binary - can start on a blank space.
;; States: start carry-1 back-to-beginning
set WhatToWrite (list (list 1 1 0) (list 1 1 0) (list -1 0 1) )
set HowToMove (list (list 0 0 -1) (list 0 0 -1) (list -1 1 1) )
set NextState (list (list -1 -1 1) (list 2 2 1) (list -1 2 2) )
;; Select line above from these two:
;; can either count by 1 each time it is run:
;; set NextState (list (list -1 -1 1) (list 2 2 1) (list -1 2 2) )
;; or count forever once started:
;; set NextState (list (list 0 0 1) (list 2 2 1) (list 0 2 2) )
set RealTimePerTick 0.2
]
if (Turing_Program_Selection = "Busy-Beaver 3-State, 2-Sym")
[
;; from the RosettaCode.org Universal Turing Machine page
;; state name: a b c
set WhatToWrite (list (list 1 0 1) (list 1 0 1) (list 1 0 1) (list -1 0 1) (list -1 0 1) )
set HowToMove (list (list 1 0 -1) (list -1 0 1) (list -1 0 0) (list 0 0 0) (list 0 0 0) )
set NextState (list (list 1 -9 2) (list 0 -9 1) (list 1 -9 -1) (list -9 -9 -9) (list -9 -9 -9) )
;; Clear the tape
ask Patches [set pcolor BlankColor]
]
;; should output 13 ones and take 107 steps to do it...
if (Turing_Program_Selection = "Busy-Beaver 4-State, 2-Sym")
[
;; from the RosettaCode.org Universal Turing Machine page
;; state name: A B C D
set WhatToWrite (list (list 1 0 1) (list 1 0 -1) (list 1 0 1) (list 1 0 -1) (list -1 0 1) )
set HowToMove (list (list 1 0 -1) (list -1 0 -1) (list 1 0 -1) (list 1 0 1) (list 0 0 0) )
set NextState (list (list 1 -9 1) (list 0 -9 2) (list -1 -9 3) (list 3 -9 0) (list -9 -9 -9) )
;; Clear the tape
ask Patches [set pcolor BlankColor]
]
;; This takes 38 steps to write 9 ones/zeroes
if (Turing_Program_Selection = "Busy-Beaver 2-State, 3-Sym")
[
;; A B
set WhatToWrite (list (list 0 1 0) (list 1 1 0) (list -1 0 1) (list -1 0 1) (list -1 0 1) )
set HowToMove (list (list 1 -1 1) (list -1 1 -1) (list 0 0 0) (list 0 0 0) (list 0 0 0) )
set NextState(list (list 1 1 -1) (list 0 1 1) (list -9 -9 -9) (list -9 -9 -9) (list -9 -9 -9) )
;; Clear the tape
ask Patches [set pcolor BlankColor]
]
;; This only makes 501 ones and stops after 134,467 steps -- it does do that !!!
if (Turing_Program_Selection = "Lazy-Beaver 5-State, 2-Sym")
[
;; from the RosettaCode.org Universal Turing Machine page
;; state name: A0 B1 C2 D3 E4
set WhatToWrite (list (list 1 0 -1) (list 1 0 1) (list 1 0 -1) (list -1 0 1) (list 1 0 1) )
set HowToMove (list (list 1 0 -1) (list 1 0 1) (list -1 0 1) (list 1 0 1) (list -1 0 1) )
set NextState (list (list 1 -9 2) (list 2 -9 3) (list 0 -9 1) (list 4 -9 -1) (list 2 -9 0) )
;; Clear the tape
ask Patches [set pcolor BlankColor]
;; Looks like it goes much more forward than back on the tape
;; so start the head just a row from the bottom:
ask turtles [setxy 0 -1 * max-pxcor + 1]
;; and go faster
set RealTimePerTick 0.02
]
;; The rest have large outputs and run for a long time, so I haven't confirmed
;; that they work as advertised...
;; This is the 5,2 record holder: 4098 ones in 47,176,870 steps.
;; With max-pxcor of 14 and offset r/w head start (below), this will
;; run off the tape at about 150,000+steps...
if (Turing_Program_Selection = "Busy-Beaver 5-State, 2-Sym")
[
;; from the RosettaCode.org Universal Turing Machine page
;; state name: A B C D E
set WhatToWrite (list (list 1 0 1) (list 1 0 1) (list 1 0 -1) (list 1 0 1) (list 1 0 -1) )
set HowToMove (list (list 1 0 -1) (list 1 0 1) (list 1 0 -1) (list -1 0 -1) (list 1 0 -1) )
set NextState (list (list 1 -9 2) (list 2 -9 1) (list 3 -9 4) (list 0 -9 3) (list -1 -9 0) )
;; Clear the tape
ask Patches [set pcolor BlankColor]
;; Writes more backward than forward, so start a few rows from the top:
ask turtles [setxy 0 max-pxcor - 3]
;; and go faster
set RealTimePerTick 0.02
]
if (Turing_Program_Selection = "Lazy-Beaver 3-State, 3-Sym")
[
;; This should write 5600 ones/zeros and take 29,403,894 steps.
;; Ran it to 175,000+ steps and only covered 1/2 of the cells (w/max-pxcor = 14)...
;; state name: A B C
set WhatToWrite (list (list 0 1 0) (list 1 -1 0) (list 0 1 0) (list -1 0 1) (list -1 0 1) )
set HowToMove (list (list 1 1 -1) (list -1 1 1) (list 1 -1 1) (list 0 0 0) (list 0 0 0) )
set NextState (list (list 1 0 0) (list 2 2 1) (list -1 0 1) (list -9 -9 -9) (list -9 -9 -9) )
;; Clear the tape
ask Patches [set pcolor BlankColor]
;; It goes much more forward than back on the tape
;; so start the head just a row from the bottom:
ask turtles [setxy 0 -1 * max-pxcor + 1]
;; and go faster
set RealTimePerTick 0.02
]
if (Turing_Program_Selection = "Busy-Beaver 3-State, 3-Sym")
[
;; This should write 374,676,383 ones/zeros and take 119,112,334,170,342,540 (!!!) steps.
;; Rn it to ~ 175,000 steps covering about 2/3 of the max-pxcor=14 cells.
;; state name: A B C
set WhatToWrite (list (list 0 1 0) (list -1 1 0) (list 0 0 0) (list -1 0 1) (list -1 0 1) )
set HowToMove (list (list 1 -1 -1) (list -1 1 -1) (list 1 1 1) (list 0 0 0) (list 0 0 0) )
set NextState (list (list 1 0 2) (list 0 1 1) (list -1 0 2) (list -9 -9 -9) (list -9 -9 -9) )
;; Clear the tape
ask Patches [set pcolor BlankColor]
;; Writes more backward than forward, so start a rowish from the top:
ask turtles [setxy 0 max-pxcor - 1]
;; and go faster
set RealTimePerTick 0.02
]
;; in all cases reset the machine state to 0:
ask turtles [set MyState 0]
set MachineState 0
;; and the ticks
reset-ticks
endNim
import strutils, tables
proc runUTM(state, halt, blank: string, tape: seq[string] = @[],
rules: seq[seq[string]]) =
var
st = state
pos = 0
tape = tape
rulesTable: Table[tuple[s0, v0: string], tuple[v1, dr, s1: string]]
if tape.len == 0: tape = @[blank]
if pos < 0: pos += tape.len
assert pos in 0..tape.high
for r in rules:
assert r.len == 5
rulesTable[(r[0], r[1])] = (r[2], r[3], r[4])
while true:
stdout.write st, '\t'
for i, v in tape:
stdout.write if i == pos: '[' & v & ']' else: ' ' & v & ' '
echo()
if st == halt: break
if not rulesTable.hasKey((st, tape[pos])): break
let (v1, dr, s1) = rulesTable[(st, tape[pos])]
tape[pos] = v1
if dr == "left":
if pos > 0: dec pos
else: tape.insert blank
if dr == "right":
inc pos
if pos >= tape.len: tape.add blank
st = s1
echo "incr machine\n"
runUTM(halt = "qf",
state = "q0",
tape = "1 1 1".split,
blank = "B",
rules = @["q0 1 1 right q0".splitWhitespace,
"q0 B 1 stay qf".splitWhitespace])
echo "\nbusy beaver\n"
runUTM(halt = "halt",
state = "a",
blank = "0",
rules = @["a 0 1 right b".splitWhitespace,
"a 1 1 left c".splitWhitespace,
"b 0 1 left a".splitWhitespace,
"b 1 1 right b".splitWhitespace,
"c 0 1 left b".splitWhitespace,
"c 1 1 stay halt".splitWhitespace])
echo "\nsorting test\n"
runUTM(halt = "STOP",
state = "A",
blank = "0",
tape = "2 2 2 1 2 2 1 2 1 2 1 2 1 2".split,
rules = @["A 1 1 right A".splitWhitespace,
"A 2 3 right B".splitWhitespace,
"A 0 0 left E".splitWhitespace,
"B 1 1 right B".splitWhitespace,
"B 2 2 right B".splitWhitespace,
"B 0 0 left C".splitWhitespace,
"C 1 2 left D".splitWhitespace,
"C 2 2 left C".splitWhitespace,
"C 3 2 left E".splitWhitespace,
"D 1 1 left D".splitWhitespace,
"D 2 2 left D".splitWhitespace,
"D 3 1 right A".splitWhitespace,
"E 1 1 left E".splitWhitespace,
"E 0 0 right STOP".splitWhitespace])
- Output:
incr machine q0 [1] 1 1 q0 1 [1] 1 q0 1 1 [1] q0 1 1 1 [B] qf 1 1 1 [1] busy beaver a [0] b 1 [0] a [1] 1 c [0] 1 1 b [0] 1 1 1 a [0] 1 1 1 1 b 1 [1] 1 1 1 b 1 1 [1] 1 1 b 1 1 1 [1] 1 b 1 1 1 1 [1] b 1 1 1 1 1 [0] a 1 1 1 1 [1] 1 c 1 1 1 [1] 1 1 halt 1 1 1 [1] 1 1 sorting test A [2] 2 2 1 2 2 1 2 1 2 1 2 1 2 B 3 [2] 2 1 2 2 1 2 1 2 1 2 1 2 B 3 2 [2] 1 2 2 1 2 1 2 1 2 1 2 B 3 2 2 [1] 2 2 1 2 1 2 1 2 1 2 B 3 2 2 1 [2] 2 1 2 1 2 1 2 1 2 B 3 2 2 1 2 [2] 1 2 1 2 1 2 1 2 B 3 2 2 1 2 2 [1] 2 1 2 1 2 1 2 B 3 2 2 1 2 2 1 [2] 1 2 1 2 1 2 B 3 2 2 1 2 2 1 2 [1] 2 1 2 1 2 B 3 2 2 1 2 2 1 2 1 [2] 1 2 1 2 B 3 2 2 1 2 2 1 2 1 2 [1] 2 1 2 B 3 2 2 1 2 2 1 2 1 2 1 [2] 1 2 B 3 2 2 1 2 2 1 2 1 2 1 2 [1] 2 B 3 2 2 1 2 2 1 2 1 2 1 2 1 [2] B 3 2 2 1 2 2 1 2 1 2 1 2 1 2 [0] C 3 2 2 1 2 2 1 2 1 2 1 2 1 [2] 0 C 3 2 2 1 2 2 1 2 1 2 1 2 [1] 2 0 D 3 2 2 1 2 2 1 2 1 2 1 [2] 2 2 0 D 3 2 2 1 2 2 1 2 1 2 [1] 2 2 2 0 D 3 2 2 1 2 2 1 2 1 [2] 1 2 2 2 0 D 3 2 2 1 2 2 1 2 [1] 2 1 2 2 2 0 D 3 2 2 1 2 2 1 [2] 1 2 1 2 2 2 0 D 3 2 2 1 2 2 [1] 2 1 2 1 2 2 2 0 D 3 2 2 1 2 [2] 1 2 1 2 1 2 2 2 0 D 3 2 2 1 [2] 2 1 2 1 2 1 2 2 2 0 D 3 2 2 [1] 2 2 1 2 1 2 1 2 2 2 0 D 3 2 [2] 1 2 2 1 2 1 2 1 2 2 2 0 D 3 [2] 2 1 2 2 1 2 1 2 1 2 2 2 0 D [3] 2 2 1 2 2 1 2 1 2 1 2 2 2 0 A 1 [2] 2 1 2 2 1 2 1 2 1 2 2 2 0 B 1 3 [2] 1 2 2 1 2 1 2 1 2 2 2 0 B 1 3 2 [1] 2 2 1 2 1 2 1 2 2 2 0 B 1 3 2 1 [2] 2 1 2 1 2 1 2 2 2 0 B 1 3 2 1 2 [2] 1 2 1 2 1 2 2 2 0 B 1 3 2 1 2 2 [1] 2 1 2 1 2 2 2 0 B 1 3 2 1 2 2 1 [2] 1 2 1 2 2 2 0 B 1 3 2 1 2 2 1 2 [1] 2 1 2 2 2 0 B 1 3 2 1 2 2 1 2 1 [2] 1 2 2 2 0 B 1 3 2 1 2 2 1 2 1 2 [1] 2 2 2 0 B 1 3 2 1 2 2 1 2 1 2 1 [2] 2 2 0 B 1 3 2 1 2 2 1 2 1 2 1 2 [2] 2 0 B 1 3 2 1 2 2 1 2 1 2 1 2 2 [2] 0 B 1 3 2 1 2 2 1 2 1 2 1 2 2 2 [0] C 1 3 2 1 2 2 1 2 1 2 1 2 2 [2] 0 C 1 3 2 1 2 2 1 2 1 2 1 2 [2] 2 0 C 1 3 2 1 2 2 1 2 1 2 1 [2] 2 2 0 C 1 3 2 1 2 2 1 2 1 2 [1] 2 2 2 0 D 1 3 2 1 2 2 1 2 1 [2] 2 2 2 2 0 D 1 3 2 1 2 2 1 2 [1] 2 2 2 2 2 0 D 1 3 2 1 2 2 1 [2] 1 2 2 2 2 2 0 D 1 3 2 1 2 2 [1] 2 1 2 2 2 2 2 0 D 1 3 2 1 2 [2] 1 2 1 2 2 2 2 2 0 D 1 3 2 1 [2] 2 1 2 1 2 2 2 2 2 0 D 1 3 2 [1] 2 2 1 2 1 2 2 2 2 2 0 D 1 3 [2] 1 2 2 1 2 1 2 2 2 2 2 0 D 1 [3] 2 1 2 2 1 2 1 2 2 2 2 2 0 A 1 1 [2] 1 2 2 1 2 1 2 2 2 2 2 0 B 1 1 3 [1] 2 2 1 2 1 2 2 2 2 2 0 B 1 1 3 1 [2] 2 1 2 1 2 2 2 2 2 0 B 1 1 3 1 2 [2] 1 2 1 2 2 2 2 2 0 B 1 1 3 1 2 2 [1] 2 1 2 2 2 2 2 0 B 1 1 3 1 2 2 1 [2] 1 2 2 2 2 2 0 B 1 1 3 1 2 2 1 2 [1] 2 2 2 2 2 0 B 1 1 3 1 2 2 1 2 1 [2] 2 2 2 2 0 B 1 1 3 1 2 2 1 2 1 2 [2] 2 2 2 0 B 1 1 3 1 2 2 1 2 1 2 2 [2] 2 2 0 B 1 1 3 1 2 2 1 2 1 2 2 2 [2] 2 0 B 1 1 3 1 2 2 1 2 1 2 2 2 2 [2] 0 B 1 1 3 1 2 2 1 2 1 2 2 2 2 2 [0] C 1 1 3 1 2 2 1 2 1 2 2 2 2 [2] 0 C 1 1 3 1 2 2 1 2 1 2 2 2 [2] 2 0 C 1 1 3 1 2 2 1 2 1 2 2 [2] 2 2 0 C 1 1 3 1 2 2 1 2 1 2 [2] 2 2 2 0 C 1 1 3 1 2 2 1 2 1 [2] 2 2 2 2 0 C 1 1 3 1 2 2 1 2 [1] 2 2 2 2 2 0 D 1 1 3 1 2 2 1 [2] 2 2 2 2 2 2 0 D 1 1 3 1 2 2 [1] 2 2 2 2 2 2 2 0 D 1 1 3 1 2 [2] 1 2 2 2 2 2 2 2 0 D 1 1 3 1 [2] 2 1 2 2 2 2 2 2 2 0 D 1 1 3 [1] 2 2 1 2 2 2 2 2 2 2 0 D 1 1 [3] 1 2 2 1 2 2 2 2 2 2 2 0 A 1 1 1 [1] 2 2 1 2 2 2 2 2 2 2 0 A 1 1 1 1 [2] 2 1 2 2 2 2 2 2 2 0 B 1 1 1 1 3 [2] 1 2 2 2 2 2 2 2 0 B 1 1 1 1 3 2 [1] 2 2 2 2 2 2 2 0 B 1 1 1 1 3 2 1 [2] 2 2 2 2 2 2 0 B 1 1 1 1 3 2 1 2 [2] 2 2 2 2 2 0 B 1 1 1 1 3 2 1 2 2 [2] 2 2 2 2 0 B 1 1 1 1 3 2 1 2 2 2 [2] 2 2 2 0 B 1 1 1 1 3 2 1 2 2 2 2 [2] 2 2 0 B 1 1 1 1 3 2 1 2 2 2 2 2 [2] 2 0 B 1 1 1 1 3 2 1 2 2 2 2 2 2 [2] 0 B 1 1 1 1 3 2 1 2 2 2 2 2 2 2 [0] C 1 1 1 1 3 2 1 2 2 2 2 2 2 [2] 0 C 1 1 1 1 3 2 1 2 2 2 2 2 [2] 2 0 C 1 1 1 1 3 2 1 2 2 2 2 [2] 2 2 0 C 1 1 1 1 3 2 1 2 2 2 [2] 2 2 2 0 C 1 1 1 1 3 2 1 2 2 [2] 2 2 2 2 0 C 1 1 1 1 3 2 1 2 [2] 2 2 2 2 2 0 C 1 1 1 1 3 2 1 [2] 2 2 2 2 2 2 0 C 1 1 1 1 3 2 [1] 2 2 2 2 2 2 2 0 D 1 1 1 1 3 [2] 2 2 2 2 2 2 2 2 0 D 1 1 1 1 [3] 2 2 2 2 2 2 2 2 2 0 A 1 1 1 1 1 [2] 2 2 2 2 2 2 2 2 0 B 1 1 1 1 1 3 [2] 2 2 2 2 2 2 2 0 B 1 1 1 1 1 3 2 [2] 2 2 2 2 2 2 0 B 1 1 1 1 1 3 2 2 [2] 2 2 2 2 2 0 B 1 1 1 1 1 3 2 2 2 [2] 2 2 2 2 0 B 1 1 1 1 1 3 2 2 2 2 [2] 2 2 2 0 B 1 1 1 1 1 3 2 2 2 2 2 [2] 2 2 0 B 1 1 1 1 1 3 2 2 2 2 2 2 [2] 2 0 B 1 1 1 1 1 3 2 2 2 2 2 2 2 [2] 0 B 1 1 1 1 1 3 2 2 2 2 2 2 2 2 [0] C 1 1 1 1 1 3 2 2 2 2 2 2 2 [2] 0 C 1 1 1 1 1 3 2 2 2 2 2 2 [2] 2 0 C 1 1 1 1 1 3 2 2 2 2 2 [2] 2 2 0 C 1 1 1 1 1 3 2 2 2 2 [2] 2 2 2 0 C 1 1 1 1 1 3 2 2 2 [2] 2 2 2 2 0 C 1 1 1 1 1 3 2 2 [2] 2 2 2 2 2 0 C 1 1 1 1 1 3 2 [2] 2 2 2 2 2 2 0 C 1 1 1 1 1 3 [2] 2 2 2 2 2 2 2 0 C 1 1 1 1 1 [3] 2 2 2 2 2 2 2 2 0 E 1 1 1 1 [1] 2 2 2 2 2 2 2 2 2 0 E 1 1 1 [1] 1 2 2 2 2 2 2 2 2 2 0 E 1 1 [1] 1 1 2 2 2 2 2 2 2 2 2 0 E 1 [1] 1 1 1 2 2 2 2 2 2 2 2 2 0 E [1] 1 1 1 1 2 2 2 2 2 2 2 2 2 0 E [0] 1 1 1 1 1 2 2 2 2 2 2 2 2 2 0 STOP 0 [1] 1 1 1 1 2 2 2 2 2 2 2 2 2 0
Pascal
The code uses a stringlist to store the rules. It also uses a stringlist to store the output. It should straightforward to follow.
The 47,000,000 step busy beaver executes in 40 secs on my computer. For greater speed, it would be better to use a fixed size array to store the output (rather than a stringlist) when simulating Turing machines programs which have a large number of steps.
Simple Incrementer
program project1;
uses
Classes, sysutils;
type
TCurrent = record
state : string;
input : char;
end;
TMovesTo = record
state : string;
output : char;
moves : char;
end;
var
ST, ET: TDateTime;
C:TCurrent;
M:TMovesTo;
Tape, Rules:TStringList;
TP:integer; //TP = Tape position
Blank : char;
j:integer;
FinalState, InitialState, R : string;
Count:integer;
Function ApplyRule(C:TCurrent):TMovesTo;
var
x,k:integer;
begin
//Find the appropriate rule and pass it as the result
For k:= 0 to Rules.Count-1 do
begin
If (k mod 5 = 0) and (Rules[k] = C.state) and (Rules[k+1] = C.input) then
begin
Result.output := Rules[k+2][1];
Result.moves := Rules[k+3][1];
Result.state := Rules[k+4];
end;
end;
end;
Procedure ChangeTape(var TapePosition:integer;N:TMovesTo);
begin
Tape[TapePosition]:=N.output;
Case N.moves of
'l':begin
TapePosition := TapePosition-1;
end;
'r':begin
TapePosition := TapePosition+1;
end;
end;
end;
function GetInput(TapePosition:integer):char;
begin
Result:=Tape[TapePosition][1];
end;
procedure ShowResult;
var
k:integer;
begin
writeln('Current State :',C.state);
writeln('Input :',C.input);
write(' Tape ');
For k:= 0 to Tape.count-1 do
begin
write(Tape[k]);
end;
writeln;
writeln('New State :',M.state);
writeln('Tape position :',TP);
writeln('-----------------------');
end;
begin
writeln('Universal Turing Machine');
writeln('------------------------');
Count:=0;
ST:=Time;
//Set up the rules
Rules := TStringList.create;
R := 'q0,1,1,right,q0,q0,B,1,stay,qf';
Rules.CommaText := R;
InitialState := 'q0';
FinalState := 'qf';
//Set up the tape
Tape:=TStringList.create;
Tape.add('1');
Tape.add('1');
Tape.add('1');
Blank := 'B';
For j:= 1 to 10 do
begin
Tape.add(Blank);
end;
//Set up the initial state
writeln('Initial state');
C.state:=InitialState;
C.input:=' ';
M.state:='';
M.output:=' ';
M.moves:=' ';
TP:=0;
//Run the machine
While (TP >= 0) and (M.State <> FinalState) do
begin
C.Input := GetInput(TP);
If M.state <> '' then
begin
C.State := M.State;
end;
M:=ApplyRule(C);
ChangeTape(TP,M);
Count:=Count+1;
ShowResult;
end;
//State the outcome.
If TP < 0 then
begin
writeln('Error! Tape has slipped off at left!');
end;
If M.state = FinalState then
begin
writeln('Program has finished');
ET:=Time;
writeln('Time taken ');
writeln(FormatDateTime('sss:zzz',ET-ST));
writeln(Count, ' steps taken');
end;
Tape.free;
Rules.free;
readln;
end.
Output
Universal Turing Machine ------------------------ Initial state Current State :q0 Input :1 Tape 111BBBBBBBBBB New State :q0 Tape position :1 ----------------------- Current State :q0 Input :1 Tape 111BBBBBBBBBB New State :q0 Tape position :2 ----------------------- Current State :q0 Input :1 Tape 111BBBBBBBBBB New State :q0 Tape position :3 ----------------------- Current State :q0 Input :B Tape 1111BBBBBBBBB New State :qf Tape position :3 ----------------------- Program has finished Time taken 00:000 4 steps taken
3 state Busy Beaver
program project1;
uses
Classes, sysutils;
type
TCurrent = record
state : string;
input : char;
end;
TMovesTo = record
state : string;
output : char;
moves : char;
end;
var
ST, ET: TDateTime;
C:TCurrent;
M:TMovesTo;
Tape, Rules:TStringList;
TP:integer; //TP = Tape position
Blank : char;
j:integer;
FinalState, InitialState, R : string;
Count:integer;
Function ApplyRule(C:TCurrent):TMovesTo;
var
x,k:integer;
begin
//Find the appropriate rule and pass it as the result
For k:= 0 to Rules.Count-1 do
begin
If (k mod 5 = 0) and (Rules[k] = C.state) and (Rules[k+1] = C.input) then
begin
Result.output := Rules[k+2][1];
Result.moves := Rules[k+3][1];
Result.state := Rules[k+4];
end;
end;
end;
Procedure ChangeTape(var TapePosition:integer;N:TMovesTo);
begin
Tape[TapePosition]:=N.output;
Case N.moves of
'l':begin
TapePosition := TapePosition-1;
end;
'r':begin
TapePosition := TapePosition+1;
end;
end;
end;
function GetInput(TapePosition:integer):char;
begin
Result:=Tape[TapePosition][1];
end;
procedure ShowResult;
var
k:integer;
begin
writeln('Current State :',C.state);
writeln('Input :',C.input);
write(' Tape ');
For k:= 0 to Tape.count-1 do
begin
write(Tape[k]);
end;
writeln;
writeln('New State :',M.state);
writeln('Tape position :',TP);
writeln('-----------------------');
end;
begin
writeln('Universal Turing Machine');
writeln('------------------------');
Count:=0;
ST:=Time;
//Set up the rules
Rules := TStringList.create;
R:= 'a,0,1,right,b,a,1,1,left,c,b,0,1,left,a,b,1,1,right,b,c,0,1,left,b,c,1,1,stay,halt';
Rules.CommaText := R;
//Set up the state names
InitialState := 'a';
FinalState := 'halt';
//Set up the tape
Blank := '0';
Tape:=TStringList.create;
For j:= 1 to 20 do
begin
Tape.add(Blank);
end;
//Set up the initial state
writeln('Initial state');
C.state:=InitialState;
C.input:=' ';
M.state:='';
M.output:=' ';
M.moves:=' ';
TP:=10;
//Run the machine
While (TP >= 0) and (M.State <> FinalState) do
begin
C.Input := GetInput(TP);
If M.state <> '' then
begin
C.State := M.State;
end;
M:=ApplyRule(C);
ChangeTape(TP,M);
Count:=Count+1;
ShowResult;
end;
//State the outcome.
If TP < 0 then
begin
writeln('Error! Tape has slipped off at left!');
end;
If M.state = FinalState then
begin
writeln('Program has finished');
ET:=Time;
writeln('Time taken ');
writeln(FormatDateTime('sss:zzz',ET-ST));
writeln(Count, ' steps taken');
end;
Tape.free;
Rules.free;
readln;
end.
output
Universal Turing Machine ------------------------ Initial state Current State :a Input :0 Tape 00000000001000000000 New State :b Tape position :11 ----------------------- Current State :b Input :0 Tape 00000000001100000000 New State :a Tape position :10 ----------------------- Current State :a Input :1 Tape 00000000001100000000 New State :c Tape position :9 ----------------------- Current State :c Input :0 Tape 00000000011100000000 New State :b Tape position :8 ----------------------- Current State :b Input :0 Tape 00000000111100000000 New State :a Tape position :7 ----------------------- Current State :a Input :0 Tape 00000001111100000000 New State :b Tape position :8 ----------------------- Current State :b Input :1 Tape 00000001111100000000 New State :b Tape position :9 ----------------------- Current State :b Input :1 Tape 00000001111100000000 New State :b Tape position :10 ----------------------- Current State :b Input :1 Tape 00000001111100000000 New State :b Tape position :11 ----------------------- Current State :b Input :1 Tape 00000001111100000000 New State :b Tape position :12 ----------------------- Current State :b Input :0 Tape 00000001111110000000 New State :a Tape position :11 ----------------------- Current State :a Input :1 Tape 00000001111110000000 New State :c Tape position :10 ----------------------- Current State :c Input :1 Tape 00000001111110000000 New State :halt Tape position :10 ----------------------- Program has finished Time taken 00:001 13 steps taken
5 state 2 symbol
//Busy Beaver 5 state 2-symbol
program project1;
uses
Classes, sysutils;
type
TCurrent = record
state : string;
input : char;
end;
TMovesTo = record
state : string;
output : char;
moves : char;
end;
var
C:TCurrent;
M:TMovesTo;
Tape, Rules : TStringList;
TP:integer; // TP = Tape position
Blank : char;
FinalState , InitialState, R: string;
ST, ET: TDateTime;
j, Count:integer;
Function ApplyRule(C:TCurrent):TMovesTo;
var
x,k:integer;
begin
//Find the appropriate rule (given the state and input) and get the result
For k:= 0 to Rules.Count-1 do
begin
If (k mod 5 = 0) and (Rules[k] = C.state) and (Rules[k+1] = C.input) then
begin
with result do
begin
output := Rules[k+2][1];
moves := Rules[k+3][1];
state := Rules[k+4];
end;
end;
end;
end;
Procedure ChangeTape(var TapePosition:integer; N:TMovesTo);
begin
Tape[TapePosition]:=N.output;
Case N.moves of
'l':begin
TapePosition := TapePosition-1;
end;
'r':begin
TapePosition := TapePosition+1;
end;
end;
end;
function GetInput(TapePosition:integer):char;
begin
Result:=Tape[TapePosition][1];
end;
procedure ShowResult;
var
k:integer;
begin
writeln('Current State :',C.state);
writeln('Input :',C.input);
write(' Tape ');
For k:= 0 to Tape.count-1 do
begin
write(Tape[k]);
end;
writeln;
writeln('New State :',M.state);
writeln('Tape position :',TP);
writeln('-----------------------');
end;
begin
writeln('Universal Turing Machine');
writeln('------------------------');
//Number of steos and time taken
Count:=0;
ST:=Time;
//Set up the rules
Rules := TStringList.create;
R:='A, 0, 1, right, B,A, 1, 1, left, C, B, 0, 1, right, C,B, 1, 1, right, B,C, 0, 1, right, D,C, 1, 0, left, E,D, 0, 1, left, A,D, 1, 1, left, D,E, 0, 1, stay, H,E, 1, 0, left, A';
Rules.CommaText := R;
InitialState := 'A';
FinalState := 'H';
Blank := '0';
//Set up the blank tape
Tape:=TStringList.create;
For j:= 1 to 15000 do
begin
Tape.add(Blank);
end;
//Set up the initial state
//I discovered by trial and error a suitable start point (TP)
writeln('Initial state');
C.state:=InitialState;
C.input:=' ';
M.state:='';
M.output:=' ';
M.moves:=' ';
TP:=14000;
//Run the machine
While (TP >= 0) and (M.State <> FinalState) do
begin
C.Input := GetInput(TP);
If M.state <> '' then
begin
C.State := M.State;
end;
M := ApplyRule(C);
ChangeTape(TP,M);
Count:=Count+1;
end;
//State the outcome.
If TP < 0 then
begin
writeln('Error! Tape has slipped off at left!');
end;
If M.state = FinalState then
begin
ShowResult;
writeln('Program has finished');
ET:=Time;
writeln('Time taken ');
writeln(FormatDateTime('sss:zzz',ET-ST));
writeln(Count, ' steps taken');
end;
Tape.free;
Rules.free;
readln;
end.
Output
New State :H Tape position :1757 ----------------------- Program has finished Time taken 40:856 47176870 steps taken
The tape looks like this : 1010010010010010010010010010010010010010010 ...
Perl
use strict;
use warnings;
sub run_utm {
my %o = @_;
my $st = $o{state} // die "init head state undefined";
my $blank = $o{blank} // die "blank symbol undefined";
my @rules = @{$o{rules}} or die "rules undefined";
my @tape = $o{tape} ? @{$o{tape}} : ($blank);
my $halt = $o{halt};
my $pos = $o{pos} // 0;
$pos += @tape if $pos < 0;
die "bad init position" if $pos >= @tape || $pos < 0;
step: while (1) {
print "$st\t";
for (0 .. $#tape) {
my $v = $tape[$_];
print $_ == $pos ? "[$v]" : " $v ";
}
print "\n";
last if $st eq $halt;
for (@rules) {
my ($s0, $v0, $v1, $dir, $s1) = @$_;
next unless $s0 eq $st and $tape[$pos] eq $v0;
$tape[$pos] = $v1;
if ($dir eq 'left') {
if ($pos == 0) { unshift @tape, $blank}
else { $pos-- }
} elsif ($dir eq 'right') {
push @tape, $blank if ++$pos >= @tape
}
$st = $s1;
next step;
}
die "no matching rules";
}
}
print "incr machine\n";
run_utm halt=>'qf',
state=>'q0',
tape=>[1,1,1],
blank=>'B',
rules=>[[qw/q0 1 1 right q0/],
[qw/q0 B 1 stay qf/]];
print "\nbusy beaver\n";
run_utm halt=>'halt',
state=>'a',
blank=>'0',
rules=>[[qw/a 0 1 right b/],
[qw/a 1 1 left c/],
[qw/b 0 1 left a/],
[qw/b 1 1 right b/],
[qw/c 0 1 left b/],
[qw/c 1 1 stay halt/]];
print "\nsorting test\n";
run_utm halt=>'STOP',
state=>'A',
blank=>'0',
tape=>[qw/2 2 2 1 2 2 1 2 1 2 1 2 1 2/],
rules=>[[qw/A 1 1 right A/],
[qw/A 2 3 right B/],
[qw/A 0 0 left E/],
[qw/B 1 1 right B/],
[qw/B 2 2 right B/],
[qw/B 0 0 left C/],
[qw/C 1 2 left D/],
[qw/C 2 2 left C/],
[qw/C 3 2 left E/],
[qw/D 1 1 left D/],
[qw/D 2 2 left D/],
[qw/D 3 1 right A/],
[qw/E 1 1 left E/],
[qw/E 0 0 right STOP/]];
Phix
with javascript_semantics enum name, initState, endState, blank, rules -- Machine definitions constant incrementer = { /*name =*/ "Simple incrementer", /*initState =*/ "q0", /*endState =*/ "qf", /*blank =*/ "B", /*rules =*/ { {"q0", "1", "1", "right", "q0"}, {"q0", "B", "1", "stay", "qf"} } } constant threeStateBB = { /*name =*/ "Three-state busy beaver", /*initState =*/ "a", /*endState =*/ "halt", /*blank =*/ "0", /*rules =*/ { {"a", "0", "1", "right", "b"}, {"a", "1", "1", "left", "c"}, {"b", "0", "1", "left", "a"}, {"b", "1", "1", "right", "b"}, {"c", "0", "1", "left", "b"}, {"c", "1", "1", "stay", "halt"} } } constant fiveStateBB = { /*name =*/ "Five-state busy beaver", /*initState =*/ "A", /*endState =*/ "H", /*blank =*/ "0", /*rules =*/ { {"A", "0", "1", "right", "B"}, {"A", "1", "1", "left", "C"}, {"B", "0", "1", "right", "C"}, {"B", "1", "1", "right", "B"}, {"C", "0", "1", "right", "D"}, {"C", "1", "0", "left", "E"}, {"D", "0", "1", "left", "A"}, {"D", "1", "1", "left", "D"}, {"E", "0", "1", "stay", "H"}, {"E", "1", "0", "left", "A"} } } procedure show(string state, integer headpos, sequence tape) printf(1," %-6s | ",{state}) for p=1 to length(tape) do printf(1,iff(p=headpos?"[%s]":" %s "),{tape[p]}) end for printf(1,"\n") end procedure -- a universal turing machine procedure UTM(sequence machine, sequence tape, integer countOnly=0) string state = machine[initState] integer headpos = 1, counter = 0 printf(1,"\n\n%s\n%s\n",{machine[name],repeat('=',length(machine[name]))}) if not countOnly then printf(1," State | Tape [head]\n---------------------\n") end if while 1 do if headpos>length(tape) then tape = append(tape,machine[blank]) elsif headpos<1 then tape = prepend(tape,machine[blank]) headpos = 1 end if if not countOnly then show(state, headpos, tape) end if for i=1 to length(machine[rules]) do sequence rule = machine[rules][i] if rule[1]=state and rule[2]=tape[headpos] then tape[headpos] = rule[3] if rule[4] == "left" then headpos -= 1 end if if rule[4] == "right" then headpos += 1 end if state = rule[5] exit end if end for counter += 1 if state=machine[endState] then exit end if end while if countOnly then printf(1,"Steps taken: %d\n",{counter}) else show(state, headpos, tape) end if end procedure atom t0 = time() UTM(incrementer, {"1", "1", "1"}) UTM(threeStateBB, {}) if platform()!=JS then -- 1min 7s UTM(fiveStateBB, {}, countOnly:=1) end if ?elapsed(time()-t0)
- Output:
Simple incrementer ================== State | Tape [head] --------------------- q0 | [1] 1 1 q0 | 1 [1] 1 q0 | 1 1 [1] q0 | 1 1 1 [B] qf | 1 1 1 [1] Three-state busy beaver ======================= State | Tape [head] --------------------- a | [0] b | 1 [0] a | [1] 1 c | [0] 1 1 b | [0] 1 1 1 a | [0] 1 1 1 1 b | 1 [1] 1 1 1 b | 1 1 [1] 1 1 b | 1 1 1 [1] 1 b | 1 1 1 1 [1] b | 1 1 1 1 1 [0] a | 1 1 1 1 [1] 1 c | 1 1 1 [1] 1 1 halt | 1 1 1 [1] 1 1 Five-state busy beaver ====================== Steps taken: 47176870 "13.3s"
Obviously the first two are near-instant but as shown the last one is a bit too much to bother with running in a browser.
PHL
module turing;
extern printf;
struct @Command {
field @Integer tape {get:tape,set:stape};
field @Integer move {get:move,set:smove};
field @Integer next {get:next,set:snext};
@Command init(@Integer tape, @Integer move, @Integer next) [
this.stape(tape);
this.smove(move);
this.snext(next);
return this;
]
};
doc 2 dimansional array structure;
struct @Rules {
field @Integer maxstates { get: maxstates, set: smaxstates };
field @Integer maxvalue { get: maxvalue, set: smaxvalue };
field @Array<@Array<@Command> > table {get: t, set: st};
@Rules init(@Integer states, @Integer values)
[
this.smaxstates(states);
this.smaxvalue(values);
this.st(new @Array<@Array<@Command> >.init(states));
return this;
]
@Void setRule(@Integer state, @Integer tape, @Command command)
[
if (null == this::t.get(state)) {
this::t.set(state, new @Array<@Command>.init(this::maxvalue));
}
this::t.get(state).set(tape, command);
]
@Command getRule(@Integer state, @Integer tape)
[
return this::t.get(state).get(tape);
]
};
@Void emulateTuring(@Rules rules, @Integer start, @Integer stop, @Array<@Integer> tape, @Integer blank) [
var tapepointer = 0;
var state = start;
doc output;
printf("Tape\tState\n");
while (state != stop) {
doc add more cells to the tape;
if (tapepointer == tape::size) tape.add(blank);
if (tapepointer == 0-1) { tape = (new @Array<@Integer>..blank).addAll(tape); tapepointer = 0; }
doc output;
for (var i = 0; i < tape::size; i=i+1) {
printf("%i", tape.get(i));
}
printf("\t%i\n", state);
for (var i = 0; i < tapepointer; i=i+1) {
printf(" ");
}
printf("^\n");
doc the value of the current cell;
var tapeval = tape.get(tapepointer);
doc the current state;
var command = rules.getRule(state, tapeval);
tape.set(tapepointer, command::tape);
tapepointer = tapepointer + command::move;
state = command::next;
}
doc output;
for (var i = 0; i < tape::size; i=i+1) {
printf("%i", tape.get(i));
}
printf("\t%i\n", state);
for (var i = 0; i < tapepointer; i=i+1) {
printf(" ");
}
printf("^\n");
]
@Integer main [
doc incrementer;
doc 2 states, 2 symbols;
var rules = new @Rules.init(2, 2);
doc q0, 1 -> 1, right, q0;
doc q0, B -> 1, stay, qf;
rules.setRule(0, 1, new @Command.init(1, 1, 0));
rules.setRule(0, 0, new @Command.init(1, 0, 1));
doc tape = [1, 1, 1];
var tape = new @Array<@Integer>..1..1..1;
doc start turing machine;
emulateTuring(rules, 0, 1, tape, 0);
doc ---------------------------------------------------;
doc three state busy beaver;
doc 4 states, 2 symbols;
rules = new @Rules.init(4, 2);
doc a, 0 -> 1, right, b
a, 1 -> 1, left, c
b, 0 -> 1, left, a
b, 1 -> 1, right, b
c, 0 -> 1, left, b
c, 1 -> 1, stay, halt
;
doc a = 0,
b = 1,
c = 2,
halt = 3;
rules.setRule(0, 0, new @Command.init(1, 1, 1));
rules.setRule(0, 1, new @Command.init(1, 0-1, 2));
rules.setRule(1, 0, new @Command.init(1, 0-1, 0));
rules.setRule(1, 1, new @Command.init(1, 1, 1));
rules.setRule(2, 0, new @Command.init(1, 0-1, 1));
rules.setRule(2, 1, new @Command.init(1, 0, 3));
doc tape = [];
tape = new @Array<@Integer>;
doc start turing machine;
emulateTuring(rules, 0, 3, tape, 0);
return 0;
]Output:
Tape State
111 0
^
111 0
^
111 0
^
1110 0
^
1111 1
^
Tape State
0 0
^
10 1
^
11 0
^
011 2
^
0111 1
^
01111 0
^
11111 1
^
11111 1
^
11111 1
^
11111 1
^
111110 1
^
111111 0
^
111111 2
^
111111 3
^
PicoLisp
# Finite state machine
(de turing (Tape Init Halt Blank Rules Verbose)
(let
(Head 1
State Init
Rule NIL
S 'start
C (length Tape))
(catch NIL
(loop
(state 'S
(start 'print
(when (=0 C)
(setq Tape (insert Head Tape Blank))
(inc 'C) ) )
(print 'lookup
(when Verbose
(for (N . I) Tape
(if (= N Head)
(print (list I))
(prin I) ) )
(prinl) )
(when (= State Halt) (throw NIL) ) )
(lookup 'do
(setq Rule
(find
'((X)
(and
(= (car X) State)
(= (cadr X) (car (nth Tape Head))) ) )
Rules ) ) )
(do 'step
(setq Tape (place Head Tape (caddr Rule))) )
(step 'print
(cond
((= (cadddr Rule) 'R) (inc 'Head))
((= (cadddr Rule) 'L) (dec 'Head)) )
(cond
((< Head 1)
(setq Tape (insert Head Tape Blank))
(inc 'C)
(one Head) )
((> Head C)
(setq Tape (insert Head Tape Blank))
(inc 'C) ) )
(setq State (last Rule)) ) ) ) ) )
Tape )
(println "Simple incrementer")
(turing '(1 1 1) 'A 'H 'B '((A 1 1 R A) (A B 1 S H)) T)
(println "Three-state busy beaver")
(turing '() 'A 'H 0
'((A 0 1 R B)
(A 1 1 L C)
(B 0 1 L A)
(B 1 1 R B)
(C 0 1 L B)
(C 1 1 S H)) T )
(println "Five-state busy beaver")
(let Tape (turing '() 'A 'H 0
'((A 0 1 R B)
(A 1 1 L C)
(B 0 1 R C)
(B 1 1 R B)
(C 0 1 R D)
(C 1 0 L E)
(D 0 1 L A)
(D 1 1 L D)
(E 0 1 S H)
(E 1 0 L A)) NIL)
(println '0s: (cnt '((X) (= 0 X)) Tape))
(println '1s: (cnt '((X) (= 1 X)) Tape)) )
(bye)- Output:
"Simple incrementer" (1)11 1(1)1 11(1) 111(B) 111(1) "Three-state busy beaver" (0) 1(0) (1)1 (0)11 (0)111 (0)1111 1(1)111 11(1)11 111(1)1 1111(1) 11111(0) 1111(1)1 111(1)11 111(1)11 "Five-state busy beaver" 0s: 8191 1s: 4098
Prolog
The universal machine
Source for this example was lightly adapted from https://bitbucket.org/ttmrichter/turing. This machine, because of Prolog's dynamic nature, has to check its configuration and the rules' compliance to the same at run-time. This is the role of all but the first of the memberchk/2 predicates. In addition, calling the user-supplied rules has to be wrapped in a once/1 wrapper because there is no way to guarantee in advance that the rules provided are deterministic. (An alternative to doing this is to simply allow perform/5 to be non-deterministic or to check for multiple results and report an error on such.)
turing(Config, Rules, TapeIn, TapeOut) :-
call(Config, IS, _, _, _, _),
perform(Config, Rules, IS, {[], TapeIn}, {Ls, Rs}),
reverse(Ls, Ls1),
append(Ls1, Rs, TapeOut).
perform(Config, Rules, State, TapeIn, TapeOut) :-
call(Config, _, FS, RS, B, Symbols),
( memberchk(State, FS) ->
TapeOut = TapeIn
; memberchk(State, RS) ->
{LeftIn, RightIn} = TapeIn,
symbol(RightIn, Symbol, RightRem, B),
memberchk(Symbol, Symbols),
once(call(Rules, State, Symbol, NewSymbol, Action, NewState)),
memberchk(NewSymbol, Symbols),
action(Action, {LeftIn, [NewSymbol|RightRem]}, {LeftOut, RightOut}, B),
perform(Config, Rules, NewState, {LeftOut, RightOut}, TapeOut) ).
symbol([], B, [], B).
symbol([Sym|Rs], Sym, Rs, _).
action(left, {Lin, Rin}, {Lout, Rout}, B) :- left(Lin, Rin, Lout, Rout, B).
action(stay, Tape, Tape, _).
action(right, {Lin, Rin}, {Lout, Rout}, B) :- right(Lin, Rin, Lout, Rout, B).
left([], Rs, [], [B|Rs], B).
left([L|Ls], Rs, Ls, [L|Rs], _).
right(L, [], [B|L], [], B).
right(L, [S|Rs], [S|L], Rs, _).
The incrementer machine
incrementer_config(IS, FS, RS, B, S) :-
IS = q0, % initial state
FS = [qf], % halting states
RS = [IS], % running states
B = 0, % blank symbol
S = [B, 1]. % valid symbols
incrementer(q0, 1, 1, right, q0).
incrementer(q0, b, 1, stay, qf).
turing(incrementer_config, incrementer, [1, 1, 1], TapeOut).
This will, on execution, fill TapeOut with [1, 1, 1, 1].
The busy beaver machine
busy_beaver_config(IS, FS, RS, B, S) :-
IS = 'A', % initial state
FS = ['HALT'], % halting states
RS = [IS, 'B', 'C'], % running states
B = 0, % blank symbol
S = [B, 1]. % valid symbols
busy_beaver('A', 0, 1, right, 'B').
busy_beaver('A', 1, 1, left, 'C').
busy_beaver('B', 0, 1, left, 'A').
busy_beaver('B', 1, 1, right, 'B').
busy_beaver('C', 0, 1, left, 'B').
busy_beaver('C', 1, 1, stay, 'HALT').
turing(busy_beaver_config, busy_beaver, [], TapeOut).
This will, on execution, fill TapeOut with [1, 1, 1, 1, 1, 1].
Python
from __future__ import print_function
def run_utm(
state = None,
blank = None,
rules = [],
tape = [],
halt = None,
pos = 0):
st = state
if not tape: tape = [blank]
if pos < 0: pos += len(tape)
if pos >= len(tape) or pos < 0: raise Error( "bad init position")
rules = dict(((s0, v0), (v1, dr, s1)) for (s0, v0, v1, dr, s1) in rules)
while True:
print(st, '\t', end=" ")
for i, v in enumerate(tape):
if i == pos: print("[%s]" % (v,), end=" ")
else: print(v, end=" ")
print()
if st == halt: break
if (st, tape[pos]) not in rules: break
(v1, dr, s1) = rules[(st, tape[pos])]
tape[pos] = v1
if dr == 'left':
if pos > 0: pos -= 1
else: tape.insert(0, blank)
if dr == 'right':
pos += 1
if pos >= len(tape): tape.append(blank)
st = s1
# EXAMPLES
print("incr machine\n")
run_utm(
halt = 'qf',
state = 'q0',
tape = list("111"),
blank = 'B',
rules = map(tuple,
["q0 1 1 right q0".split(),
"q0 B 1 stay qf".split()]
)
)
print("\nbusy beaver\n")
run_utm(
halt = 'halt',
state = 'a',
blank = '0',
rules = map(tuple,
["a 0 1 right b".split(),
"a 1 1 left c".split(),
"b 0 1 left a".split(),
"b 1 1 right b".split(),
"c 0 1 left b".split(),
"c 1 1 stay halt".split()]
)
)
print("\nsorting test\n")
run_utm(halt = 'STOP',
state = 'A',
blank = '0',
tape = "2 2 2 1 2 2 1 2 1 2 1 2 1 2".split(),
rules = map(tuple,
["A 1 1 right A".split(),
"A 2 3 right B".split(),
"A 0 0 left E".split(),
"B 1 1 right B".split(),
"B 2 2 right B".split(),
"B 0 0 left C".split(),
"C 1 2 left D".split(),
"C 2 2 left C".split(),
"C 3 2 left E".split(),
"D 1 1 left D".split(),
"D 2 2 left D".split(),
"D 3 1 right A".split(),
"E 1 1 left E".split(),
"E 0 0 right STOP".split()]
)
)
Racket
#lang racket
;;;=============================================================
;;; Due to heavy use of pattern matching we define few macros
;;;=============================================================
(define-syntax-rule (define-m f m ...)
(define f (match-lambda m ... (x x))))
(define-syntax-rule (define-m* f m ...)
(define f (match-lambda** m ...)))
;;;=============================================================
;;; The definition of a functional type Tape,
;;; representing infinite tape with O(1) operations:
;;; put, get, shift-right and shift-left.
;;;=============================================================
(struct Tape (the-left-part ; i-1 i-2 i-3 ...
the-current-record ; i
the-right-part)) ; i+1 i+2 i+3 ...
;; the initial record on the tape
(define-m initial-tape
[(cons h t) (Tape '() h t)])
;; shifts caret to the right
(define (snoc a b) (cons b a))
(define-m shift-right
[(Tape '() '() (cons h t)) (Tape '() h t)] ; left end
[(Tape l x '()) (Tape (snoc l x) '() '())] ; right end
[(Tape l x (cons h t)) (Tape (snoc l x) h t)]) ; general case
;; shifts caret to the left
(define-m flip-tape [(Tape l x r) (Tape r x l)])
(define shift-left
(compose flip-tape shift-right flip-tape))
;; returns the current record on the tape
(define-m get [(Tape _ v _) v])
;; writes to the current position on the tape
(define-m* put
[('() t) t]
[(v (Tape l _ r)) (Tape l v r)])
;; Shows the list representation of the tape (≤ O(n)).
;; A tape is shown as (... a b c (d) e f g ...)
;; where (d) marks the current position of the caret.
(define (revappend a b) (foldl cons b a))
(define-m show-tape
[(Tape '() '() '()) '()]
[(Tape l '() r) (revappend l (cons '() r))]
[(Tape l v r) (revappend l (cons (list v) r))])
;;;-------------------------------------------------------------------
;;; The Turing Machine interpreter
;;;
;; interpretation of output triple for a given tape
(define-m* interprete
[((list v 'right S) tape) (list S (shift-right (put v tape)))]
[((list v 'left S) tape) (list S (shift-left (put v tape)))]
[((list v 'stay S) tape) (list S (put v tape))]
[((list S _) tape) (list S tape)])
;; Runs the program.
;; The initial state is set to start.
;; The initial tape is given as a list of records.
;; The initial position is the leftmost symbol of initial record.
(define (run-turing prog t0 start)
((fixed-point
(match-lambda
[`(,S ,T) (begin
(printf "~a\t~a\n" S (show-tape T))
(interprete (prog `(,S ,(get T))) T))]))
(list start (initial-tape t0))))
;; a general fixed point operator
(define ((fixed-point f) x)
(let F ([x x] [fx (f x)])
(if (equal? x fx)
fx
(F fx (f fx)))))
;; A macro for definition of a Turing-Machines.
;; Transforms to a function which accepts a list of initial
;; tape records as input and returns the tape after stopping.
(define-syntax-rule (Turing-Machine #:start start (a b c d e) ...)
(λ (l)
(displayln "STATE\tTAPE")
((match-lambda [(list _ t) (flatten (show-tape t))])
(run-turing
(match-lambda ['(a b) '(c d e)] ... [x x])
l start))))
The resulting Turing Machine is a function that maps the initial tape record to the final one, so that several machines could run one after another or composed as any other functions
Examples:
The simple incrementer:
(define INC
(Turing-Machine #:start 'q0
[q0 1 1 right q0]
[q0 () 1 stay qf]))
> (INC '(1 1 1)) STATE TAPE q0 ((1) 1 1) q0 (1 (1) 1) q0 (1 1 (1)) q0 (1 1 1 ()) qf (1 1 1 (1)) (1 1 1 1)
The incrementer for binary numbers
(define ADD1
(Turing-Machine #:start 'Start
[Start 1 1 right Start]
[Start 0 0 right Start]
[Start () () left Add]
[Add 0 1 stay End]
[Add 1 0 left Add]
[Add () 1 stay End]))
> (ADD1 '(1 1 0)) STATE TAPE Start ((1) 1 0) Start (1 (1) 0) Start (1 1 (0)) Start (1 1 0 ()) Add (1 1 (0)) End (1 1 (1)) (1 1 1) > (define ADD2 (compose ADD1 ADD1)) > (ADD2 '(1 1 0)) STATE TAPE Start ((1) 1 0) Start (1 (1) 0) Start (1 1 (0)) Start (1 1 0 ()) Add (1 1 (0)) End (1 1 (1)) STATE TAPE Start ((1) 1 1) Start (1 (1) 1) Start (1 1 (1)) Start (1 1 1 ()) Add (1 1 (1)) Add (1 (1) 0) Add ((1) 0 0) Add (() 0 0 0) End ((1) 0 0 0) (1 0 0 0)
The busy beaver
(define BEAVER
(Turing-Machine #:start 'a
[a () 1 right b]
[a 1 1 left c]
[b () 1 left a]
[b 1 1 right b]
[c () 1 left b]
[c 1 1 stay halt]))
> (BEAVER '(())) STATE TAPE a () b (1 ()) a ((1) 1) c (() 1 1) b (() 1 1 1) a (() 1 1 1 1) b (1 (1) 1 1 1) b (1 1 (1) 1 1) b (1 1 1 (1) 1) b (1 1 1 1 (1)) b (1 1 1 1 1 ()) a (1 1 1 1 (1) 1) c (1 1 1 (1) 1 1) halt (1 1 1 (1) 1 1) (1 1 1 1 1 1)
The sorting machine
(define SORT
(Turing-Machine #:start 'A
[A 1 1 right A]
[A 2 3 right B]
[A () () left E]
[B 1 1 right B]
[B 2 2 right B]
[B () () left C]
[C 1 2 left D]
[C 2 2 left C]
[C 3 2 left E]
[D 1 1 left D]
[D 2 2 left D]
[D 3 1 right A]
[E 1 1 left E]
[E () () right STOP]))
> (SORT '(2 1 2 2 2 1 1)) STATE TAPE A ((2) 1 2 2 2 1 1) B (3 (1) 2 2 2 1 1) B (3 1 (2) 2 2 1 1) B (3 1 2 (2) 2 1 1) B (3 1 2 2 (2) 1 1) B (3 1 2 2 2 (1) 1) B (3 1 2 2 2 1 (1)) B (3 1 2 2 2 1 1 ()) C (3 1 2 2 2 1 (1)) D (3 1 2 2 2 (1) 2) D (3 1 2 2 (2) 1 2) D (3 1 2 (2) 2 1 2) D (3 1 (2) 2 2 1 2) D (3 (1) 2 2 2 1 2) D ((3) 1 2 2 2 1 2) A (1 (1) 2 2 2 1 2) A (1 1 (2) 2 2 1 2) B (1 1 3 (2) 2 1 2) B (1 1 3 2 (2) 1 2) B (1 1 3 2 2 (1) 2) B (1 1 3 2 2 1 (2)) B (1 1 3 2 2 1 2 ()) C (1 1 3 2 2 1 (2)) C (1 1 3 2 2 (1) 2) D (1 1 3 2 (2) 2 2) D (1 1 3 (2) 2 2 2) D (1 1 (3) 2 2 2 2) A (1 1 1 (2) 2 2 2) B (1 1 1 3 (2) 2 2) B (1 1 1 3 2 (2) 2) B (1 1 1 3 2 2 (2)) B (1 1 1 3 2 2 2 ()) C (1 1 1 3 2 2 (2)) C (1 1 1 3 2 (2) 2) C (1 1 1 3 (2) 2 2) C (1 1 1 (3) 2 2 2) E (1 1 (1) 2 2 2 2) E (1 (1) 1 2 2 2 2) E ((1) 1 1 2 2 2 2) E (() 1 1 1 2 2 2 2) STOP ((1) 1 1 2 2 2 2) (1 1 1 2 2 2 2)
Raku
(formerly Perl 6)
sub run_utm(:$state! is copy, :$blank!, :@rules!, :@tape = [$blank], :$halt, :$pos is copy = 0) {
$pos += @tape if $pos < 0;
die "Bad initial position" unless $pos ~~ ^@tape;
STEP: loop {
print "$state\t";
for ^@tape {
my $v = @tape[$_];
print $_ == $pos ?? "[$v]" !! " $v ";
}
print "\n";
last if $state eq $halt;
for @rules -> @rule {
my ($s0, $v0, $v1, $dir, $s1) = @rule;
next unless $s0 eq $state and @tape[$pos] eq $v0;
@tape[$pos] = $v1;
given $dir {
when 'left' {
if $pos == 0 { unshift @tape, $blank }
else { $pos-- }
}
when 'right' {
push @tape, $blank if ++$pos >= @tape;
}
}
$state = $s1;
next STEP;
}
die 'No matching rules';
}
}
say "incr machine";
run_utm :halt<qf>,
:state<q0>,
:tape[1,1,1],
:blank<B>,
:rules[
[< q0 1 1 right q0 >],
[< q0 B 1 stay qf >]
];
say "\nbusy beaver";
run_utm :halt<halt>,
:state<a>,
:blank<0>,
:rules[
[< a 0 1 right b >],
[< a 1 1 left c >],
[< b 0 1 left a >],
[< b 1 1 right b >],
[< c 0 1 left b >],
[< c 1 1 stay halt >]
];
say "\nsorting test";
run_utm :halt<STOP>,
:state<A>,
:blank<0>,
:tape[< 2 2 2 1 2 2 1 2 1 2 1 2 1 2 >],
:rules[
[< A 1 1 right A >],
[< A 2 3 right B >],
[< A 0 0 left E >],
[< B 1 1 right B >],
[< B 2 2 right B >],
[< B 0 0 left C >],
[< C 1 2 left D >],
[< C 2 2 left C >],
[< C 3 2 left E >],
[< D 1 1 left D >],
[< D 2 2 left D >],
[< D 3 1 right A >],
[< E 1 1 left E >],
[< E 0 0 right STOP >]
];
- Output:
incr machine q0 [1] 1 1 q0 1 [1] 1 q0 1 1 [1] q0 1 1 1 [B] qf 1 1 1 [1] busy beaver a [0] b 1 [0] a [1] 1 c [0] 1 1 b [0] 1 1 1 a [0] 1 1 1 1 b 1 [1] 1 1 1 b 1 1 [1] 1 1 b 1 1 1 [1] 1 b 1 1 1 1 [1] b 1 1 1 1 1 [0] a 1 1 1 1 [1] 1 c 1 1 1 [1] 1 1 halt 1 1 1 [1] 1 1 sorting test A [2] 2 2 1 2 2 1 2 1 2 1 2 1 2 B 3 [2] 2 1 2 2 1 2 1 2 1 2 1 2 B 3 2 [2] 1 2 2 1 2 1 2 1 2 1 2 B 3 2 2 [1] 2 2 1 2 1 2 1 2 1 2 B 3 2 2 1 [2] 2 1 2 1 2 1 2 1 2 B 3 2 2 1 2 [2] 1 2 1 2 1 2 1 2 B 3 2 2 1 2 2 [1] 2 1 2 1 2 1 2 B 3 2 2 1 2 2 1 [2] 1 2 1 2 1 2 B 3 2 2 1 2 2 1 2 [1] 2 1 2 1 2 B 3 2 2 1 2 2 1 2 1 [2] 1 2 1 2 B 3 2 2 1 2 2 1 2 1 2 [1] 2 1 2 B 3 2 2 1 2 2 1 2 1 2 1 [2] 1 2 B 3 2 2 1 2 2 1 2 1 2 1 2 [1] 2 B 3 2 2 1 2 2 1 2 1 2 1 2 1 [2] B 3 2 2 1 2 2 1 2 1 2 1 2 1 2 [0] C 3 2 2 1 2 2 1 2 1 2 1 2 1 [2] 0 C 3 2 2 1 2 2 1 2 1 2 1 2 [1] 2 0 D 3 2 2 1 2 2 1 2 1 2 1 [2] 2 2 0 D 3 2 2 1 2 2 1 2 1 2 [1] 2 2 2 0 D 3 2 2 1 2 2 1 2 1 [2] 1 2 2 2 0 D 3 2 2 1 2 2 1 2 [1] 2 1 2 2 2 0 D 3 2 2 1 2 2 1 [2] 1 2 1 2 2 2 0 D 3 2 2 1 2 2 [1] 2 1 2 1 2 2 2 0 D 3 2 2 1 2 [2] 1 2 1 2 1 2 2 2 0 D 3 2 2 1 [2] 2 1 2 1 2 1 2 2 2 0 D 3 2 2 [1] 2 2 1 2 1 2 1 2 2 2 0 D 3 2 [2] 1 2 2 1 2 1 2 1 2 2 2 0 D 3 [2] 2 1 2 2 1 2 1 2 1 2 2 2 0 D [3] 2 2 1 2 2 1 2 1 2 1 2 2 2 0 A 1 [2] 2 1 2 2 1 2 1 2 1 2 2 2 0 B 1 3 [2] 1 2 2 1 2 1 2 1 2 2 2 0 B 1 3 2 [1] 2 2 1 2 1 2 1 2 2 2 0 B 1 3 2 1 [2] 2 1 2 1 2 1 2 2 2 0 B 1 3 2 1 2 [2] 1 2 1 2 1 2 2 2 0 B 1 3 2 1 2 2 [1] 2 1 2 1 2 2 2 0 B 1 3 2 1 2 2 1 [2] 1 2 1 2 2 2 0 B 1 3 2 1 2 2 1 2 [1] 2 1 2 2 2 0 B 1 3 2 1 2 2 1 2 1 [2] 1 2 2 2 0 B 1 3 2 1 2 2 1 2 1 2 [1] 2 2 2 0 B 1 3 2 1 2 2 1 2 1 2 1 [2] 2 2 0 B 1 3 2 1 2 2 1 2 1 2 1 2 [2] 2 0 B 1 3 2 1 2 2 1 2 1 2 1 2 2 [2] 0 B 1 3 2 1 2 2 1 2 1 2 1 2 2 2 [0] C 1 3 2 1 2 2 1 2 1 2 1 2 2 [2] 0 C 1 3 2 1 2 2 1 2 1 2 1 2 [2] 2 0 C 1 3 2 1 2 2 1 2 1 2 1 [2] 2 2 0 C 1 3 2 1 2 2 1 2 1 2 [1] 2 2 2 0 D 1 3 2 1 2 2 1 2 1 [2] 2 2 2 2 0 D 1 3 2 1 2 2 1 2 [1] 2 2 2 2 2 0 D 1 3 2 1 2 2 1 [2] 1 2 2 2 2 2 0 D 1 3 2 1 2 2 [1] 2 1 2 2 2 2 2 0 D 1 3 2 1 2 [2] 1 2 1 2 2 2 2 2 0 D 1 3 2 1 [2] 2 1 2 1 2 2 2 2 2 0 D 1 3 2 [1] 2 2 1 2 1 2 2 2 2 2 0 D 1 3 [2] 1 2 2 1 2 1 2 2 2 2 2 0 D 1 [3] 2 1 2 2 1 2 1 2 2 2 2 2 0 A 1 1 [2] 1 2 2 1 2 1 2 2 2 2 2 0 B 1 1 3 [1] 2 2 1 2 1 2 2 2 2 2 0 B 1 1 3 1 [2] 2 1 2 1 2 2 2 2 2 0 B 1 1 3 1 2 [2] 1 2 1 2 2 2 2 2 0 B 1 1 3 1 2 2 [1] 2 1 2 2 2 2 2 0 B 1 1 3 1 2 2 1 [2] 1 2 2 2 2 2 0 B 1 1 3 1 2 2 1 2 [1] 2 2 2 2 2 0 B 1 1 3 1 2 2 1 2 1 [2] 2 2 2 2 0 B 1 1 3 1 2 2 1 2 1 2 [2] 2 2 2 0 B 1 1 3 1 2 2 1 2 1 2 2 [2] 2 2 0 B 1 1 3 1 2 2 1 2 1 2 2 2 [2] 2 0 B 1 1 3 1 2 2 1 2 1 2 2 2 2 [2] 0 B 1 1 3 1 2 2 1 2 1 2 2 2 2 2 [0] C 1 1 3 1 2 2 1 2 1 2 2 2 2 [2] 0 C 1 1 3 1 2 2 1 2 1 2 2 2 [2] 2 0 C 1 1 3 1 2 2 1 2 1 2 2 [2] 2 2 0 C 1 1 3 1 2 2 1 2 1 2 [2] 2 2 2 0 C 1 1 3 1 2 2 1 2 1 [2] 2 2 2 2 0 C 1 1 3 1 2 2 1 2 [1] 2 2 2 2 2 0 D 1 1 3 1 2 2 1 [2] 2 2 2 2 2 2 0 D 1 1 3 1 2 2 [1] 2 2 2 2 2 2 2 0 D 1 1 3 1 2 [2] 1 2 2 2 2 2 2 2 0 D 1 1 3 1 [2] 2 1 2 2 2 2 2 2 2 0 D 1 1 3 [1] 2 2 1 2 2 2 2 2 2 2 0 D 1 1 [3] 1 2 2 1 2 2 2 2 2 2 2 0 A 1 1 1 [1] 2 2 1 2 2 2 2 2 2 2 0 A 1 1 1 1 [2] 2 1 2 2 2 2 2 2 2 0 B 1 1 1 1 3 [2] 1 2 2 2 2 2 2 2 0 B 1 1 1 1 3 2 [1] 2 2 2 2 2 2 2 0 B 1 1 1 1 3 2 1 [2] 2 2 2 2 2 2 0 B 1 1 1 1 3 2 1 2 [2] 2 2 2 2 2 0 B 1 1 1 1 3 2 1 2 2 [2] 2 2 2 2 0 B 1 1 1 1 3 2 1 2 2 2 [2] 2 2 2 0 B 1 1 1 1 3 2 1 2 2 2 2 [2] 2 2 0 B 1 1 1 1 3 2 1 2 2 2 2 2 [2] 2 0 B 1 1 1 1 3 2 1 2 2 2 2 2 2 [2] 0 B 1 1 1 1 3 2 1 2 2 2 2 2 2 2 [0] C 1 1 1 1 3 2 1 2 2 2 2 2 2 [2] 0 C 1 1 1 1 3 2 1 2 2 2 2 2 [2] 2 0 C 1 1 1 1 3 2 1 2 2 2 2 [2] 2 2 0 C 1 1 1 1 3 2 1 2 2 2 [2] 2 2 2 0 C 1 1 1 1 3 2 1 2 2 [2] 2 2 2 2 0 C 1 1 1 1 3 2 1 2 [2] 2 2 2 2 2 0 C 1 1 1 1 3 2 1 [2] 2 2 2 2 2 2 0 C 1 1 1 1 3 2 [1] 2 2 2 2 2 2 2 0 D 1 1 1 1 3 [2] 2 2 2 2 2 2 2 2 0 D 1 1 1 1 [3] 2 2 2 2 2 2 2 2 2 0 A 1 1 1 1 1 [2] 2 2 2 2 2 2 2 2 0 B 1 1 1 1 1 3 [2] 2 2 2 2 2 2 2 0 B 1 1 1 1 1 3 2 [2] 2 2 2 2 2 2 0 B 1 1 1 1 1 3 2 2 [2] 2 2 2 2 2 0 B 1 1 1 1 1 3 2 2 2 [2] 2 2 2 2 0 B 1 1 1 1 1 3 2 2 2 2 [2] 2 2 2 0 B 1 1 1 1 1 3 2 2 2 2 2 [2] 2 2 0 B 1 1 1 1 1 3 2 2 2 2 2 2 [2] 2 0 B 1 1 1 1 1 3 2 2 2 2 2 2 2 [2] 0 B 1 1 1 1 1 3 2 2 2 2 2 2 2 2 [0] C 1 1 1 1 1 3 2 2 2 2 2 2 2 [2] 0 C 1 1 1 1 1 3 2 2 2 2 2 2 [2] 2 0 C 1 1 1 1 1 3 2 2 2 2 2 [2] 2 2 0 C 1 1 1 1 1 3 2 2 2 2 [2] 2 2 2 0 C 1 1 1 1 1 3 2 2 2 [2] 2 2 2 2 0 C 1 1 1 1 1 3 2 2 [2] 2 2 2 2 2 0 C 1 1 1 1 1 3 2 [2] 2 2 2 2 2 2 0 C 1 1 1 1 1 3 [2] 2 2 2 2 2 2 2 0 C 1 1 1 1 1 [3] 2 2 2 2 2 2 2 2 0 E 1 1 1 1 [1] 2 2 2 2 2 2 2 2 2 0 E 1 1 1 [1] 1 2 2 2 2 2 2 2 2 2 0 E 1 1 [1] 1 1 2 2 2 2 2 2 2 2 2 0 E 1 [1] 1 1 1 2 2 2 2 2 2 2 2 2 0 E [1] 1 1 1 1 2 2 2 2 2 2 2 2 2 0 E [0] 1 1 1 1 1 2 2 2 2 2 2 2 2 2 0 STOP 0 [1] 1 1 1 1 2 2 2 2 2 2 2 2 2 0
REXX
Programming notes: the tape is essentially infinite in two directions, but the tape starts at location one (unity), and
may extend to -∞ and +∞ (subject to virtual memory limitations), something short of ≈ 2 billion bytes.
Minimal error checking is done, but if no rule is found to be applicable, an appropriate error message is issued.
incrementer machine
/*REXX program executes a Turing machine based on initial state, tape, and rules. */
state = 'q0' /*the initial Turing machine state. */
term = 'qf' /*a state that is used for a halt. */
blank = 'B' /*this character is a "true" blank. */
call Turing_rule 'q0 1 1 right q0' /*define a rule for the Turing machine.*/
call Turing_rule 'q0 B 1 stay qf' /* " " " " " " " */
call Turing_init 1 1 1 /*initialize the tape to some string(s)*/
call TM /*go and invoke the Turning machine. */
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
TM: !=1; bot=1; top=1; @er= '***error***' /*start at the tape location 1. */
say /*might as well display a blank line. */
do cycle=1 until state==term /*process Turing machine instructions.*/
do k=1 for rules /* " " " rules. */
parse var rule.k rState rTape rWrite rMove rNext . /*pick pieces. */
if state\==rState | @.!\==rTape then iterate /*wrong rule ? */
@.!=rWrite /*right rule; write it ───► the tape. */
if rMove== 'left' then !=!-1 /*Are we moving left? Then subtract 1*/
if rMove=='right' then !=!+1 /* " " " right? " add 1*/
bot=min(bot, !); top=max(top, !) /*find the tape bottom and top. */
state=rNext; iterate cycle /*use this for the next state; and */
end /*k*/
say @er 'unknown state:' state; leave /*oops, we have an unknown state error.*/
end /*cycle*/
$= /*start with empty string (the tape). */
do t=bot to top; _=@.t
if _==blank then _=' ' /*do we need to translate a true blank?*/
$=$ || pad || _ /*construct char by char, maybe pad it.*/
end /*t*/ /* [↑] construct the tape's contents*/
L=length($) /*obtain length of " " " */
if L==0 then $= "[tape is blank.]" /*make an empty tape visible to user.*/
if L>1000 then $=left($, 1000) ... /*truncate tape to 1k bytes, append ···*/
say "tape's contents:" $ /*show the tape's contents (or 1st 1k).*/
say "tape's length: " L /* " " " length. */
say 'Turning machine used ' rules " rules in " cycle ' cycles.'
return
/*──────────────────────────────────────────────────────────────────────────────────────*/
Turing_init: @.=blank; parse arg x; do j=1 for words(x); @.j=word(x,j); end /*j*/
return
/*──────────────────────────────────────────────────────────────────────────────────────*/
Turing_rule: if symbol('RULES')=="LIT" then rules=0; rules=rules+1
pad=left('', length( word( arg(1),2 ) ) \==1 ) /*padding for rule*/
rule.rules=arg(1); say right('rule' rules, 20) "═══►" rule.rules
return
output
rule 1 ═══► q0 1 1 right q0
rule 2 ═══► q0 B 1 stay qf
tape's contents: 1111
tape's length: 4
Turning machine used 2 rules in 4 cycles.
three-state busy beaver
/*REXX program executes a Turing machine based on initial state, tape, and rules. */
state = 'a' /*the initial Turing machine state. */
term = 'halt' /*a state that is used for a halt. */
blank = 0 /*this character is a "true" blank. */
call Turing_rule 'a 0 1 right b' /*define a rule for the Turing machine.*/
call Turing_rule 'a 1 1 left c' /* " " " " " " " */
call Turing_rule 'b 0 1 left a' /* " " " " " " " */
call Turing_rule 'b 1 1 right b' /* " " " " " " " */
call Turing_rule 'c 0 1 left b' /* " " " " " " " */
call Turing_rule 'c 1 1 stay halt' /* " " " " " " " */
call Turing_init /*initialize the tape to some string(s)*/
call TM /*go and invoke the Turning machine. */
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
TM: ∙∙∙
output
rule 1 ═══► a 0 1 right b
rule 2 ═══► a 1 1 left c
rule 3 ═══► b 0 1 left a
rule 4 ═══► b 1 1 right b
rule 5 ═══► c 0 1 left b
rule 6 ═══► c 1 1 stay halt
tape's contents: 111111
tape's length: 6
Turning machine used 6 rules in 13 cycles.
five-state busy beaver
/*REXX program executes a Turing machine based on initial state, tape, and rules. */
state = 'A' /*initialize the Turing machine state.*/
term = 'H' /*a state that is used for the halt. */
blank = 0 /*this character is a "true" blank. */
call Turing_rule 'A 0 1 right B' /*define a rule for the Turing machine.*/
call Turing_rule 'A 1 1 left C' /* " " " " " " " */
call Turing_rule 'B 0 1 right C' /* " " " " " " " */
call Turing_rule 'B 1 1 right B' /* " " " " " " " */
call Turing_rule 'C 0 1 right D' /* " " " " " " " */
call Turing_rule 'C 1 0 left E' /* " " " " " " " */
call Turing_rule 'D 0 1 left A' /* " " " " " " " */
call Turing_rule 'D 1 1 left D' /* " " " " " " " */
call Turing_rule 'E 0 1 stay H' /* " " " " " " " */
call Turing_rule 'E 1 0 left A' /* " " " " " " " */
call Turing_init /*initialize the tape to some string(s)*/
call TM /*go and invoke the Turning machine. */
exit /*stick a fork in it, we're done.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
TM: ∙∙∙
output
rule 1 ═══► A 0 1 right B
rule 2 ═══► A 1 1 left C
rule 3 ═══► B 0 1 right C
rule 4 ═══► B 1 1 right B
rule 5 ═══► C 0 1 right D
rule 6 ═══► C 1 0 left E
rule 7 ═══► D 0 1 left A
rule 8 ═══► D 1 1 left D
rule 9 ═══► E 0 1 stay H
rule 10 ═══► E 1 0 left A
tape's contents: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 ...
tape's length is: 12289
Turning machine used 10 rules in 47176870 cycles.
stress sort
/*REXX program executes a Turing machine based on initial state, tape, and rules. */
state = 'A' /*the initial Turing machine state. */
term = 'halt' /*a state that is used for the halt. */
blank = 0 /*this character is a "true" blank. */
call Turing_rule 'A 1 1 right A' /*define a rule for the Turing machine.*/
call Turing_rule 'A 2 3 right B' /* " " " " " " " */
call Turing_rule 'A 0 0 left E' /* " " " " " " " */
call Turing_rule 'B 1 1 right B' /* " " " " " " " */
call Turing_rule 'B 2 2 right B' /* " " " " " " " */
call Turing_rule 'B 0 0 left C' /* " " " " " " " */
call Turing_rule 'C 1 2 left D' /* " " " " " " " */
call Turing_rule 'C 2 2 left C' /* " " " " " " " */
call Turing_rule 'C 3 2 left E' /* " " " " " " " */
call Turing_rule 'D 1 1 left D' /* " " " " " " " */
call Turing_rule 'D 2 2 left D' /* " " " " " " " */
call Turing_rule 'D 3 1 right A' /* " " " " " " " */
call Turing_rule 'E 1 1 left E' /* " " " " " " " */
call Turing_rule 'E 0 0 right halt' /* " " " " " " " */
call Turing_init 1 2 2 1 2 2 1 2 1 2 1 2 1 2 /*initialize the tape to some string(s)*/
call TM /*go and invoke the Turning machine. */
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
TM: ∙∙∙
output
rule 1 ═══► A 1 1 right A
rule 2 ═══► A 2 3 right B
rule 3 ═══► A 0 0 left E
rule 4 ═══► B 1 1 right B
rule 5 ═══► B 2 2 right B
rule 6 ═══► B 0 0 left C
rule 7 ═══► C 1 2 left D
rule 8 ═══► C 2 2 left C
rule 9 ═══► C 3 2 left E
rule 10 ═══► D 1 1 left D
rule 11 ═══► D 2 2 left D
rule 12 ═══► D 3 1 right A
rule 13 ═══► E 1 1 left E
rule 14 ═══► E 0 0 right halt
tape's contents: 11111122222222
tape's length: 16
Turning machine used 14 rules in 118 cycles.
Ruby
The universal machine
class Turing
class Tape
def initialize(symbols, blank, starting_tape)
@symbols = symbols
@blank = blank
@tape = starting_tape
@index = 0
end
def read
retval = @tape[@index]
unless retval
retval = @tape[@index] = @blank
end
raise "invalid symbol '#{retval}' on tape" unless @tape.member?(retval)
return retval
end
def write(symbol)
@tape[@index] = symbol
end
def right
@index += 1
end
def left
if @index == 0
@tape.unshift @blank
else
@index -= 1
end
end
def stay
# nop
end
def get_tape
return @tape
end
end
def initialize(symbols, blank,
initial_state, halt_states, running_states,
rules, starting_tape = [])
@tape = Tape.new(symbols, blank, starting_tape)
@initial_state = initial_state
@halt_states = halt_states
@running_states = running_states
@rules = rules
@halted = false
end
def run
raise "machine already halted" if @halted
state = @initial_state
while (true)
break if @halt_states.member? state
raise "unknown state '#{state}'" unless @running_states.member? state
symbol = @tape.read
outsym, action, state = @rules[state][symbol]
@tape.write outsym
@tape.send action
end
@halted = true
return @tape.get_tape
end
end
The incrementer machine
incrementer_rules = {
:q0 => { 1 => [1, :right, :q0],
:b => [1, :stay, :qf]}
}
t = Turing.new([:b, 1], # permitted symbols
:b, # blank symbol
:q0, # starting state
[:qf], # terminating states
[:q0], # running states
incrementer_rules, # operating rules
[1, 1, 1]) # starting tape
print t.run, "\n"
The busy beaver machine
busy_beaver_rules = {
:a => { 0 => [1, :right, :b],
1 => [1, :left, :c]},
:b => { 0 => [1, :left, :a],
1 => [1, :right, :b]},
:c => { 0 => [1, :left, :b],
1 => [1, :stay, :halt]}
}
t = Turing.new([0, 1], # permitted symbols
0, # blank symbol
:a, # starting state
[:halt], # terminating states
[:a, :b, :c], # running states
busy_beaver_rules, # operating rules
[]) # starting tape
print t.run, "\n"
Rust
use std::collections::VecDeque;
use std::fmt::{Display, Formatter, Result};
fn main() {
println!("Simple incrementer");
let rules_si = vec!(
Rule::new("q0", '1', '1', Direction::Right, "q0"),
Rule::new("q0", 'B', '1', Direction::Stay, "qf")
);
let states_si = vec!("q0", "qf");
let terminating_states_si = vec!("qf");
let permissible_symbols_si = vec!('B', '1');
let mut tm_si = TM::new(states_si, "q0", terminating_states_si, permissible_symbols_si, 'B', rules_si, "111");
while !tm_si.is_done() {
println!("{}", tm_si);
tm_si.step();
}
println!("___________________");
println!("Three-state busy beaver");
let rules_bb3 = vec!(
Rule::new("a", '0', '1', Direction::Right, "b"),
Rule::new("a", '1', '1', Direction::Left, "c"),
Rule::new("b", '0', '1', Direction::Left, "a"),
Rule::new("b", '1', '1', Direction::Right, "b"),
Rule::new("c", '0', '1', Direction::Left, "b"),
Rule::new("c", '1', '1', Direction::Stay, "halt"),
);
let states_bb3 = vec!("a", "b", "c", "halt");
let terminating_states_bb3 = vec!("halt");
let permissible_symbols_bb3 = vec!('0', '1');
let mut tm_bb3 = TM::new(states_bb3 ,"a", terminating_states_bb3, permissible_symbols_bb3, '0', rules_bb3, "0");
while !tm_bb3.is_done() {
println!("{}", tm_bb3);
tm_bb3.step();
}
println!("{}", tm_bb3);
println!("___________________");
println!("Five-state busy beaver");
let rules_bb5 = vec!(
Rule::new("A", '0', '1', Direction::Right, "B"),
Rule::new("A", '1', '1', Direction::Left, "C"),
Rule::new("B", '0', '1', Direction::Right, "C"),
Rule::new("B", '1', '1', Direction::Right, "B"),
Rule::new("C", '0', '1', Direction::Right, "D"),
Rule::new("C", '1', '0', Direction::Left, "E"),
Rule::new("D", '0', '1', Direction::Left, "A"),
Rule::new("D", '1', '1', Direction::Left, "D"),
Rule::new("E", '0', '1', Direction::Stay, "H"),
Rule::new("E", '1', '0', Direction::Left, "A"),
);
let states_bb5 = vec!("A", "B", "C", "D", "E", "H");
let terminating_states_bb5 = vec!("H");
let permissible_symbols_bb5 = vec!('0', '1');
let mut tm_bb5 = TM::new(states_bb5 ,"A", terminating_states_bb5, permissible_symbols_bb5, '0', rules_bb5, "0");
let mut steps = 0;
while !tm_bb5.is_done() {
tm_bb5.step();
steps += 1;
}
println!("Steps: {}", steps);
println!("Band lenght: {}", tm_bb5.band.len());
}
struct TM<'a> {
state: &'a str,
terminating_states: Vec<&'a str>,
rules: Vec<Rule<'a>>,
band: VecDeque<char>,
head: usize,
blank: char,
}
struct Rule<'a> {
state: &'a str,
read: char,
write: char,
dir: Direction,
new_state: &'a str,
}
enum Direction{
Left,
Right,
Stay,
}
impl<'a> TM<'a> {
fn new(_states: Vec<&'a str>, initial_state: &'a str, terminating_states: Vec<&'a str>, _permissible_symbols: Vec<char>, blank: char, rules: Vec<Rule<'a>>, input: &str) -> Self {
Self { state: initial_state, terminating_states, rules, band: input.chars().collect::<VecDeque<_>>(), head: 0, blank }
}
fn is_done(&self) -> bool {
self.terminating_states.contains(&self.state)
}
fn step(&mut self) {
let field = self.band.get(self.head).unwrap();
let rule = self.rules.iter().find(|rule| rule.state == self.state && &rule.read == field).unwrap();
let field = self.band.get_mut(self.head).unwrap();
*field = rule.write;
self.state = rule.new_state;
match rule.dir {
Direction::Left => {
if self.head == 0 {
self.band.push_front(self.blank)
} else {
self.head -= 1;
}
},
Direction::Right => {
if self.head == self.band.len() - 1 {
self.band.push_back(self.blank)
}
self.head += 1;
},
Direction::Stay => {},
}
}
}
impl<'a> Display for TM<'a> {
fn fmt(&self, f: &mut Formatter<'_>) -> Result {
let band = self.band.iter().enumerate().map(|(i, c)| {
if i == self.head {
format!("[{}]", c)
} else {
format!(" {} ", c)
}
}).fold(String::new(), |acc, val| acc + &val);
write!(f, "{}\t{}", self.state, band)
}
}
impl<'a> Rule<'a> {
fn new(state: &'a str, read: char, write: char, dir: Direction, new_state: &'a str) -> Self {
Self { state, read, write, dir, new_state }
}
}
- Output:
Simple incrementer q0 [1] 1 1 q0 1 [1] 1 q0 1 1 [1] q0 1 1 1 [B] ___________________ Three-state busy beaver a [0] b 1 [0] a [1] 1 c [0] 1 1 b [0] 1 1 1 a [0] 1 1 1 1 b 1 [1] 1 1 1 b 1 1 [1] 1 1 b 1 1 1 [1] 1 b 1 1 1 1 [1] b 1 1 1 1 1 [0] a 1 1 1 1 [1] 1 c 1 1 1 [1] 1 1 halt 1 1 1 [1] 1 1 ___________________ Five-state busy beaver Steps: 47176870 Band lenght: 12289
Scala
A simple implementation of Universal Turing Machine in Scala:
package utm.scala
import scala.annotation.tailrec
import scala.language.implicitConversions
/**
* Implementation of Universal Turing Machine in Scala that can simulate an arbitrary
* Turing machine on arbitrary input
*
* @author Abdulla Abdurakhmanov (https://github.com/abdolence/utms)
*/
class UniversalTuringMachine[S](
val rules: List[UTMRule[S]],
val initialState: S,
val finalStates: Set[S],
val blankSymbol: String,
val inputTapeVals: Iterable[String],
printEveryIter: Int = 1
) {
private val initialTape = UTMTape( inputTapeVals.toVector, 0, blankSymbol )
@tailrec
private def iterate( state: S, curIteration: Int, tape: UTMTape ): UTMTape = {
if (curIteration % printEveryIter == 0) {
print( s"${curIteration}: ${state}: " )
tape.printTape()
}
if (finalStates.contains( state )) {
println( s"Finished in the final state: ${state}" )
tape.printTape()
tape
} else {
rules.find(rule => rule.state == state && rule.fromSymbol == tape.current() ) match {
case Some( rule ) => {
val updatedTape = tape.updated( rule.toSymbol, rule.action )
iterate( rule.toState, curIteration + 1, updatedTape )
}
case _ => {
println(
s"Finished: no suitable rules found for ${state}/${tape.current()}"
)
tape.printTape()
tape
}
}
}
}
def run(): UTMTape =
iterate( state = initialState, curIteration = 0, tape = initialTape )
}
/**
* Universal Turing Machine actions
*/
sealed trait UTMAction
object UTMAction {
case object left extends UTMAction
case object right extends UTMAction
case object stay extends UTMAction
}
/**
* Universal Turing Machine rule definition
*/
case class UTMRule[S]( state: S, fromSymbol: String, toSymbol: String, action: UTMAction, toState: S )
object UTMRule {
implicit def tupleToUTMLRule[S](
tuple: ( S, String, String, UTMAction, S )
): UTMRule[S] =
UTMRule[S]( tuple._1, tuple._2, tuple._3, tuple._4, tuple._5 )
}
/**
* Universal Turing Machine Tape
*/
case class UTMTape( content: Vector[String], position: Int, blankSymbol: String ) {
private def updateContentAtPos( symbol: String ) = {
if (position >= content.length) {
content :+ symbol
} else if (position < 0) {
symbol +: content
} else if (content( position ) != symbol)
content.updated( position, symbol )
else
content
}
private[scala] def updated( symbol: String, action: UTMAction ): UTMTape = {
val updatedTape =
this.copy( content = updateContentAtPos( symbol ), position = action match {
case UTMAction.left => position - 1
case UTMAction.right => position + 1
case UTMAction.stay => position
} )
if (updatedTape.position < 0) {
updatedTape.copy(
content = blankSymbol +: updatedTape.content,
position = 0
)
} else if (updatedTape.position >= updatedTape.content.length) {
updatedTape.copy( content = updatedTape.content :+ blankSymbol )
} else
updatedTape
}
private[scala] def current(): String = {
if (content.isDefinedAt( position ))
content( position )
else
blankSymbol
}
def printTape(): Unit = {
print( "[" )
if (position < 0)
print( "˅" )
content.zipWithIndex.foreach {
case ( symbol, index ) =>
if (position == index)
print( "˅" )
else
print( " " )
print( s"$symbol" )
}
if (position >= content.length)
print( "˅" )
println( "]" )
}
}
object UniversalTuringMachine extends App {
main()
def main(): Unit = {
import UTMAction._
def createIncrementMachine() = {
sealed trait IncrementStates
case object q0 extends IncrementStates
case object qf extends IncrementStates
new UniversalTuringMachine[IncrementStates](
rules = List( ( q0, "1", "1", right, q0 ), ( q0, "B", "1", stay, qf ) ),
initialState = q0,
finalStates = Set( qf ),
blankSymbol = "B",
inputTapeVals = Seq( "1", "1", "1" )
).run()
}
def createThreeStateBusyBeaver() = {
sealed trait ThreeStateBusyStates
case object a extends ThreeStateBusyStates
case object b extends ThreeStateBusyStates
case object c extends ThreeStateBusyStates
case object halt extends ThreeStateBusyStates
new UniversalTuringMachine[ThreeStateBusyStates](
rules = List(
( a, "0", "1", right, b ),
( a, "1", "1", left, c ),
( b, "0", "1", left, a ),
( b, "1", "1", right, b ),
( c, "0", "1", left, b ),
( c, "1", "1", stay, halt )
),
initialState = a,
finalStates = Set( halt ),
blankSymbol = "0",
inputTapeVals = Seq()
).run()
}
def createFiveState2SymBusyBeaverMachine() = {
sealed trait FiveBeaverStates
case object FA extends FiveBeaverStates
case object FB extends FiveBeaverStates
case object FC extends FiveBeaverStates
case object FD extends FiveBeaverStates
case object FE extends FiveBeaverStates
case object FH extends FiveBeaverStates
new UniversalTuringMachine[FiveBeaverStates](
rules = List(
( FA, "0", "1", right, FB ),
( FA, "1", "1", left, FC ),
( FB, "0", "1", right, FC ),
( FB, "1", "1", right, FB ),
( FC, "0", "1", right, FD ),
( FC, "1", "0", left, FE ),
( FD, "0", "1", left, FA ),
( FD, "1", "1", left, FD ),
( FE, "0", "1", stay, FH ),
( FE, "1", "0", left, FA )
),
initialState = FA,
finalStates = Set( FH ),
blankSymbol = "0",
inputTapeVals = Seq(),
printEveryIter = 100000
).run()
}
createIncrementMachine()
createThreeStateBusyBeaver()
// careful here, 47 mln iterations
createFiveState2SymBusyBeaverMachine()
}
}
- Output:
0: q0(): [˅1 1 1] 1: q0(): [ 1˅1 1] 2: q0(): [ 1 1˅1] 3: q0(): [ 1 1 1˅B] 4: qf(): [ 1 1 1˅1] Finished in the final state: qf() [ 1 1 1˅1] 0: a(): [˅] 1: b(): [ 1˅0] 2: a(): [˅1 1] 3: c(): [˅0 1 1] 4: b(): [˅0 1 1 1] 5: a(): [˅0 1 1 1 1] 6: b(): [ 1˅1 1 1 1] 7: b(): [ 1 1˅1 1 1] 8: b(): [ 1 1 1˅1 1] 9: b(): [ 1 1 1 1˅1] 10: b(): [ 1 1 1 1 1˅0] 11: a(): [ 1 1 1 1˅1 1] 12: c(): [ 1 1 1˅1 1 1] 13: halt(): [ 1 1 1˅1 1 1] Finished in the final state: halt()
Scheme
The Implementation
;----------------------------------------------------------------------------------------------
; The tape is a doubly-linked list of "cells". Each cell is a pair in which the cdr points
; to the cell on its right, and the car is a vector containing: 0: the value of the cell;
; 1: pointer to the cell on this cell's left; 2: #t if the cell has never been written.
; Make a new cell with the given contents, but linked to no other cell(s).
; (This is the only place that a cell can be marked as un-written.)
(define make-cell
(lambda (val . opt-unwrit)
(list (vector val '() (if (pair? opt-unwrit) (car opt-unwrit) #f)))))
; Return the un-written flag of the cell.
(define cell-unwrit?
(lambda (cell)
(vector-ref (car cell) 2)))
; Return the value of the cell.
(define cell-get
(lambda (cell)
(vector-ref (car cell) 0)))
; Store the value of the cell.
; Clears the un-written flag of the cell.
(define cell-set!
(lambda (cell val)
(vector-set! (car cell) 0 val)
(vector-set! (car cell) 2 #f)))
; Return the cell to the right of the given cell on the tape.
; Returns () if there is no cell to the right.
(define cell-right
(lambda (cell)
(cdr cell)))
; Return the cell to the left of the given cell on the tape.
; Returns () if there is no cell to the left.
(define cell-left
(lambda (cell)
(vector-ref (car cell) 1)))
; Return the cell to the right of the given cell on the tape.
; Extends the tape with the give blank symbol if there is no cell to the right.
; Optionally, passes the given un-written flag to make-cell (if needed).
(define cell-extend-right
(lambda (cell blank . opt-unwrit)
(if (null? (cdr cell))
(let ((new (if (pair? opt-unwrit) (make-cell blank (car opt-unwrit)) (make-cell blank))))
(vector-set! (car new) 1 cell)
(set-cdr! cell new)
new)
(cell-right cell))))
; Return the cell to the left of the given cell on the tape.
; Extends the tape with the give blank symbol if there is no cell to the left.
; Optionally, passes the given un-written flag to make-cell (if needed).
(define cell-extend-left
(lambda (cell blank . opt-unwrit)
(if (null? (vector-ref (car cell) 1))
(let ((new (if (pair? opt-unwrit) (make-cell blank (car opt-unwrit)) (make-cell blank))))
(set-cdr! new cell)
(vector-set! (car cell) 1 new)
new)
(cell-left cell))))
; Make a new tape whose cells contain the values in the given list.
; Optionally, pad the tape per the given blank symbol, left-padding and right-padding amounts.
(define make-tape
(lambda (values . opt-pads)
(unless (pair? values) (error 'make-tape "values argument is not a list" pads))
(let* ((tape (make-cell (car values)))
(last (do ((values (cdr values) (cdr values))
(cell tape (cell-extend-right cell (car values))))
((null? values) cell))))
(when (pair? opt-pads)
(let ((blank (list-ref opt-pads 0))
(left (list-ref opt-pads 1))
(right (list-ref opt-pads 2)))
(unless (and (integer? left) (integer? right))
(error 'make-tape "padding arguments must be integers" opt-pads))
(do ((count 0 (1+ count))
(cell last (cell-extend-right cell blank #t)))
((>= count right)))
(do ((count 0 (1+ count))
(cell tape (cell-extend-left cell blank #t)))
((>= count left)))))
tape)))
; Make a deep copy of the given tape.
; Note: Only copies from the given cell forward.
(define tape-copy
(lambda (tape)
(let ((copy (make-cell (cell-get tape))))
(do ((tape (cdr tape) (cdr tape))
(cell copy (cell-extend-right cell (cell-get tape))))
((null? tape)))
copy)))
; Return the first cell on a tape.
; Optionally, leading blank symbols are not included (will return last cell of blank tape).
(define tape-fst
(lambda (cell . opt-blank)
(let ((fst (do ((fst cell (cell-left fst))) ((null? (cell-left fst)) fst))))
(if (null? opt-blank)
fst
(do ((fst fst (cell-right fst)))
((or (null? (cell-right fst)) (not (eq? (car opt-blank) (cell-get fst)))) fst))))))
; Return the last cell on a tape.
; Optionally, trailing blank symbols are not included (will return first cell of blank tape).
(define tape-lst
(lambda (cell . opt-blank)
(let ((lst (do ((lst cell (cell-right lst))) ((null? (cell-right lst)) lst))))
(if (null? opt-blank)
lst
(do ((lst lst (cell-left lst)))
((or (null? (cell-left lst)) (not (eq? (car opt-blank) (cell-get lst)))) lst))))))
; Return true if the given tape is empty. (I.e. contains nothing but blank symbols.)
(define tape-empty?
(lambda (cell blank)
(let ((fst (tape-fst cell blank)))
(and (null? (cell-right fst)) (eq? blank (cell-get fst))))))
; Convert the contents of a tape to a string.
; Place a mark around the indicated cell (if any match).
; Prints the entire contents regardless of which cell is given.
; Optionally, leading and trailing instances of the given blank symbol are suppressed.
; The values of un-written cells are not shown, though space for them is included.
(define tape->string
(lambda (cell mark . opt-blank)
(let ((strlst (list #\[))
(marked-prev #f)
(fst (if (null? opt-blank) (tape-fst cell) (tape-fst cell (car opt-blank))))
(lst (if (null? opt-blank) (tape-lst cell) (tape-lst cell (car opt-blank)))))
(do ((cell fst (cell-right cell)))
((eq? cell (cell-right lst)))
(let* ((mark-now (eq? cell mark))
(fmtstr (cond (mark-now " {~a}") (marked-prev " ~a") (else " ~a")))
(value (if (and (not mark-now) (cell-unwrit? cell)) " " (cell-get cell))))
(set! strlst (append strlst (string->list (format fmtstr value))))
(set! marked-prev mark-now)))
(list->string (append strlst (string->list (if marked-prev " ]" " ]")))))))
;----------------------------------------------------------------------------------------------
; A Turing Machine contains the 7-tuple that formally defines it, stored in an array to
; make access relatively fast. The transitions are stored in an association list keyed by
; the pair (q_i . s_j) for ease of lookup.
; Make a new Turing Machine from the given arguments:
; Symbols list, blank symbol, input symbols list, states list, initial state, final (accepting)
; states list, and list of transitions. A transition is a 5-element list of: state (q_i),
; symbol read (s_i), symbol to write (s_ij), direction to move (d_ij), and next state (q_ij).
(define make-turing
(lambda (symbols blank inputs states initial finals . transitions)
; Raise error if any element in list lst is not in list set.
(define all-list-in-set
(lambda (set lst msg)
(for-each (lambda (val) (unless (memq val set) (error 'make-turing msg val))) lst)))
; Raise error if the given transition is not correctly formed.
(define transition-validate
(lambda (tran)
(when (or (not (list? tran)) (not (= 5 (length tran))))
(error 'make-turing "transition not a length-5 list" tran))
(let ((q_i (list-ref tran 0))
(s_j (list-ref tran 1))
(s_ij (list-ref tran 2))
(d_ij (list-ref tran 3))
(q_ij (list-ref tran 4)))
(unless (memq q_i states) (error 'make-turing "q_i not in states" q_i))
(unless (memq s_j symbols) (error 'make-turing "s_j not in symbols" s_j))
(unless (memq s_ij symbols) (error 'make-turing "s_ij not in symbols" s_ij))
(unless (memq d_ij '(L R N)) (error 'make-turing "d_ij not in {L R N}" d_ij))
(unless (memq q_ij states) (error 'make-turing "q_ij not in states" q_ij)))))
; Convert the given transitions list into an alist of transitions keyed by (q_i . s_j).
(define transitions-alist
(lambda (trns)
(cond ((null? trns) '())
(else (cons (cons (cons (caar trns) (cadar trns)) (list (cddar trns)))
(transitions-alist (cdr trns)))))))
; Validate all the arguments.
(unless (list? symbols) (error 'make-turing "symbols not a list" symbols))
(unless (memq blank symbols) (error 'make-turing "blank not in symbols" blank))
(all-list-in-set symbols inputs "inputs not all in symbols")
(unless (list? states) (error 'make-turing "states not a list" states))
(unless (memq initial states) (error 'make-turing "initial not in states" initial))
(all-list-in-set states finals "finals not all in states")
(for-each (lambda (tran) (transition-validate tran)) transitions)
; Construct and return the Turing Machine tuple vector.
(let ((tuple (make-vector 7)))
(vector-set! tuple 0 symbols)
(vector-set! tuple 1 blank)
(vector-set! tuple 2 inputs)
(vector-set! tuple 3 states)
(vector-set! tuple 4 initial)
(vector-set! tuple 5 finals)
(vector-set! tuple 6 (transitions-alist transitions))
tuple)))
; Return the symbols of a Turing Machine.
(define-syntax turing-symbols (syntax-rules () ((_ tm) (vector-ref tm 0))))
; Return the blank symbol of a Turing Machine.
(define-syntax turing-blank (syntax-rules () ((_ tm) (vector-ref tm 1))))
; Return the input symbols of a Turing Machine.
(define-syntax turing-inputs (syntax-rules () ((_ tm) (vector-ref tm 2))))
; Return the states of a Turing Machine.
(define-syntax turing-states (syntax-rules () ((_ tm) (vector-ref tm 3))))
; Return the initial state of a Turing Machine.
(define-syntax turing-initial (syntax-rules () ((_ tm) (vector-ref tm 4))))
; Return the final states of a Turing Machine.
(define-syntax turing-finals (syntax-rules () ((_ tm) (vector-ref tm 5))))
; Return the transitions of a Turing Machine.
(define-syntax turing-transitions (syntax-rules () ((_ tm) (vector-ref tm 6))))
; Return the q_i (current state) of alist element transition.
(define-syntax tran-q_i (syntax-rules () ((_ atran) (car (car atran)))))
; Return the s_j (symbol read from the tape) of alist element transition.
(define-syntax tran-s_j (syntax-rules () ((_ atran) (cdr (car atran)))))
; Return the s_ij (symbol written) of alist element transition.
(define-syntax tran-s_ij (syntax-rules () ((_ atran) (car (cadr atran)))))
; Return the d_ij (direction of move) of alist element transition.
(define-syntax tran-d_ij (syntax-rules () ((_ atran) (cadr (cadr atran)))))
; Return the q_ij (state transition) of alist element transition.
(define-syntax tran-q_ij (syntax-rules () ((_ atran) (caddr (cadr atran)))))
; Lookup the transition matching the given state and symbol in the given Turing Machine.
(define atrns-lookup
(lambda (state symbol tm)
(assoc (cons state symbol) (turing-transitions tm))))
; Convert the given Turing Machine transition to a string.
(define tran->string
(lambda (atran)
(format "(~a ~a ~a ~a ~a)"
(tran-q_i atran) (tran-s_j atran) (tran-s_ij atran) (tran-d_ij atran) (tran-q_ij atran))))
; Convert the given Turing Machine definition to a string.
; Options (zero or more) are, in order: component prefix string (default "");
; component suffix string (default ""); component separator string (default newline).
(define turing->string
(lambda (tm . opts)
(let ((prestr (if (> (length opts) 0) (list-ref opts 0) ""))
(sufstr (if (> (length opts) 1) (list-ref opts 1) ""))
(sepstr (if (> (length opts) 2) (list-ref opts 2) (make-string 1 #\newline)))
(strlst '()))
(set! strlst (append strlst (string->list
(format "~a~a~a~a" prestr (turing-symbols tm) sufstr sepstr))))
(set! strlst (append strlst (string->list
(format "~a~a~a~a" prestr (turing-blank tm) sufstr sepstr))))
(set! strlst (append strlst (string->list
(format "~a~a~a~a" prestr (turing-inputs tm) sufstr sepstr))))
(set! strlst (append strlst (string->list
(format "~a~a~a~a" prestr (turing-states tm) sufstr sepstr))))
(set! strlst (append strlst (string->list
(format "~a~a~a~a" prestr (turing-initial tm) sufstr sepstr))))
(set! strlst (append strlst (string->list
(format "~a~a~a~a" prestr (turing-finals tm) sufstr
(if (> (length (turing-transitions tm)) 0) sepstr "")))))
(do ((index 0 (1+ index)))
((>= index (length (turing-transitions tm))))
(set! strlst (append strlst (string->list
(format "~a~a~a~a" prestr (tran->string (list-ref (turing-transitions tm) index)) sufstr
(if (< index (1- (length (turing-transitions tm)))) sepstr ""))))))
(list->string strlst))))
;----------------------------------------------------------------------------------------------
; Run the given Turing Machine on the given input tape.
; If specified, display log of progress. Optionally, abort after given number of iterations.
; Returns the count of iterations, the accepting state (if one; else void), and the output
; tape as multiple values.
(define turing-run
(lambda (tm cell show-log? . opt-abort)
; Validate contents of input tape. (Leading/trailing blanks allowed; internals are not.)
(unless (tape-empty? cell (turing-blank tm))
(let ((fst (tape-fst cell (turing-blank tm)))
(lst (tape-lst cell (turing-blank tm))))
(if (eq? fst lst)
(unless (memq (cell-get fst) (turing-symbols tm))
(error 'turing-run "input tape has disallowed content" (cell-get fst)))
(do ((cell fst (cell-right cell)))
((eq? cell (cell-right lst)))
(unless (memq (cell-get cell) (turing-inputs tm))
(error 'turing-run "input tape has disallowed content" (cell-get cell)))))))
; Initialize state and head.
(let ((state (turing-initial tm)) (head cell) (atran #f)
(abort (and (pair? opt-abort) (integer? (car opt-abort))
(> (car opt-abort) 0) (car opt-abort))))
; Loop until no transition matches state/symbol or reached a final state.
(do ((count 0 (1+ count))
(atran (atrns-lookup state (cell-get head) tm)
(atrns-lookup state (cell-get head) tm)))
((or (not atran) (memq state (turing-finals tm)) (and abort (>= count abort)))
; Display final progress (optional).
(when show-log?
(let* ((string (format "~a" state))
(strlen (string-length string))
(padlen (max 1 (- 25 strlen)))
(strpad (make-string padlen #\ )))
(printf "~a~a~a~%" string strpad (tape->string cell head))))
; Return resultant count, accepting state (or void), and tape.
(values count (if (memq state (turing-finals tm)) state (void)) head))
; Display progress (optional).
(when show-log?
(let* ((string (format "~a ~a -> ~a ~a ~a" state (cell-get head)
(tran-s_ij atran) (tran-d_ij atran) (tran-q_ij atran)))
(strlen (string-length string))
(padlen (max 1 (- 25 strlen)))
(strpad (make-string padlen #\ )))
(printf "~a~a~a~%" string strpad (tape->string cell head))))
; Iterate.
(cell-set! head (tran-s_ij atran))
(set! state (tran-q_ij atran))
(set! head (case (tran-d_ij atran)
((L) (cell-extend-left head (turing-blank tm)))
((R) (cell-extend-right head (turing-blank tm)))
((N) head)))))))
;----------------------------------------------------------------------------------------------
Test Runner
;----------------------------------------------------------------------------------------------
; Run specified tests: A caption string, a Turing machine, a list of tests, and options (if
; 'notm present, do not output the Turing Machine definition (otherwise display it); if 'supp
; present, suppress leading/trailing blanks; 'mark present, mark the output tape; if 'supp
; present, suppress leading/trailing blanks; if 'leng present, print only the length of the
; output tape, not the contents of either; if 'show present, show an empty input tape (by
; default empty inputs are not shown)). A test is a list of: limit count (0 = unlimited),
; #t to log progress, and the input tape.
(define run-tm-tests
(lambda (caption tm test-lst . opts)
(printf "~%~a...~%" caption)
(unless (memq 'notm opts) (printf "~%~a~%" (turing->string tm)))
(let ((input #f))
(let loop ((tests test-lst))
(unless (null? tests)
(newline)
(set! input (tape-copy (caddar tests)))
(let-values (((count accepting output)
(turing-run tm (caddar tests) (cadar tests) (caar tests))))
(if (memq 'leng opts)
(printf "count = ~d~%accept = ~a~%output length = ~d~%"
count accepting (length output))
(let ((instr (if (memq 'supp opts)
(tape->string input #f (turing-blank tm))
(tape->string input #f)))
(outstr (if (memq 'supp opts)
(tape->string output (if (memq 'mark opts) output #f)
(turing-blank tm))
(tape->string output (if (memq 'mark opts) output #f)))))
(printf "count = ~d~%accept = ~a~%" count accepting)
(when (or (memq 'show opts) (not (tape-empty? input (turing-blank tm))))
(printf "input = ~a~%" instr))
(printf "output = ~a~%" outstr))))
(loop (cdr tests)))))))
;----------------------------------------------------------------------------------------------
The Task
(run-tm-tests
"Simple incrementer"
(make-turing
'(B 1)
'B
'(1)
'(q0 qf)
'q0
'(qf)
'(q0 1 1 R q0)
'(q0 B 1 N qf))
(list
(list 0 #t (make-tape '(1 1 1)))
(list 0 #t (make-tape '(B)))
) 'notm 'mark)
(run-tm-tests
"Three-state busy beaver"
(make-turing
'(0 1)
'0
'()
'(a b c halt)
'a
'(halt)
'(a 0 1 R b)
'(a 1 1 L c)
'(b 0 1 L a)
'(b 1 1 R b)
'(c 0 1 L b)
'(c 1 1 N halt))
(list
(list 0 #t (make-tape '(0) 0 3 2)) ; padding determined empirically
) 'notm 'mark)
(run-tm-tests
"5-state 2-symbol probable busy beaver"
(make-turing
'(0 1)
'0
'()
'(A B C D E H)
'A
'(H)
'(A 0 1 R B)
'(A 1 1 L C)
'(B 0 1 R C)
'(B 1 1 R B)
'(C 0 1 R D)
'(C 1 0 L E)
'(D 0 1 L A)
'(D 1 1 L D)
'(E 0 1 N H)
'(E 1 0 L A))
(list
(list 0 #f (make-tape '(0)))
) 'notm 'leng)
- Output:
Simple incrementer...
q0 1 -> 1 R q0 [ {1} 1 1 ]
q0 1 -> 1 R q0 [ 1 {1} 1 ]
q0 1 -> 1 R q0 [ 1 1 {1} ]
q0 B -> 1 N qf [ 1 1 1 {B} ]
qf [ 1 1 1 {1} ]
count = 4
accept = qf
input = [ 1 1 1 ]
output = [ 1 1 1 {1} ]
q0 B -> 1 N qf [ {B} ]
qf [ {1} ]
count = 1
accept = qf
output = [ {1} ]
Three-state busy beaver...
a 0 -> 1 R b [ {0} ]
b 0 -> 1 L a [ 1 {0} ]
a 1 -> 1 L c [ {1} 1 ]
c 0 -> 1 L b [ {0} 1 1 ]
b 0 -> 1 L a [ {0} 1 1 1 ]
a 0 -> 1 R b [ {0} 1 1 1 1 ]
b 1 -> 1 R b [ 1 {1} 1 1 1 ]
b 1 -> 1 R b [ 1 1 {1} 1 1 ]
b 1 -> 1 R b [ 1 1 1 {1} 1 ]
b 1 -> 1 R b [ 1 1 1 1 {1} ]
b 0 -> 1 L a [ 1 1 1 1 1 {0} ]
a 1 -> 1 L c [ 1 1 1 1 {1} 1 ]
c 1 -> 1 N halt [ 1 1 1 {1} 1 1 ]
halt [ 1 1 1 {1} 1 1 ]
count = 13
accept = halt
output = [ 1 1 1 {1} 1 1 ]
5-state 2-symbol probable busy beaver...
count = 47176870
accept = H
output length = 12289
More Examples
(run-tm-tests
"Sorting test"
(make-turing
'(0 1 2 3)
'0
'(1 2)
'(A B C D E STOP)
'A
'(STOP)
'(A 1 1 R A)
'(A 2 3 R B)
'(A 0 0 L E)
'(B 1 1 R B)
'(B 2 2 R B)
'(B 0 0 L C)
'(C 1 2 L D)
'(C 2 2 L C)
'(C 3 2 L E)
'(D 1 1 L D)
'(D 2 2 L D)
'(D 3 1 R A)
'(E 1 1 L E)
'(E 0 0 R STOP))
(list
(list 0 #t (make-tape '(2 2 2 1 2 2 1 2 1 2 1 2 1 2) 0 1 1)) ; padding determined empirically
) 'notm 'supp)
(run-tm-tests
"Duplicate sequence of 1s"
(make-turing
'(0 1)
'0
'(1)
'(s1 s2 s3 s4 s5 H)
's1
'(H)
'(s1 0 0 N H)
'(s1 1 0 R s2)
'(s2 0 0 R s3)
'(s2 1 1 R s2)
'(s3 0 1 L s4)
'(s3 1 1 R s3)
'(s4 0 0 L s5)
'(s4 1 1 L s4)
'(s5 0 1 R s1)
'(s5 1 1 L s5))
(list
(list 0 #t (make-tape '(1 1 1) 0 0 4)) ; padding determined empirically
) 'notm 'supp)
(run-tm-tests
"Turing's first example from On Computable Numbers"
(make-turing
'(_ 0 1)
'_
'(_)
'(b c e f)
'b
'()
'(b _ 0 R c)
'(c _ _ R e)
'(e _ 1 R f)
'(f _ _ R b))
(list
(list 20 #t (make-tape '(_)))
) 'notm 'mark)
(run-tm-tests
"Palindrome checker"
(make-turing
'(_ 1 2)
'_
'(1 2)
'(br r1 e1 r2 e2 wl odd even)
'br
'(odd even)
; branch to look for 1 or 2 at end
'(br 1 _ R r1)
'(br 2 _ R r2)
'(br _ _ N even)
; walk right to end for 1
'(r1 1 1 R r1)
'(r1 2 2 R r1)
'(r1 _ _ L e1)
; check end symbol for 1
'(e1 1 _ L wl)
'(e1 _ _ N odd)
; walk right to end for 2
'(r2 2 2 R r2)
'(r2 1 1 R r2)
'(r2 _ _ L e2)
; check end symbol for 2
'(e2 2 _ L wl)
'(e2 _ _ N odd)
; walk left to beginning
'(wl 1 1 L wl)
'(wl 2 2 L wl)
'(wl _ _ R br))
(list
(list 0 #t (make-tape '(1 2 1)))
(list 0 #t (make-tape '(1 2 2)))
(list 0 #t (make-tape '(1 1)))
(list 0 #t (make-tape '(2 1)))
(list 0 #t (make-tape '(1)))
(list 0 #f (make-tape '(2 1 1 1 2)))
(list 0 #f (make-tape '(2 1 1 2 2)))
(list 0 #f (make-tape '(1 1 2 2 1 1)))
(list 0 #f (make-tape '(1 1 2 2 1 2)))
) 'notm 'mark)
- Output:
Sorting test...
A 2 -> 3 R B [ {2} 2 2 1 2 2 1 2 1 2 1 2 1 2 ]
B 2 -> 2 R B [ 3 {2} 2 1 2 2 1 2 1 2 1 2 1 2 ]
B 2 -> 2 R B [ 3 2 {2} 1 2 2 1 2 1 2 1 2 1 2 ]
B 1 -> 1 R B [ 3 2 2 {1} 2 2 1 2 1 2 1 2 1 2 ]
B 2 -> 2 R B [ 3 2 2 1 {2} 2 1 2 1 2 1 2 1 2 ]
B 2 -> 2 R B [ 3 2 2 1 2 {2} 1 2 1 2 1 2 1 2 ]
B 1 -> 1 R B [ 3 2 2 1 2 2 {1} 2 1 2 1 2 1 2 ]
B 2 -> 2 R B [ 3 2 2 1 2 2 1 {2} 1 2 1 2 1 2 ]
B 1 -> 1 R B [ 3 2 2 1 2 2 1 2 {1} 2 1 2 1 2 ]
B 2 -> 2 R B [ 3 2 2 1 2 2 1 2 1 {2} 1 2 1 2 ]
B 1 -> 1 R B [ 3 2 2 1 2 2 1 2 1 2 {1} 2 1 2 ]
B 2 -> 2 R B [ 3 2 2 1 2 2 1 2 1 2 1 {2} 1 2 ]
B 1 -> 1 R B [ 3 2 2 1 2 2 1 2 1 2 1 2 {1} 2 ]
B 2 -> 2 R B [ 3 2 2 1 2 2 1 2 1 2 1 2 1 {2} ]
B 0 -> 0 L C [ 3 2 2 1 2 2 1 2 1 2 1 2 1 2 {0} ]
C 2 -> 2 L C [ 3 2 2 1 2 2 1 2 1 2 1 2 1 {2} 0 ]
C 1 -> 2 L D [ 3 2 2 1 2 2 1 2 1 2 1 2 {1} 2 0 ]
D 2 -> 2 L D [ 3 2 2 1 2 2 1 2 1 2 1 {2} 2 2 0 ]
D 1 -> 1 L D [ 3 2 2 1 2 2 1 2 1 2 {1} 2 2 2 0 ]
D 2 -> 2 L D [ 3 2 2 1 2 2 1 2 1 {2} 1 2 2 2 0 ]
D 1 -> 1 L D [ 3 2 2 1 2 2 1 2 {1} 2 1 2 2 2 0 ]
D 2 -> 2 L D [ 3 2 2 1 2 2 1 {2} 1 2 1 2 2 2 0 ]
D 1 -> 1 L D [ 3 2 2 1 2 2 {1} 2 1 2 1 2 2 2 0 ]
D 2 -> 2 L D [ 3 2 2 1 2 {2} 1 2 1 2 1 2 2 2 0 ]
D 2 -> 2 L D [ 3 2 2 1 {2} 2 1 2 1 2 1 2 2 2 0 ]
D 1 -> 1 L D [ 3 2 2 {1} 2 2 1 2 1 2 1 2 2 2 0 ]
D 2 -> 2 L D [ 3 2 {2} 1 2 2 1 2 1 2 1 2 2 2 0 ]
D 2 -> 2 L D [ 3 {2} 2 1 2 2 1 2 1 2 1 2 2 2 0 ]
D 3 -> 1 R A [ {3} 2 2 1 2 2 1 2 1 2 1 2 2 2 0 ]
A 2 -> 3 R B [ 1 {2} 2 1 2 2 1 2 1 2 1 2 2 2 0 ]
B 2 -> 2 R B [ 1 3 {2} 1 2 2 1 2 1 2 1 2 2 2 0 ]
B 1 -> 1 R B [ 1 3 2 {1} 2 2 1 2 1 2 1 2 2 2 0 ]
B 2 -> 2 R B [ 1 3 2 1 {2} 2 1 2 1 2 1 2 2 2 0 ]
B 2 -> 2 R B [ 1 3 2 1 2 {2} 1 2 1 2 1 2 2 2 0 ]
B 1 -> 1 R B [ 1 3 2 1 2 2 {1} 2 1 2 1 2 2 2 0 ]
B 2 -> 2 R B [ 1 3 2 1 2 2 1 {2} 1 2 1 2 2 2 0 ]
B 1 -> 1 R B [ 1 3 2 1 2 2 1 2 {1} 2 1 2 2 2 0 ]
B 2 -> 2 R B [ 1 3 2 1 2 2 1 2 1 {2} 1 2 2 2 0 ]
B 1 -> 1 R B [ 1 3 2 1 2 2 1 2 1 2 {1} 2 2 2 0 ]
B 2 -> 2 R B [ 1 3 2 1 2 2 1 2 1 2 1 {2} 2 2 0 ]
B 2 -> 2 R B [ 1 3 2 1 2 2 1 2 1 2 1 2 {2} 2 0 ]
B 2 -> 2 R B [ 1 3 2 1 2 2 1 2 1 2 1 2 2 {2} 0 ]
B 0 -> 0 L C [ 1 3 2 1 2 2 1 2 1 2 1 2 2 2 {0} ]
C 2 -> 2 L C [ 1 3 2 1 2 2 1 2 1 2 1 2 2 {2} 0 ]
C 2 -> 2 L C [ 1 3 2 1 2 2 1 2 1 2 1 2 {2} 2 0 ]
C 2 -> 2 L C [ 1 3 2 1 2 2 1 2 1 2 1 {2} 2 2 0 ]
C 1 -> 2 L D [ 1 3 2 1 2 2 1 2 1 2 {1} 2 2 2 0 ]
D 2 -> 2 L D [ 1 3 2 1 2 2 1 2 1 {2} 2 2 2 2 0 ]
D 1 -> 1 L D [ 1 3 2 1 2 2 1 2 {1} 2 2 2 2 2 0 ]
D 2 -> 2 L D [ 1 3 2 1 2 2 1 {2} 1 2 2 2 2 2 0 ]
D 1 -> 1 L D [ 1 3 2 1 2 2 {1} 2 1 2 2 2 2 2 0 ]
D 2 -> 2 L D [ 1 3 2 1 2 {2} 1 2 1 2 2 2 2 2 0 ]
D 2 -> 2 L D [ 1 3 2 1 {2} 2 1 2 1 2 2 2 2 2 0 ]
D 1 -> 1 L D [ 1 3 2 {1} 2 2 1 2 1 2 2 2 2 2 0 ]
D 2 -> 2 L D [ 1 3 {2} 1 2 2 1 2 1 2 2 2 2 2 0 ]
D 3 -> 1 R A [ 1 {3} 2 1 2 2 1 2 1 2 2 2 2 2 0 ]
A 2 -> 3 R B [ 1 1 {2} 1 2 2 1 2 1 2 2 2 2 2 0 ]
B 1 -> 1 R B [ 1 1 3 {1} 2 2 1 2 1 2 2 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 3 1 {2} 2 1 2 1 2 2 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 3 1 2 {2} 1 2 1 2 2 2 2 2 0 ]
B 1 -> 1 R B [ 1 1 3 1 2 2 {1} 2 1 2 2 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 3 1 2 2 1 {2} 1 2 2 2 2 2 0 ]
B 1 -> 1 R B [ 1 1 3 1 2 2 1 2 {1} 2 2 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 3 1 2 2 1 2 1 {2} 2 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 3 1 2 2 1 2 1 2 {2} 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 3 1 2 2 1 2 1 2 2 {2} 2 2 0 ]
B 2 -> 2 R B [ 1 1 3 1 2 2 1 2 1 2 2 2 {2} 2 0 ]
B 2 -> 2 R B [ 1 1 3 1 2 2 1 2 1 2 2 2 2 {2} 0 ]
B 0 -> 0 L C [ 1 1 3 1 2 2 1 2 1 2 2 2 2 2 {0} ]
C 2 -> 2 L C [ 1 1 3 1 2 2 1 2 1 2 2 2 2 {2} 0 ]
C 2 -> 2 L C [ 1 1 3 1 2 2 1 2 1 2 2 2 {2} 2 0 ]
C 2 -> 2 L C [ 1 1 3 1 2 2 1 2 1 2 2 {2} 2 2 0 ]
C 2 -> 2 L C [ 1 1 3 1 2 2 1 2 1 2 {2} 2 2 2 0 ]
C 2 -> 2 L C [ 1 1 3 1 2 2 1 2 1 {2} 2 2 2 2 0 ]
C 1 -> 2 L D [ 1 1 3 1 2 2 1 2 {1} 2 2 2 2 2 0 ]
D 2 -> 2 L D [ 1 1 3 1 2 2 1 {2} 2 2 2 2 2 2 0 ]
D 1 -> 1 L D [ 1 1 3 1 2 2 {1} 2 2 2 2 2 2 2 0 ]
D 2 -> 2 L D [ 1 1 3 1 2 {2} 1 2 2 2 2 2 2 2 0 ]
D 2 -> 2 L D [ 1 1 3 1 {2} 2 1 2 2 2 2 2 2 2 0 ]
D 1 -> 1 L D [ 1 1 3 {1} 2 2 1 2 2 2 2 2 2 2 0 ]
D 3 -> 1 R A [ 1 1 {3} 1 2 2 1 2 2 2 2 2 2 2 0 ]
A 1 -> 1 R A [ 1 1 1 {1} 2 2 1 2 2 2 2 2 2 2 0 ]
A 2 -> 3 R B [ 1 1 1 1 {2} 2 1 2 2 2 2 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 3 {2} 1 2 2 2 2 2 2 2 0 ]
B 1 -> 1 R B [ 1 1 1 1 3 2 {1} 2 2 2 2 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 3 2 1 {2} 2 2 2 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 3 2 1 2 {2} 2 2 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 3 2 1 2 2 {2} 2 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 3 2 1 2 2 2 {2} 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 3 2 1 2 2 2 2 {2} 2 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 3 2 1 2 2 2 2 2 {2} 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 3 2 1 2 2 2 2 2 2 {2} 0 ]
B 0 -> 0 L C [ 1 1 1 1 3 2 1 2 2 2 2 2 2 2 {0} ]
C 2 -> 2 L C [ 1 1 1 1 3 2 1 2 2 2 2 2 2 {2} 0 ]
C 2 -> 2 L C [ 1 1 1 1 3 2 1 2 2 2 2 2 {2} 2 0 ]
C 2 -> 2 L C [ 1 1 1 1 3 2 1 2 2 2 2 {2} 2 2 0 ]
C 2 -> 2 L C [ 1 1 1 1 3 2 1 2 2 2 {2} 2 2 2 0 ]
C 2 -> 2 L C [ 1 1 1 1 3 2 1 2 2 {2} 2 2 2 2 0 ]
C 2 -> 2 L C [ 1 1 1 1 3 2 1 2 {2} 2 2 2 2 2 0 ]
C 2 -> 2 L C [ 1 1 1 1 3 2 1 {2} 2 2 2 2 2 2 0 ]
C 1 -> 2 L D [ 1 1 1 1 3 2 {1} 2 2 2 2 2 2 2 0 ]
D 2 -> 2 L D [ 1 1 1 1 3 {2} 2 2 2 2 2 2 2 2 0 ]
D 3 -> 1 R A [ 1 1 1 1 {3} 2 2 2 2 2 2 2 2 2 0 ]
A 2 -> 3 R B [ 1 1 1 1 1 {2} 2 2 2 2 2 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 1 3 {2} 2 2 2 2 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 1 3 2 {2} 2 2 2 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 1 3 2 2 {2} 2 2 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 1 3 2 2 2 {2} 2 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 1 3 2 2 2 2 {2} 2 2 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 1 3 2 2 2 2 2 {2} 2 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 1 3 2 2 2 2 2 2 {2} 2 0 ]
B 2 -> 2 R B [ 1 1 1 1 1 3 2 2 2 2 2 2 2 {2} 0 ]
B 0 -> 0 L C [ 1 1 1 1 1 3 2 2 2 2 2 2 2 2 {0} ]
C 2 -> 2 L C [ 1 1 1 1 1 3 2 2 2 2 2 2 2 {2} 0 ]
C 2 -> 2 L C [ 1 1 1 1 1 3 2 2 2 2 2 2 {2} 2 0 ]
C 2 -> 2 L C [ 1 1 1 1 1 3 2 2 2 2 2 {2} 2 2 0 ]
C 2 -> 2 L C [ 1 1 1 1 1 3 2 2 2 2 {2} 2 2 2 0 ]
C 2 -> 2 L C [ 1 1 1 1 1 3 2 2 2 {2} 2 2 2 2 0 ]
C 2 -> 2 L C [ 1 1 1 1 1 3 2 2 {2} 2 2 2 2 2 0 ]
C 2 -> 2 L C [ 1 1 1 1 1 3 2 {2} 2 2 2 2 2 2 0 ]
C 2 -> 2 L C [ 1 1 1 1 1 3 {2} 2 2 2 2 2 2 2 0 ]
C 3 -> 2 L E [ 1 1 1 1 1 {3} 2 2 2 2 2 2 2 2 0 ]
E 1 -> 1 L E [ 1 1 1 1 {1} 2 2 2 2 2 2 2 2 2 0 ]
E 1 -> 1 L E [ 1 1 1 {1} 1 2 2 2 2 2 2 2 2 2 0 ]
E 1 -> 1 L E [ 1 1 {1} 1 1 2 2 2 2 2 2 2 2 2 0 ]
E 1 -> 1 L E [ 1 {1} 1 1 1 2 2 2 2 2 2 2 2 2 0 ]
E 1 -> 1 L E [ {1} 1 1 1 1 2 2 2 2 2 2 2 2 2 0 ]
E 0 -> 0 R STOP [ {0} 1 1 1 1 1 2 2 2 2 2 2 2 2 2 0 ]
STOP [ 0 {1} 1 1 1 1 2 2 2 2 2 2 2 2 2 0 ]
count = 128
accept = STOP
input = [ 2 2 2 1 2 2 1 2 1 2 1 2 1 2 ]
output = [ 1 1 1 1 1 2 2 2 2 2 2 2 2 2 ]
Duplicate sequence of 1s...
s1 1 -> 0 R s2 [ {1} 1 1 ]
s2 1 -> 1 R s2 [ 0 {1} 1 ]
s2 1 -> 1 R s2 [ 0 1 {1} ]
s2 0 -> 0 R s3 [ 0 1 1 {0} ]
s3 0 -> 1 L s4 [ 0 1 1 0 {0} ]
s4 0 -> 0 L s5 [ 0 1 1 {0} 1 ]
s5 1 -> 1 L s5 [ 0 1 {1} 0 1 ]
s5 1 -> 1 L s5 [ 0 {1} 1 0 1 ]
s5 0 -> 1 R s1 [ {0} 1 1 0 1 ]
s1 1 -> 0 R s2 [ 1 {1} 1 0 1 ]
s2 1 -> 1 R s2 [ 1 0 {1} 0 1 ]
s2 0 -> 0 R s3 [ 1 0 1 {0} 1 ]
s3 1 -> 1 R s3 [ 1 0 1 0 {1} ]
s3 0 -> 1 L s4 [ 1 0 1 0 1 {0} ]
s4 1 -> 1 L s4 [ 1 0 1 0 {1} 1 ]
s4 0 -> 0 L s5 [ 1 0 1 {0} 1 1 ]
s5 1 -> 1 L s5 [ 1 0 {1} 0 1 1 ]
s5 0 -> 1 R s1 [ 1 {0} 1 0 1 1 ]
s1 1 -> 0 R s2 [ 1 1 {1} 0 1 1 ]
s2 0 -> 0 R s3 [ 1 1 0 {0} 1 1 ]
s3 1 -> 1 R s3 [ 1 1 0 0 {1} 1 ]
s3 1 -> 1 R s3 [ 1 1 0 0 1 {1} ]
s3 0 -> 1 L s4 [ 1 1 0 0 1 1 {0} ]
s4 1 -> 1 L s4 [ 1 1 0 0 1 {1} 1 ]
s4 1 -> 1 L s4 [ 1 1 0 0 {1} 1 1 ]
s4 0 -> 0 L s5 [ 1 1 0 {0} 1 1 1 ]
s5 0 -> 1 R s1 [ 1 1 {0} 0 1 1 1 ]
s1 0 -> 0 N H [ 1 1 1 {0} 1 1 1 ]
H [ 1 1 1 {0} 1 1 1 ]
count = 28
accept = H
input = [ 1 1 1 ]
output = [ 1 1 1 0 1 1 1 ]
Turing's first example from On Computable Numbers...
b _ -> 0 R c [ {_} ]
c _ -> _ R e [ 0 {_} ]
e _ -> 1 R f [ 0 _ {_} ]
f _ -> _ R b [ 0 _ 1 {_} ]
b _ -> 0 R c [ 0 _ 1 _ {_} ]
c _ -> _ R e [ 0 _ 1 _ 0 {_} ]
e _ -> 1 R f [ 0 _ 1 _ 0 _ {_} ]
f _ -> _ R b [ 0 _ 1 _ 0 _ 1 {_} ]
b _ -> 0 R c [ 0 _ 1 _ 0 _ 1 _ {_} ]
c _ -> _ R e [ 0 _ 1 _ 0 _ 1 _ 0 {_} ]
e _ -> 1 R f [ 0 _ 1 _ 0 _ 1 _ 0 _ {_} ]
f _ -> _ R b [ 0 _ 1 _ 0 _ 1 _ 0 _ 1 {_} ]
b _ -> 0 R c [ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ {_} ]
c _ -> _ R e [ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ 0 {_} ]
e _ -> 1 R f [ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ 0 _ {_} ]
f _ -> _ R b [ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ 0 _ 1 {_} ]
b _ -> 0 R c [ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ {_} ]
c _ -> _ R e [ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ 0 {_} ]
e _ -> 1 R f [ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ 0 _ {_} ]
f _ -> _ R b [ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ 0 _ 1 {_} ]
b [ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ {_} ]
count = 20
accept = #<void>
output = [ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ 0 _ 1 _ {_} ]
Palindrome checker...
br 1 -> _ R r1 [ {1} 2 1 ]
r1 2 -> 2 R r1 [ _ {2} 1 ]
r1 1 -> 1 R r1 [ _ 2 {1} ]
r1 _ -> _ L e1 [ _ 2 1 {_} ]
e1 1 -> _ L wl [ _ 2 {1} _ ]
wl 2 -> 2 L wl [ _ {2} _ _ ]
wl _ -> _ R br [ {_} 2 _ _ ]
br 2 -> _ R r2 [ _ {2} _ _ ]
r2 _ -> _ L e2 [ _ _ {_} _ ]
e2 _ -> _ N odd [ _ {_} _ _ ]
odd [ _ {_} _ _ ]
count = 10
accept = odd
input = [ 1 2 1 ]
output = [ _ {_} _ _ ]
br 1 -> _ R r1 [ {1} 2 2 ]
r1 2 -> 2 R r1 [ _ {2} 2 ]
r1 2 -> 2 R r1 [ _ 2 {2} ]
r1 _ -> _ L e1 [ _ 2 2 {_} ]
e1 [ _ 2 {2} _ ]
count = 4
accept = #<void>
input = [ 1 2 2 ]
output = [ _ 2 {2} _ ]
br 1 -> _ R r1 [ {1} 1 ]
r1 1 -> 1 R r1 [ _ {1} ]
r1 _ -> _ L e1 [ _ 1 {_} ]
e1 1 -> _ L wl [ _ {1} _ ]
wl _ -> _ R br [ {_} _ _ ]
br _ -> _ N even [ _ {_} _ ]
even [ _ {_} _ ]
count = 6
accept = even
input = [ 1 1 ]
output = [ _ {_} _ ]
br 2 -> _ R r2 [ {2} 1 ]
r2 1 -> 1 R r2 [ _ {1} ]
r2 _ -> _ L e2 [ _ 1 {_} ]
e2 [ _ {1} _ ]
count = 3
accept = #<void>
input = [ 2 1 ]
output = [ _ {1} _ ]
br 1 -> _ R r1 [ {1} ]
r1 _ -> _ L e1 [ _ {_} ]
e1 _ -> _ N odd [ {_} _ ]
odd [ {_} _ ]
count = 3
accept = odd
input = [ 1 ]
output = [ {_} _ ]
count = 21
accept = odd
input = [ 2 1 1 1 2 ]
output = [ _ _ {_} _ _ _ ]
count = 15
accept = #<void>
input = [ 2 1 1 2 2 ]
output = [ _ _ 1 {2} _ _ ]
count = 28
accept = even
input = [ 1 1 2 2 1 1 ]
output = [ _ _ _ {_} _ _ _ ]
count = 7
accept = #<void>
input = [ 1 1 2 2 1 2 ]
output = [ _ 1 2 2 1 {2} _ ]
SequenceL
This implemetnation is based on the Computing Machines introduced in Turing's 1936 paper ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM. With the exception of "skeleton tables".
SequenceL Code:
//region Imports
import <Utilities/Conversion.sl>;
import <Utilities/Sequence.sl>;
//endregion
//region Types
MCONFIG ::= (Label: char(1), Symbols: char(2), Operations: char(2), FinalConfig: char(1));
STATE ::= (CurrentConfig: char(1), CurrentPosition: int(0), Tape: char(1));
INPUT_DATA ::= (Iterations: int(0), InitialTape: char(1), StartingPosition: int(0), InitialConfig: char(1), MConfigs: MCONFIG(1));
//endregion
//region Constants
SPACE_CHAR := '_';
DELIMITTER := '|';
NULL_CONFIG := (Label: "", Symbols: [], Operations: [], FinalConfig: "");
TRACE_HEADER := ["Config:\t| Place:\t| Tape:"];
//endregion
//region Helpers
StateToString(state(0)) :=
state.CurrentConfig ++
" \t\t| " ++ intToString(state.CurrentPosition) ++
" \t| " ++ state.Tape;
StateToArrowString(state(0)) :=
state.Tape ++ "\n" ++
duplicate(' ', state.CurrentPosition - 1) ++ "|\n" ++
duplicate(' ', state.CurrentPosition - 1) ++ state.CurrentConfig ++ "\n";
HeadOfEach(strings(2))[i] :=
head(strings[i]);
RemoveCharacter(character(0), string(1))[i] :=
string[i] when not(string[i] = character);
GetFSquares(Tape(1))[i] :=
Tape[i] when (i mod 2) = 1;
//endregion
//region Parsing
ParseConfig(Line(1)) :=
let
entries := split(Line, DELIMITTER);
label := entries[1];
symbols := split(entries[2], ',');
operations := split(entries[3], ',');
finalConfig := entries[4];
in
((Label: label, Symbols: symbols, Operations: operations, FinalConfig: finalConfig) when not((Line[1] = '/') and (Line[2] = '/')))
when size(Line) > 0;
ParseTextFile(Text(1)) :=
let
noSpaces := RemoveCharacter('\t', RemoveCharacter('\r', RemoveCharacter(' ', Text)));
lines := split(noSpaces, '\n');
iterations := stringToInt(lines[1]);
initialTape := lines[2];
initialPosition := stringToInt(lines[3]);
initialConfig := lines[4];
mConfigs := ParseConfig(lines[5 ... size(lines)]);
in
(Iterations: iterations, InitialTape: initialTape, StartingPosition: initialPosition, InitialConfig: initialConfig, MConfigs: mConfigs);
//endregion
//region Config Finding
Matches: char(0) * char(2) -> bool;
Matches(currentSymbol(0), symbols(2)) :=
true when size(symbols) = 0 //some(equalListNT("", symbols))
else
true when currentSymbol = SPACE_CHAR and some(equalListNT("none", symbols))
else
true when not(currentSymbol = SPACE_CHAR) and some(equalListNT("any", symbols))
else
true when some(currentSymbol = HeadOfEach(symbols))
else
false;
GetCurrentSymbol(State(0)) :=
State.Tape[State.CurrentPosition] when size(State.Tape) >= State.CurrentPosition and State.CurrentPosition > 0
else
SPACE_CHAR;
GetConfigHelper(label(1), symbol(0), mConfigs(1))[i] :=
mConfigs[i] when equalList(mConfigs[i].Label, label) and Matches(symbol, mConfigs[i].Symbols);
GetConfig(label(1), symbol(0), mConfigs(1)) :=
let
searchResults := GetConfigHelper(label, symbol, mConfigs);
in
NULL_CONFIG when size(searchResults) = 0
else
searchResults[1];
//endregion
//region Operations
TrimTapeEnd(tape(1), position(0)) :=
tape when position = size(tape)
else
tape when not(last(tape) = SPACE_CHAR)
else
TrimTapeEnd(allButLast(tape), position);
ApplyOperations(State(0), Operations(2)) :=
let
newState := ApplyOperation(State, head(Operations));
in
State when size(Operations) = 0
else
ApplyOperations(newState, tail(Operations));
ApplyOperation(State(0), Operation(1)) :=
let
newTape :=
PrintOperation(head(tail(Operation)), State.CurrentPosition, State.Tape) when head(Operation) = 'P'
else
EraseOperation(State.CurrentPosition, State.Tape) when head(Operation) = 'E'
else
[SPACE_CHAR] ++ State.Tape when head(Operation) = 'L' and State.CurrentPosition = 1
else
State.Tape ++ [SPACE_CHAR] when head(Operation) = 'R' and State.CurrentPosition = size(State.Tape)
else
State.Tape;
newPosition :=
1 when head(Operation) = 'L' and State.CurrentPosition = 1
else
State.CurrentPosition + 1 when head(Operation) = 'R'
else
State.CurrentPosition - 1 when head(Operation) = 'L'
else
State.CurrentPosition;
trimmedTape := TrimTapeEnd(newTape, newPosition);
in
State when size(Operation) = 0
else
(CurrentPosition: newPosition, Tape: trimmedTape);
PrintOperation(Symbol(0), Position(0), Tape(1)) :=
let
diff := Position - size(Tape) when Position > size(Tape) else 0;
expandedTape := Tape ++ duplicate(SPACE_CHAR, diff);
finalTape := setElementAt(expandedTape, Position, Symbol);
in
finalTape;
EraseOperation(Position(0), Tape(1)) :=
PrintOperation(SPACE_CHAR, Position, Tape);
//endregion
//region Execution
RunMachine(Text(1), Flag(1)) :=
let
input := ParseTextFile(Text);
initialState := (CurrentConfig: input.InitialConfig, CurrentPosition: input.StartingPosition, Tape: input.InitialTape);
processed := Process(initialState, input.MConfigs, input.Iterations);
processedWithTrace := ProcessWithTrace(initialState, input.MConfigs, input.Iterations);
in
"\n" ++ delimit(TRACE_HEADER ++ StateToString(processedWithTrace), '\n') when equalList(Flag, "trace")
else
"\n" ++ delimit(StateToArrowString(processedWithTrace), '\n') when equalList(Flag, "arrow-trace")
else
processed.Tape when equalList(Flag, "tape")
else
TrimTapeEnd(GetFSquares(processed.Tape), 1) when equalList(Flag, "f-squares")
else
boolToString(DoesMachineHalt(initialState, input.MConfigs, input.Iterations)) when equalList(Flag, "halts")
else
StateToString(processed);
DoesMachineHalt(InitialState(0), mConfigs(1), Iterations(0)) :=
let
resultState := Process(InitialState, mConfigs, Iterations);
in
equalList(resultState.CurrentConfig, "halt");
ProcessWithTrace(InitialState(0), mConfigs(1), Iterations(0)) :=
[InitialState] when Iterations <= 0 or size(InitialState.CurrentConfig) = 0 or equalList(InitialState.CurrentConfig, "halt")
else
[InitialState] ++ ProcessWithTrace(Iterate(InitialState, mConfigs), mConfigs, Iterations - 1);
Process(InitialState(0), mConfigs(1), Iterations(0)) :=
InitialState when Iterations = 0 or size(InitialState.CurrentConfig) = 0 or equalList(InitialState.CurrentConfig, "halt")
else
Process(Iterate(InitialState, mConfigs), mConfigs, Iterations - 1);
Iterate(State(0), mConfigs(1)) :=
let
currentConfig := GetConfig(State.CurrentConfig, GetCurrentSymbol(State), mConfigs);
newState := Execute(State, currentConfig);
in
newState;
Execute(State(0), mConfig(0)) :=
let
newState := ApplyOperations(State, mConfig.Operations);
in
(CurrentConfig: mConfig.FinalConfig, CurrentPosition: newState.CurrentPosition, Tape: newState.Tape);
//endregionC++ Driver Code:
#include <iostream>
#include <fstream>
#include <string>
#include <cerrno>
#include "SL_Generated.h"
int cores = 0;
string fileName = "../../INPUT/irrational.tm";
string flag = "tape";
string fileContents = "";
using namespace std;
std::string get_file_contents(const char *filename);
int main( int argc, char** argv )
{
if(argc >= 2)
{
fileName = argv[1];
}
if(argc >= 3)
{
flag = argv[2];
}
if(argc >= 4)
{
cores = atoi(argv[3]);
}
int flagDims[] = { flag.length(), 0};
Sequence<char> flagSeq((void*)(flag.c_str()), flagDims);
fileContents = get_file_contents(fileName.c_str());
int inputDims[] = { fileContents.length(), 0};
Sequence<char> input((void*)(fileContents.c_str()), inputDims);
Sequence<char> result;
sl_init(cores);
sl_RunMachine(input, flagSeq, cores, result);
cout<<result<<endl;
sl_done();
return 0;
}
std::string get_file_contents(const char *filename)
{
std::ifstream in(filename, std::ios::in | std::ios::binary);
if (in)
{
std::string contents;
in.seekg(0, std::ios::end);
contents.resize(in.tellg());
in.seekg(0, std::ios::beg);
in.read(&contents[0], contents.size());
in.close();
return(contents);
}
throw(errno);
}
Turing Machine Files
File explanation:
An input file consists of the following:
a) The first line contains an integer specifying the number of iterations to execute the machine.
b) The second line specifies the initial tape of the machine.
c) The third line contains an integer specifying the initial position of the head of the machine.
d) The fourth line specifies the initial m-config of the machine.
e) The remaining lines contain the rules of the machine or a comment, one on each line. Blank lines are ignored.
i. A line is specified to be a comment if it starts with two forward slashes (//).
Every rule consists of four parts separated by a pipe (|):
a) The first part is the name of the m-config declaration.
b) The second part is the specification of the scanned symbols. This is a, possibly empty, list of symbols, variable names, and keywords.
c) The third part is the specification of the actions to perform. This is a, possibly empty, list of print commands, erase commands, and move commands.
d) The fourth part specifies the target m-config.
Incrementer
60 111 1 q0 q0 | 1 | P1, R | q0 q0 | none | P1 | halt
Busy beaver
60 1 a a| none | P1, R | b a| 1 | P1, L | c b| none | P1, L | a b| 1 | P1, R | b c| none | P1, L | b c| 1 | P1 | halt
Sort
50000000 1221221211 1 A A|1|P1,R|A A|2|P3,R|B A|none|E,L|E B|1|P1,R|B B|2|P2,R|B B|none|E,L|C C|1|P2,L|D C|2|P2,L|C C|3|P2,L|E D|1|P1,L|D D|2|P2,L|D D|3|P1,R|A E|1|P1,L|E E|none|E,R|halt
Irrational Number
100 1 b //Computes the Irrational Number .0010110111011110111110111111..... //Sets up the beginning of the tape. b | | Pe, R, Pe, R, P0, R, R, P0, L, L | o //Marks all of the 1's with x o | 1 | R, Px, L, L, L | o o | 0 | | q //Goes to the end and prints a '1'. q | 0, 1 | R, R | q q | none | P1, L | p //Goes to the first x and then Handles erasing the 'x' that marks a '1'. p | x | E, R | q p | e | R | f p | none | L, L | p //Tacks a '0' to the end. f | any | R, R | f f | none | P0, L, L | o
- Output:
Busy beaver
utm.exe busybeaver.tm arrow-trace
|
a
1_
|
b
11
|
a
_11
|
c
_111
|
b
_1111
|
a
11111
|
b
11111
|
b
11111
|
b
11111
|
b
11111_
|
b
111111
|
a
111111
|
c
111111
|
halt
Sidef
func run_utm(state="", blank="", rules=[], tape=[blank], halt="", pos=0) {
if (pos < 0) {
pos += tape.len;
}
if (pos !~ tape.range) {
die "Bad initial position";
}
loop {
print "#{state}\t";
tape.range.each { |i|
var v = tape[i];
print (i == pos ? "[#{v}]" : " #{v} ");
};
print "\n";
if (state == halt) {
break;
}
rules.each { |rule|
var (s0, v0, v1, dir, s1) = rule...;
if ((s0 != state) || (tape[pos] != v0)) {
next;
}
tape[pos] = v1;
given(dir) {
when ('left') {
if (pos == 0) { tape.unshift(blank) }
else { --pos };
}
when ('right') {
if (++pos >= tape.len) {
tape.append(blank)
}
}
}
state = s1;
goto :NEXT;
}
die 'No matching rules';
@:NEXT;
}
}
print "incr machine\n";
run_utm(
halt: 'qf',
state: 'q0',
tape: %w(1 1 1),
blank: 'B',
rules: [
%w(q0 1 1 right q0),
%w(q0 B 1 stay qf),
]);
say "\nbusy beaver";
run_utm(
halt: 'halt',
state: 'a',
blank: '0',
rules: [
%w(a 0 1 right b),
%w(a 1 1 left c),
%w(b 0 1 left a),
%w(b 1 1 right b),
%w(c 0 1 left b),
%w(c 1 1 stay halt),
]);
say "\nsorting test";
run_utm(
halt: 'STOP',
state: 'A',
blank: '0',
tape: %w(2 2 2 1 2 2 1 2 1 2 1 2 1 2),
rules: [
%w(A 1 1 right A),
%w(A 2 3 right B),
%w(A 0 0 left E),
%w(B 1 1 right B),
%w(B 2 2 right B),
%w(B 0 0 left C),
%w(C 1 2 left D),
%w(C 2 2 left C),
%w(C 3 2 left E),
%w(D 1 1 left D),
%w(D 2 2 left D),
%w(D 3 1 right A),
%w(E 1 1 left E),
%w(E 0 0 right STOP),
]);
Standard ML
In this implementation, tapes are represented like this:
- Either a tape is empty,
- the head points on a field where on the right and on the left hand side may be other symbols,
- the head may be n fields away (left or right) of the non-emty written area.
Each cell may be written (SOME) or blank (NONE). A written field may not be written blank again.
(*** Signatures ***)
signature TAPE = sig
datatype move = Left | Right | Stay
type ''a tape
val empty : ''a tape
val moveLeft : ''a tape -> ''a tape
val moveRight : ''a tape -> ''a tape
val move : ''a tape -> move -> ''a tape
val getSymbol : ''a tape -> ''a option
val write : ''a option -> ''a tape -> ''a tape
val leftOf : ''a tape -> ''a option list (* Symbols left of the head in reverse order *)
val rightOf : ''a tape -> ''a option list (* Symbols right of of the head *)
end
signature MACHINE = sig
structure Tape : TAPE
type state = int
(* ''a is band alphabet type *)
type ''a transitions = (state * ''a option Vector.vector) ->
(state * (Tape.move * ''a option) Vector.vector)
type ''a configuration = { state : state, tapes : ''a Tape.tape Vector.vector }
type ''a machine = {
alphabet : ''a Vector.vector, (* (not used) *)
states : state Vector.vector, (* (not used) *)
start : state, (* element of stats *)
final : state, (* element of stats *)
transitions : ''a transitions, (* transitions *)
tapes : ''a Tape.tape Vector.vector (* vector of the initial tapes *)
}
end
signature UNIVERSALMACHINE = sig
structure Machine : MACHINE
val start : ''a Machine.machine -> ''a Machine.configuration
(* Find a holding configuration (limited by n steps)
* Execute the handler for each step *)
val simulate : (''a Machine.configuration -> unit) -> ''a Machine.machine -> int option -> ''a Machine.configuration option
end
(*** Implementation ***)
structure Tape :> TAPE = struct
(*
* NONE => blank field
* SOME a => written field
*)
type ''a symbol = ''a option
datatype move = Left | Right | Stay
(*
* Four cases:
* 1 The tape is complete empty.
* 2 The head is in the written area.
* On the right and on the left handside are symbols,
* On the head is a symbol.
* 3/4 The head is (n-1) fields over the right/left edge of the written area.
* There is at least one entry and a rest list of entries.
*)
datatype ''a tape =
Empty
| Middle of ''a symbol list * ''a symbol * ''a symbol list
| LeftOf of ''a symbol * ''a symbol list * int
| RightOf of ''a symbol * ''a symbol list * int
val empty = Empty
fun rep a 0 = nil
| rep a n = a :: rep a (n-1)
fun leftOf (Empty) = nil
| leftOf (Middle (ls, _, _)) = ls
| leftOf (RightOf (r, rs, i)) = rep NONE i @ r :: rs
| leftOf (LeftOf _) = nil
fun rightOf (Empty) = nil
| rightOf (Middle (_, _, rs)) = rs
| rightOf (RightOf _) = nil
| rightOf (LeftOf (l, ls, i)) = rep NONE i @ l :: ls
fun write (NONE) t = t (* Cannot write a blank field! *)
| write a t = Middle (leftOf t, a, rightOf t)
fun getSymbol (Middle (_, m, _)) = m
| getSymbol _ = NONE (* blank *)
fun moveRight (Empty) = Empty
| moveRight (Middle (ls, m, nil)) = RightOf (m, ls, 0)
| moveRight (Middle (ls, m, r::rs)) = Middle (m::ls, r, rs)
| moveRight (RightOf (l, ls, n)) = RightOf (l, ls, n+1)
| moveRight (LeftOf (r, rs, 0)) = Middle (nil, r, rs)
| moveRight (LeftOf (r, rs, n)) = LeftOf (r, rs, n-1)
fun moveLeft (Empty) = Empty
| moveLeft (Middle (nil, m, rs)) = LeftOf (m, rs, 0)
| moveLeft (Middle (l::ls, m, rs)) = Middle (ls, l, m::rs)
| moveLeft (RightOf (l, ls, 0)) = Middle (ls, l, nil)
| moveLeft (RightOf (l, ls, n)) = RightOf (l, ls, n-1)
| moveLeft (LeftOf (r, rs, n)) = LeftOf (r, rs, n+1)
fun move tape Stay = tape
| move tape Right = moveRight tape
| move tape Left = moveLeft tape
(* Test *)
local
val tape : int tape = empty (* [] *)
val tape = moveRight tape (* [] *)
val NONE = getSymbol tape
val tape = write (SOME 42) tape (* [42] *)
val (SOME 42) = getSymbol tape
val tape = moveRight tape (* 42, [] *)
val tape = moveRight tape (* 42, , [] *)
val NONE = getSymbol tape
val tape = moveLeft tape (* 42, [] *)
val NONE = getSymbol tape
val tape = moveLeft tape (* [42] *)
val (SOME 42) = getSymbol tape
val tape = write NONE tape (* [42] *) (* !!! *)
val (SOME 42) = getSymbol tape
val tape = moveLeft tape (* [], 42 *)
val tape = moveLeft tape (* [], , 42 *)
val tape = write (SOME 47) tape (* [47], , 42 *)
val (SOME 47) = getSymbol tape
val tape = moveRight tape (* 47, [], 42 *)
val NONE = getSymbol tape
val tape = moveRight tape (* 47, , [42] *)
val (SOME 42) = getSymbol tape
in end
end
structure Machine :> MACHINE = struct
structure Tape = Tape
type state = int
(* ''a is band alphabet type *)
type ''a transitions = (state * ''a option Vector.vector) ->
(state * (Tape.move * ''a option) Vector.vector)
type ''a configuration = { state : state, tapes : ''a Tape.tape Vector.vector }
type ''a machine = {
alphabet : ''a Vector.vector,
states : state Vector.vector,
start : state,
final : state,
transitions : ''a transitions,
tapes : ''a Tape.tape Vector.vector
}
end
structure UniversalMachine :> UNIVERSALMACHINE = struct
structure Machine = Machine
fun start ({ start, tapes, ... } : ''a Machine.machine) : ''a Machine.configuration = {
state = start,
tapes = tapes
}
fun doTransition ({ state, tapes } : ''a Machine.configuration)
((state', actions) : (Machine.state * (Machine.Tape.move * ''a option) Vector.vector))
: ''a Machine.configuration = {
state = state',
tapes = Vector.mapi (fn (i, tape) =>
let val (move, write) = Vector.sub (actions, i)
val tape' = Machine.Tape.write write tape
val tape'' = Machine.Tape.move tape' move
in tape'' end) tapes
}
fun getSymbols ({ tapes, ... } : ''a Machine.configuration) : ''a option Vector.vector =
Vector.map (Machine.Tape.getSymbol) tapes
fun step ({ transitions, ... } : ''a Machine.machine) (conf : ''a Machine.configuration) : ''a Machine.configuration =
doTransition conf (transitions (#state conf, getSymbols conf))
fun isFinal ({final, ...} : ''a Machine.machine) ({state, ...} : ''a Machine.configuration) : bool =
final = state
fun iter term (SOME 0) f s = NONE
| iter term (SOME n) f s = if term s then SOME s else iter term (SOME (n-1)) f (f s)
| iter term NONE f s = if term s then SOME s else iter term NONE f (f s)
fun simulate handler (machine : ''a Machine.machine) optcount =
let val endconf = iter (isFinal machine) optcount (fn conf => (handler conf; step machine conf)) (start machine)
in case endconf of NONE => NONE | SOME conf => (handler conf; endconf) end
end
structure ExampleMachines = struct
structure Machine = UniversalMachine.Machine
(* Tranform the 5-Tuple notation into the vector function *)
fun makeTransitions nil : ''a Machine.transitions = (fn (t, vec) => (print (Int.toString t); raise Subscript))
| makeTransitions ((s : Machine.state,
read : ''a option,
write : ''a option,
move : Machine.Tape.move,
s' : Machine.state) :: ts) =
fn (t, vec) =>
if s=t andalso vec=(Vector.fromList [read])
then (s', Vector.fromList [(move, write)])
else makeTransitions ts (t, vec)
(* `createTape xs` creates an tape initialized by xs, where the head stands on the first element of xs *)
fun createTape' nil = Machine.Tape.empty
| createTape' (x::xs) = Machine.Tape.moveLeft (Machine.Tape.write x (createTape' xs))
fun createTape xs = Machine.Tape.moveRight (createTape' (rev xs))
(* Convert a tape into a string to print it. It needs a function that converts each symbol to string *)
fun tapeToStr (symStr : ''a -> string) (tape : ''a Machine.Tape.tape) : string =
let val left : ''a option list = rev (Machine.Tape.leftOf tape)
val right : ''a option list = Machine.Tape.rightOf tape
val current : ''a option = Machine.Tape.getSymbol tape
val symToStr : ''a option -> string = (fn (NONE) => "#" | (SOME a) => symStr a)
in
String.concatWith " " ((map symToStr left) @ [ "|" ^ symToStr current ^ "|" ] @ (map symToStr right))
end
(* Convert a vector to a list *)
fun vectToList vect = List.tabulate (Vector.length vect, fn i => Vector.sub (vect, i))
(* Do this before every step and after the last step. *)
fun handler (symToStr : ''a -> string) ({state, tapes} : ''a Machine.configuration) : unit =
let
val str = "State " ^ Int.toString state ^ "\n" ^
String.concat (vectToList (Vector.mapi (fn (i, tape) => "Tape #" ^ Int.toString i ^ ": " ^
tapeToStr symToStr tape ^ "\n") tapes))
in
print str
end
(* Simulate and make result into string *)
fun simulate (symToStr : ''a -> string) (machine : ''a Machine.machine)
(optcount : int option) : string =
case (UniversalMachine.simulate (handler symToStr) machine optcount) of
NONE => "Did not terminate."
| SOME ({state, tapes} : ''a Machine.configuration) => "Terminated."
(* Now finaly the machines! *)
val incrementer : unit Machine.machine = {
alphabet = Vector.fromList [()],
states = Vector.fromList [0, 1],
start = 0,
final = 1,
tapes = Vector.fromList [createTape (map SOME [(), (), (), ()])],
transitions = makeTransitions [
(0, SOME (), SOME (), Machine.Tape.Right, 0),
(0, NONE, SOME (), Machine.Tape.Stay, 1)]
}
val busybeaver : unit Machine.machine = {
alphabet = Vector.fromList [()],
states = Vector.fromList [0, 1, 2, 3],
start = 0,
final = 3,
tapes = Vector.fromList [Machine.Tape.empty],
transitions = makeTransitions [
(0, NONE, SOME (), Machine.Tape.Right, 1),
(0, SOME (), SOME (), Machine.Tape.Left, 2),
(1, NONE, SOME (), Machine.Tape.Left, 0),
(1, SOME (), SOME (), Machine.Tape.Right, 1),
(2, NONE, SOME (), Machine.Tape.Left, 1),
(2, SOME (), SOME (), Machine.Tape.Stay, 3)]
}
val sorting : int Machine.machine = {
alphabet = Vector.fromList [1,2,3],
states = Vector.fromList [0,1,2,3,4,5],
start = 1,
final = 0,
tapes = Vector.fromList [createTape (map SOME [2, 1, 2, 2, 1, 1])],
transitions = makeTransitions [
(1, SOME 1, SOME 1, Machine.Tape.Right, 1),
(1, SOME 2, SOME 3, Machine.Tape.Right, 2),
(1, NONE, NONE, Machine.Tape.Left, 5),
(2, SOME 1, SOME 1, Machine.Tape.Right, 2),
(2, SOME 2, SOME 2, Machine.Tape.Right, 2),
(2, NONE, NONE, Machine.Tape.Left, 3),
(3, SOME 1, SOME 2, Machine.Tape.Left, 3),
(3, SOME 2, SOME 2, Machine.Tape.Left, 3),
(3, SOME 3, SOME 2, Machine.Tape.Left, 5),
(4, SOME 1, SOME 1, Machine.Tape.Left, 4),
(4, SOME 2, SOME 2, Machine.Tape.Left, 4),
(4, SOME 3, SOME 1, Machine.Tape.Right, 1),
(5, SOME 1, SOME 1, Machine.Tape.Left, 5),
(5, NONE, NONE, Machine.Tape.Right, 0)]
}
end
(** Invoke Simulations **)
local
open ExampleMachines
val unitToString = (fn () => "()")
fun simulate_unit machine optcount = print (simulate unitToString machine optcount ^ "\n")
fun simulate_int machine optcount = print (simulate Int.toString machine optcount ^ "\n")
in
val () = print "Simulate incrementer...\n\n"
val () = simulate_unit incrementer NONE
val () = print "\nSimulate Busy Beaver...\n\n"
val () = simulate_unit busybeaver NONE
val () = print "\nSimulate Sorting...\n\n"
val () = simulate_int sorting NONE
end
- Output:
Simulate incrementer... State 0 Tape #0: |()| () () () State 0 Tape #0: () |()| () () State 0 Tape #0: () () |()| () State 0 Tape #0: () () () |()| State 0 Tape #0: () () () () |#| State 1 Tape #0: () () () () |()| Terminated. Simulate Busy Beaver... State 0 Tape #0: |#| State 1 Tape #0: () |#| State 0 Tape #0: |()| () State 2 Tape #0: |#| () () State 1 Tape #0: |#| () () () State 0 Tape #0: |#| () () () () State 1 Tape #0: () |()| () () () State 1 Tape #0: () () |()| () () State 1 Tape #0: () () () |()| () State 1 Tape #0: () () () () |()| State 1 Tape #0: () () () () () |#| State 0 Tape #0: () () () () |()| () State 2 Tape #0: () () () |()| () () State 3 Tape #0: () () () |()| () () Terminated. Simulate Sorting... State 1 Tape #0: |1| 1 2 2 1 2 State 1 Tape #0: 1 |1| 2 2 1 2 State 1 Tape #0: 1 1 |2| 2 1 2 State 2 Tape #0: 1 1 3 |2| 1 2 State 2 Tape #0: 1 1 3 2 |1| 2 State 2 Tape #0: 1 1 3 2 1 |2| State 2 Tape #0: 1 1 3 2 1 2 |#| State 3 Tape #0: 1 1 3 2 1 |2| State 3 Tape #0: 1 1 3 2 |1| 2 State 3 Tape #0: 1 1 3 |2| 2 2 State 3 Tape #0: 1 1 |3| 2 2 2 State 5 Tape #0: 1 |1| 2 2 2 2 State 5 Tape #0: |1| 1 2 2 2 2 State 5 Tape #0: |#| 1 1 2 2 2 2 State 0 Tape #0: |1| 1 2 2 2 2 Terminated.
Tcl
proc turing {states initial terminating symbols blank tape rules {doTrace 1}} {
set state $initial
set idx 0
set tape [split $tape ""]
if {[llength $tape] == 0} {
set tape [list $blank]
}
foreach rule $rules {
lassign $rule state0 sym0 sym1 move state1
set R($state0,$sym0) [list $sym1 $move $state1]
}
while {$state ni $terminating} {
set sym [lindex $tape $idx]
lassign $R($state,$sym) sym1 move state1
if {$doTrace} {
### Print the state, great for debugging
puts "[join $tape ""]\t$state->$state1"
puts "[string repeat { } $idx]^"
}
lset tape $idx $sym1
switch $move {
left {
if {[incr idx -1] < 0} {
set idx 0
set tape [concat [list $blank] $tape]
}
}
right {
if {[incr idx] == [llength $tape]} {
lappend tape $blank
}
}
}
set state $state1
}
return [join $tape ""]
}
Demonstrating:
puts "Simple incrementer"
puts TAPE=[turing {q0 qf} q0 qf {1 B} B "111" {
{q0 1 1 right q0}
{q0 B 1 stay qf}
}]
puts "Three-state busy beaver"
puts TAPE=[turing {a b c halt} a halt {0 1} 0 "" {
{a 0 1 right b}
{a 1 1 left c}
{b 0 1 left a}
{b 1 1 right b}
{c 0 1 left b}
{c 1 1 stay halt}
}]
puts "Sorting stress test"
# We suppress the trace output for this so as to keep the output short
puts TAPE=[turing {A B C D E H} A H {0 1 2 3} 0 "12212212121212" {
{A 1 1 right A}
{A 2 3 right B}
{A 0 0 left E}
{B 1 1 right B}
{B 2 2 right B}
{B 0 0 left C}
{C 1 2 left D}
{C 2 2 left C}
{C 3 2 left E}
{D 1 1 left D}
{D 2 2 left D}
{D 3 1 right A}
{E 1 1 left E}
{E 0 0 right H}
} no]
- Output:
Simple incrementer
111 q0->q0
^
111 q0->q0
^
111 q0->q0
^
111B q0->qf
^
TAPE=1111
Three-state busy beaver
0 a->b
^
10 b->a
^
11 a->c
^
011 c->b
^
0111 b->a
^
01111 a->b
^
11111 b->b
^
11111 b->b
^
11111 b->b
^
11111 b->b
^
111110 b->a
^
111111 a->c
^
111111 c->halt
^
TAPE=111111
Sorting stress test
TAPE=0111111222222220
UNIX Shell
#!/usr/bin/env bash
main() {
printf 'Simple Incrementer\n'
printf '1 1 1' | run_utm q0 qf B q0,1,1,R,q0 q0,B,1,S,qf
printf '\nThree-state busy beaver\n'
run_utm a halt 0 \
a,0,1,R,b a,1,1,L,c b,0,1,L,a b,1,1,R,b c,0,1,L,b c,1,1,S,halt \
</dev/null
}
run_utm() {
local initial=$1 final=$2 blank=$3
shift 3
local rules=("$@") tape
mapfile -t -d' ' tape
if (( ! ${#tape[@]} )); then
tape=( "$blank" )
fi
local state=$initial
local head=0
while [[ $state != $final ]]; do
print_state "$state" "$head" "${tape[@]}"
local symbol=${tape[head]}
local found=0 rule from input output move to
for rule in "${rules[@]}"; do
IFS=, read from input output move to <<<"$rule"
if [[ $state == $from && $symbol == $input ]]; then
found=1
break
fi
done
if (( ! found )); then
printf >&2 "Configuration error: no match for state=$state input=$sym\n"
return 1
fi
tape[head]=$output
state=$to
case "$move" in
L) if (( ! head-- )); then
head=0
tape=("$blank" "${tape[@]}")
fi
;;
R) if (( ++head >= ${#tape[@]} )); then
tape+=("$blank")
fi
;;
esac
done
print_state "$state" "$head" "${tape[@]}"
}
print_state() {
local state=$1 head=$2
shift 2
local tape=("$@")
printf '%s' "$state"
printf ' %s' "${tape[@]}"
printf '\r'
(( t = ${#state} + 1 + 3 * head ))
printf '\e['"$t"'C<\e[C>\n'
}
main "$@"
- Output:
Simple Incrementer q0 <1> 1 1 q0 1 <1> 1 q0 1 1 <1> q0 1 1 1 <B> qf 1 1 1 <1> Three-state busy beaver a <0> b 1 <0> a <1> 1 c <0> 1 1 b <0> 1 1 1 a <0> 1 1 1 1 b 1 <1> 1 1 1 b 1 1 <1> 1 1 b 1 1 1 <1> 1 b 1 1 1 1 <1> b 1 1 1 1 1 <0> a 1 1 1 1 <1> 1 c 1 1 1 <1> 1 1 halt 1 1 1 <1> 1 1
VBA
Option Base 1
Public Enum sett
name_ = 1
initState
endState
blank
rules
End Enum
Public incrementer As Variant, threeStateBB As Variant, fiveStateBB As Variant
'-- Machine definitions
Private Sub init()
incrementer = Array("Simple incrementer", _
"q0", _
"qf", _
"B", _
Array( _
Array("q0", "1", "1", "right", "q0"), _
Array("q0", "B", "1", "stay", "qf")))
threeStateBB = Array("Three-state busy beaver", _
"a", _
"halt", _
"0", _
Array( _
Array("a", "0", "1", "right", "b"), _
Array("a", "1", "1", "left", "c"), _
Array("b", "0", "1", "left", "a"), _
Array("b", "1", "1", "right", "b"), _
Array("c", "0", "1", "left", "b"), _
Array("c", "1", "1", "stay", "halt")))
fiveStateBB = Array("Five-state busy beaver", _
"A", _
"H", _
"0", _
Array( _
Array("A", "0", "1", "right", "B"), _
Array("A", "1", "1", "left", "C"), _
Array("B", "0", "1", "right", "C"), _
Array("B", "1", "1", "right", "B"), _
Array("C", "0", "1", "right", "D"), _
Array("C", "1", "0", "left", "E"), _
Array("D", "0", "1", "left", "A"), _
Array("D", "1", "1", "left", "D"), _
Array("E", "0", "1", "stay", "H"), _
Array("E", "1", "0", "left", "A")))
End Sub
Private Sub show(state As String, headpos As Long, tape As Collection)
Debug.Print " "; state; String$(7 - Len(state), " "); "| ";
For p = 1 To tape.Count
Debug.Print IIf(p = headpos, "[" & tape(p) & "]", " " & tape(p) & " ");
Next p
Debug.Print
End Sub
'-- a universal turing machine
Private Sub UTM(machine As Variant, tape As Collection, Optional countOnly As Long = 0)
Dim state As String: state = machine(initState)
Dim headpos As Long: headpos = 1
Dim counter As Long, rule As Variant
Debug.Print machine(name_); vbCrLf; String$(Len(machine(name_)), "=")
If Not countOnly Then Debug.Print " State | Tape [head]" & vbCrLf & "---------------------"
Do While True
If headpos > tape.Count Then
tape.Add machine(blank)
Else
If headpos < 1 Then
tape.Add machine(blank), Before:=1
headpos = 1
End If
End If
If Not countOnly Then show state, headpos, tape
For i = LBound(machine(rules)) To UBound(machine(rules))
rule = machine(rules)(i)
If rule(1) = state And rule(2) = tape(headpos) Then
tape.Remove headpos
If headpos > tape.Count Then
tape.Add rule(3)
Else
tape.Add rule(3), Before:=headpos
End If
If rule(4) = "left" Then headpos = headpos - 1
If rule(4) = "right" Then headpos = headpos + 1
state = rule(5)
Exit For
End If
Next i
counter = counter + 1
If counter Mod 100000 = 0 Then
Debug.Print counter
DoEvents
DoEvents
End If
If state = machine(endState) Then Exit Do
Loop
DoEvents
If countOnly Then
Debug.Print "Steps taken: ", counter
Else
show state, headpos, tape
Debug.Print
End If
End Sub
Public Sub main()
init
Dim tap As New Collection
tap.Add "1": tap.Add "1": tap.Add "1"
UTM incrementer, tap
Set tap = New Collection
UTM threeStateBB, tap
Set tap = New Collection
UTM fiveStateBB, tap, countOnly:=-1
End Sub- Output:
Simple incrementer ================== State | Tape [head] --------------------- q0 | [1] 1 1 q0 | 1 [1] 1 q0 | 1 1 [1] q0 | 1 1 1 [B] qf | 1 1 1 [1] Three-state busy beaver ======================= State | Tape [head] --------------------- a | [0] b | 1 [0] a | [1] 1 c | [0] 1 1 b | [0] 1 1 1 a | [0] 1 1 1 1 b | 1 [1] 1 1 1 b | 1 1 [1] 1 1 b | 1 1 1 [1] 1 b | 1 1 1 1 [1] b | 1 1 1 1 1 [0] a | 1 1 1 1 [1] 1 c | 1 1 1 [1] 1 1 halt | 1 1 1 [1] 1 1 Five-state busy beaver ====================== Steps taken: 47176870
Wren
import "/dynamic" for Enum, Tuple, Struct
import "/fmt" for Fmt
var Dir = Enum.create("Dir", ["LEFT", "RIGHT", "STAY"])
var Rule = Tuple.create("Rule", ["state1", "symbol1", "symbol2", "dir", "state2"])
var Tape = Struct.create("Tape", ["symbol", "left", "right"])
class Turing {
construct new(states, finalStates, symbols, blank, state, tapeInput, rules) {
_states = states
_finalStates = finalStates
_symbols = symbols
_blank = blank
_state = state
_tape = null
_transitions = List.filled(_states.count, null)
for (i in 0..._states.count) _transitions[i] = List.filled(_symbols.count, null)
for (i in 0...tapeInput.count) {
move_(Dir.RIGHT)
_tape.symbol = tapeInput[i]
}
if (tapeInput.count == 0) move_(Dir.RIGHT)
while (_tape.left) _tape = _tape.left
for (i in 0...rules.count) {
var rule = rules[i]
_transitions[stateIndex_(rule.state1)][symbolIndex_(rule.symbol1)] = rule
}
}
stateIndex_(state) {
var i = _states.indexOf(state)
return (i >= 0) ? i : 0
}
symbolIndex_(symbol) {
var i = _symbols.indexOf(symbol)
return (i >= 0) ? i : 0
}
move_(dir) {
var orig = _tape
if (dir == Dir.RIGHT) {
if (orig && orig.right) {
_tape = orig.right
} else {
_tape = Tape.new(_blank, null, null)
if (orig) {
_tape.left = orig
orig.right = _tape
}
}
} else if (dir == Dir.LEFT) {
if (orig && orig.left) {
_tape = orig.left
} else {
_tape = Tape.new(_blank, null, null)
if (orig) {
_tape.right = orig
orig.left = _tape
}
}
} else if (dir == Dir.STAY) {}
}
printState() {
Fmt.write("$-10s ", _state)
var t = _tape
while (t.left) t = t.left
while (t) {
if (t == _tape) {
System.write("[%(t.symbol)]")
} else {
System.write(" %(t.symbol) ")
}
t = t.right
}
System.print()
}
run(maxLines) {
var lines = 0
while (true) {
printState()
for (finalState in _finalStates) {
if (finalState == _state) return
}
lines = lines + 1
if (lines == maxLines) {
System.print("(Only the first %(maxLines) lines displayed)")
return
}
var rule = _transitions[stateIndex_(_state)][symbolIndex_(_tape.symbol)]
_tape.symbol = rule.symbol2
move_(rule.dir)
_state = rule.state2
}
}
}
System.print("Simple incrementer")
Turing.new(
["q0", "qf"], // states
["qf"], // finalStates
["B", "1"], // symbols
"B", // blank
"q0", // state
["1", "1", "1"], // tapeInput
[ // rules
Rule.new("q0", "1", "1", Dir.RIGHT, "q0"),
Rule.new("q0", "B", "1", Dir.STAY, "qf")
]
).run(20)
System.print("\nThree-state busy beaver")
Turing.new(
["a", "b", "c", "halt"], // states
["halt"], // finalStates
["0", "1"], // symbols
"0", // blank
"a", // state
[], // tapeInput
[ // rules
Rule.new("a", "0", "1", Dir.RIGHT, "b"),
Rule.new("a", "1", "1", Dir.LEFT, "c"),
Rule.new("b", "0", "1", Dir.LEFT, "a"),
Rule.new("b", "1", "1", Dir.RIGHT, "b"),
Rule.new("c", "0", "1", Dir.LEFT, "b"),
Rule.new("c", "1", "1", Dir.STAY, "halt")
]
).run(20)
System.print("\nFive-state two-symbol probable busy beaver")
Turing.new(
["A", "B", "C", "D", "E", "H"], // states
["H"], // finalStates
["0", "1"], // symbols
"0", // blank
"A", // state
[], // tapeInput
[ // rules
Rule.new("A", "0", "1", Dir.RIGHT, "B"),
Rule.new("A", "1", "1", Dir.LEFT, "C"),
Rule.new("B", "0", "1", Dir.RIGHT, "C"),
Rule.new("B", "1", "1", Dir.RIGHT, "B"),
Rule.new("C", "0", "1", Dir.RIGHT, "D"),
Rule.new("C", "1", "0", Dir.LEFT, "E"),
Rule.new("D", "0", "1", Dir.LEFT, "A"),
Rule.new("D", "1", "1", Dir.LEFT, "D"),
Rule.new("E", "0", "1", Dir.STAY, "H"),
Rule.new("E", "1", "0", Dir.LEFT, "A")
]
).run(20)- Output:
Simple incrementer q0 [1] 1 1 q0 1 [1] 1 q0 1 1 [1] q0 1 1 1 [B] qf 1 1 1 [1] Three-state busy beaver a [0] b 1 [0] a [1] 1 c [0] 1 1 b [0] 1 1 1 a [0] 1 1 1 1 b 1 [1] 1 1 1 b 1 1 [1] 1 1 b 1 1 1 [1] 1 b 1 1 1 1 [1] b 1 1 1 1 1 [0] a 1 1 1 1 [1] 1 c 1 1 1 [1] 1 1 halt 1 1 1 [1] 1 1 Five-state two-symbol probable busy beaver A [0] B 1 [0] C 1 1 [0] D 1 1 1 [0] A 1 1 [1] 1 C 1 [1] 1 1 E [1] 0 1 1 A [0] 0 0 1 1 B 1 [0] 0 1 1 C 1 1 [0] 1 1 D 1 1 1 [1] 1 D 1 1 [1] 1 1 D 1 [1] 1 1 1 D [1] 1 1 1 1 D [0] 1 1 1 1 1 A [0] 1 1 1 1 1 1 B 1 [1] 1 1 1 1 1 B 1 1 [1] 1 1 1 1 B 1 1 1 [1] 1 1 1 B 1 1 1 1 [1] 1 1 (Only the first 20 lines displayed)
Yabasic
// Machine definitions
name = 1 : initState = 2 : endState = 3 : blank = 4 : countOnly = true
incrementer$ = "Simple incrementer,q0,qf,B"
incrementer$ = incrementer$ + ",q0,1,1,right,q0,q0,B,1,stay,qf"
threeStateBB$ = "Three-state busy beaver,a,halt,0"
data "a,0,1,right,b"
data "a,1,1,left,c"
data "b,0,1,left,a"
data "b,1,1,right,b"
data "c,0,1,left,b"
data "c,1,1,stay,halt"
data ""
do
read a$
if a$ = "" break
threeStateBB$ = threeStateBB$ + "," + a$
loop
fiveStateBB$ = "Five-state busy beaver,A,H,0"
data "A,0,1,right,B"
data "A,1,1,left,C"
data "B,0,1,right,C"
data "B,1,1,right,B"
data "C,0,1,right,D"
data "C,1,0,left,E"
data "D,0,1,left,A"
data "D,1,1,left,D"
data "E,0,1,stay,H"
data "E,1,0,left,A"
data ""
do
read a$
if a$ = "" break
fiveStateBB$ = fiveStateBB$ + "," + a$
loop
clear screen
// Display a representation of the tape and machine state on the screen
sub show(state$, headPos, tape$)
local pos
print " ", state$, "\t| ";
for pos = 1 to len(tape$)
if pos = headPos then print "[", mid$(tape$, pos, 1), "] "; else print " ", mid$(tape$, pos, 1), " "; end if
next
print
end sub
sub string.rep$(s$, n)
local i, r$
for i = 1 to n
r$ = r$ + s$
next
return r$
end sub
// Simulate a turing machine
sub UTM(mach$, tape$, countOnly)
local state$, headPos, counter, machine$(1), n, m, rule
m = len(tape$)
n = token(mach$, machine$(), ",")
state$ = machine$(initState)
n = n - blank
headPos = 1
print "\n\n", machine$(name)
print string.rep$("=", len(machine$(name))), "\n"
if not countOnly print " State", "\t| Tape [head]\n----------------------"
repeat
if mid$(tape$, headPos, 1) = " " mid$(tape$, headPos, 1) = machine$(blank)
if not countOnly show(state$, headPos, tape$)
for rule = blank + 1 to n step 5
if machine$(rule) = state$ and machine$(rule + 1) = mid$(tape$, headPos, 1) then
mid$(tape$, headPos, 1) = machine$(rule + 2)
if machine$(rule + 3) = "left" then
headPos = headPos - 1
if headPos < 1 then
headPos = 1
tape$ = " " + tape$
end if
end if
if machine$(rule + 3) = "right" then
headPos = headPos + 1
if headPos > m then
m = m + 1
tape$ = tape$ + " "
end if
end if
state$ = machine$(rule + 4)
break
end if
next
counter = counter + 1
until(state$ = machine$(endState))
if countOnly then print "Steps taken: ", counter else show(state$, headPos, tape$) end if
end sub
// Main procedure
UTM(incrementer$, "111")
UTM(threeStateBB$, " ")
UTM(fiveStateBB$, " ", countOnly)zkl
This uses a dictionary/hash to hold the tape, limiting the length to 64k.
var [const] D=Dictionary; // short cut
// blank symbol and terminating state(s) are Void
var Lt=-1, Sy=0, Rt=1; // Left, Stay, Right
fcn printTape(tape,pos){
tape.keys.apply("toInt").sort()
.pump(String,'wrap(i){ ((pos==i) and "(%s)" or " %s ").fmt(tape[i]) })
.println();
}
fcn turing(state,[D]tape,[Int]pos,[D]rules,verbose=True,n=0){
if(not state){
print("%d steps. Length %d. Tape: ".fmt(n,tape.len()));
printTape(tape,Void);
return(tape);
}
r:=rules[state][tape[pos] = tape.find(pos)];
if(verbose) printTape(tape,pos);
tape[pos]=r[0];
return(self.fcn(r[2],tape,pos+r[1],rules,verbose,n+1));
}D is a dictionary, SD is a small fixed (at runtime) dictionary
println("Simple incrementer");
turing("q0",D(0,Rt, 1,Rt, 2,Rt),0, // Incrementer
D("q0",D(1,T(1,Rt,"q0"), Void,T(1,Sy,Void)) ) );
println("\nThree-state busy beaver");
turing("a",D(),0, // Three-state busy beaver
SD("a",D(Void,T(1,Rt,"b"), 1,T(1,Lt,"c")),
"b",D(Void,T(1,Lt,"a"), 1,T(1,Rt,"b")),
"c",D(Void,T(1,Lt,"b"), 1,T(1,Sy,Void)) ) );
println("\nSort");
turing("A",D(T(2,2,2,1,2,2,1,2,1,2,1,2,1,2).enumerate()),0,
SD("A",D(1,T(1,Rt,"A"), 2,T(3,Rt,"B"), Void,T(Void,Lt,"E")),
"B",D(1,T(1,Rt,"B"), 2,T(2,Rt,"B"), Void,T(Void,Lt,"C")),
"C",D(1,T(2,Lt,"D"), 2,T(2,Lt,"C"), 3, T(2, Lt,"E")),
"D",D(1,T(1,Lt,"D"), 2,T(2,Lt,"D"), 3, T(1, Rt,"A")),
"E",D(1,T(1,Lt,"E"), Void,T(Void,Rt,Void)) ) ,False);
println("\nFive-state busy beaver");
turing("A",D(),0,
SD("A",D(Void,T(1,Rt,"B"), 1,T(1, Lt,"C")),
"B",D(Void,T(1,Rt,"C"), 1,T(1, Rt,"B")),
"C",D(Void,T(1,Rt,"D"), 1,T(Void,Lt,"E")),
"D",D(Void,T(1,Lt,"A"), 1,T(1, Lt,"D")),
"E",D(Void,T(1,Sy,Void), 1,T(Void,Lt,"A")) ) ,False);- Output:
Simple incrementer (1) 1 1 1 (1) 1 1 1 (1) 1 1 1 (Void) 4 steps. Lenght 4. Tape: 1 1 1 1 Three-state busy beaver (Void) 1 (Void) (1) 1 (Void) 1 1 (Void) 1 1 1 (Void) 1 1 1 1 1 (1) 1 1 1 1 1 (1) 1 1 1 1 1 (1) 1 1 1 1 1 (1) 1 1 1 1 1 (Void) 1 1 1 1 (1) 1 1 1 1 (1) 1 1 13 steps. Lenght 6. Tape: 1 1 1 1 1 1 Sort 128 steps. Length 16. Tape: Void 1 1 1 1 1 2 2 2 2 2 2 2 2 2 Void Five-state busy beaver 47176870 steps. Length 12289. Tape: 1 Void 1 Void Void 1 Void Void 1 Void Void 1 Void Void 1 Void Void 1 Void Void 1 Void Void 1 Void Void 1 Void Void 1 Void Void 1 Void Void 1 Void Void 1 Void Void .... Read: Void==0 as 0 is the blank symbol
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