# Elementary cellular automaton/Random Number Generator

Elementary cellular automaton/Random Number Generator is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Rule 30 is considered to be chaotic enough to generate good pseudo-random numbers. As a matter of fact, rule 30 is used by the Mathematica software for its default random number generator.

Steven Wolfram's recommendation for random number generation from rule 30 consists in extracting successive bits in a fixed position in the array of cells, as the automaton changes state.

The purpose of this task is to demonstrate this. With the code written in the parent task, which you don't need to re-write here, show the ten first bytes that emerge from this recommendation. To be precise, you will start with a state of all cells but one equal to zero, and you'll follow the evolution of the particular cell whose state was initially one. Then you'll regroup those bits by packets of eight, reconstituting bytes with the first bit being the most significant.

You can pick which ever length you want for the initial array but it should be visible in the code so that your output can be reproduced with an other language.

For extra-credits, you will make this algorithm run as fast as possible in your language, for instance with an extensive use of bitwise logic.

Reference

## C

64-bits array size, cyclic borders.

`#include <stdio.h>#include <limits.h> typedef unsigned long long ull;#define N (sizeof(ull) * CHAR_BIT)#define B(x) (1ULL << (x)) void evolve(ull state, int rule){	int i, p, q, b; 	for (p = 0; p < 10; p++) {		for (b = 0, q = 8; q--; ) {			ull st = state;			b |= (st&1) << q; 			for (state = i = 0; i < N; i++)				if (rule & B(7 & (st>>(i-1) | st<<(N+1-i))))					state |= B(i);		}		printf(" %d", b);	}	putchar('\n');	return;} int main(void){	evolve(1, 30);	return 0;}`
Output:
` 220 197 147 174 117 97 149 171 100 151`

## C++

We'll re-write the code of the parent task here.

`#include <bitset>#include <stdio.h> #define SIZE	           80#define RULE               30#define RULE_TEST(x)       (RULE & 1 << (7 & (x))) void evolve(std::bitset<SIZE> &s) {    int i;    std::bitset<SIZE> t(0);    t[SIZE-1] = RULE_TEST( s[0] << 2 | s[SIZE-1] << 1 | s[SIZE-2] );    t[     0] = RULE_TEST( s[1] << 2 | s[     0] << 1 | s[SIZE-1] );    for (i = 1; i < SIZE-1; i++)	t[i] = RULE_TEST( s[i+1] << 2 | s[i] << 1 | s[i-1] );    for (i = 0; i < SIZE; i++) s[i] = t[i];}void show(std::bitset<SIZE> s) {    int i;    for (i = SIZE; i--; ) printf("%c", s[i] ? '#' : ' ');    printf("|\n");}unsigned char byte(std::bitset<SIZE> &s) {    unsigned char b = 0;    int i;    for (i=8; i--; ) {	b |= s[0] << i; 	evolve(s);    }    return b;} int main() {    int i;    std::bitset<SIZE> state(1);    for (i=10; i--; )	printf("%u%c", byte(state), i ? ' ' : '\n');    return 0;}`
Output:
`220 197 147 174 117 97 149 171 240 241`

## D

Translation of: C

Adapted from the C version, with improvements and bug fixes. Optimized for performance as requested in the task description. This is a lazy range.

`import std.stdio, std.range, std.typecons; struct CellularRNG {    private uint current;    private immutable uint rule;    private ulong state;     this(in ulong state_, in uint rule_) pure nothrow @safe @nogc {        this.state = state_;        this.rule = rule_;        popFront;    }     public enum bool empty = false;    @property uint front() pure nothrow @safe @nogc { return current; }     void popFront() pure nothrow @safe @nogc {        enum uint nBit = 8;        enum uint NU = ulong.sizeof * nBit;        current = 0;         foreach_reverse (immutable i; 0 .. nBit) {            immutable state2 = state;            current |= (state2 & 1) << i;             state = 0;            /*static*/ foreach (immutable j; staticIota!(0, NU)) {                // To avoid undefined behavior with out-of-range shifts.                static if (j > 0)                    immutable aux1 = state2 >> (j - 1);                else                    immutable aux1 = state2 >> 63;                 static if (j == 0)                    immutable aux2 = state2 << 1;                else static if (j == 1)                    immutable aux2 = state2 << 63;                else                    immutable aux2 = state2 << (NU + 1 - j);                 immutable aux = 7 & (aux1 | aux2);                if (rule & (1UL << aux))                    state |= 1UL << j;            }        }    }} void main() {    CellularRNG(1, 30).take(10).writeln;    CellularRNG(1, 30).drop(2_000_000).front.writeln;}`
Output:
```[220, 197, 147, 174, 117, 97, 149, 171, 100, 151]
44```

Run-time: less than two seconds with the ldc2 compiler.

## Go

Translation of: C
`package main import "fmt" const n = 64 func pow2(x uint) uint64 {    return uint64(1) << x} func evolve(state uint64, rule int) {    for p := 0; p < 10; p++ {        b := uint64(0)        for q := 7; q >= 0; q-- {            st := state            b |= (st & 1) << uint(q)            state = 0            for i := uint(0); i < n; i++ {                var t1, t2, t3 uint64                if i > 0 {                    t1 = st >> (i - 1)                } else {                    t1 = st >> 63                }                if i == 0 {                    t2 = st << 1                } else if i == 1 {                    t2 = st << 63                 } else {                    t2 = st << (n + 1 - i)                }                t3 = 7 & (t1 | t2)                if (uint64(rule) & pow2(uint(t3))) != 0 {                    state |= pow2(i)                }            }        }        fmt.Printf("%d ", b)    }    fmt.Println()} func main() {    evolve(1, 30)}`
Output:
```220 197 147 174 117 97 149 171 100 151
```

Assume the comonadic solution given at Elementary cellular automaton#Haskell is packed in a module `CellularAutomata`

`import CellularAutomata (runCA, rule, fromList)import Data.List (unfoldr)import Control.Comonad rnd = fromBits <\$> unfoldr (pure . splitAt 8) bits  where size = 80        bits = extract <\$> runCA (rule 30) (fromList (1:replicate size 0)) fromBits = foldl (\res x -> 2*res + x) 0`
Output:
```λ> take 10 rnd
[220,197,147,174,117,97,149,171,240,241]```

Using the rule 30 CA it is possible to determine the `RandomGen` instance which could be utilized by the `Random` class:

`import System.Random instance RandomGen (Cycle Int) where  next c = let x = c =>> step (rule 30) in (fromBits (view x), x)  split c = (c, fromList (reverse (view c)))`
```λ> let r30 = fromList [1,0,1,0,1,0,1,0,1,0,1,0,1] :: Cycle Int

λ> take 15 \$ randoms r30
[7509,4949,2517,2229,2365,2067,6753,5662,5609,7576,2885,3017,2912,5081,2356]

λ> take 30 \$ randomRs ('A','J') r30
"DHJHHFJHBDDFCBHACHDEHDHFBAEJFE"```

We can compare it with standard generator on a small integer range, using simple bin counter:

```λ> let bins lst = [ (n, length (filter (==n) lst)) | n <- nub lst]

λ> bins . take 10000 . randomRs ('A','J') \$ r30
[('D',1098),('H',1097),('J',1093),('F',850),('B',848),('C',1014),('A',1012),('E',1011),('G',1253),('I',724)]

λ> bins . take 10000 . randomRs ('A','J') <\$> getStdGen
[('G',975),('B',1035),('F',970),('J',1034),('I',956),('H',984),('C',1009),('E',1023),('A',1009),('D',1005)]```

## J

ca is a cellular automata class. The rng class inherits ca and extends it with bit and byte verbs to sample the ca.

` coclass'ca'DOC =: 'locale creation: (RULE ; INITIAL_STATE) conew ''ca'''create =: 3 :'''RULE STATE'' =: y'next =: 3 :'STATE =: RULE (((8\$2) #: [) {~ [: #. [: -. [: |: |.~"1 0&_1 0 [email protected]]) STATE'coclass'base' coclass'rng'coinsert'ca'bit =: 3 :'([ next) ({. STATE)'byte =: [: #. [: , [: bit"0 (i.8)"_coclass'base' `

Having installed these into a j session we create and use the mathematica prng.

```
m =: (30 ; 64 {. 1) conew 'rng'
byte__m"0 i.10
220 197 147 174 117 97 149 171 100 151
```

## Kotlin

Translation of: C
`// version 1.1.51 const val N = 64 fun pow2(x: Int) = 1L shl x fun evolve(state: Long, rule: Int) {    var state2 = state    for (p in 0..9) {        var b = 0        for (q in 7 downTo 0) {            val st = state2            b = (b.toLong() or ((st and 1L) shl q)).toInt()            state2 = 0L            for (i in 0 until N) {                val t = ((st ushr (i - 1)) or (st shl (N + 1 - i)) and 7L).toInt()                if ((rule.toLong() and pow2(t)) != 0L) state2 = state2 or pow2(i)            }        }        print(" \$b")    }    println()} fun main(args: Array<String>) {    evolve(1, 30)}`
Output:
``` 220 197 147 174 117 97 149 171 100 151
```

## Perl

Translation of: Perl 6
`package Automaton {    sub new {    my \$class = shift;    my \$rule = [ reverse split //, sprintf "%08b", shift ];    return bless { rule => \$rule, cells => [ @_ ] }, \$class;    }    sub next {    my \$this = shift;    my @previous = @{\$this->{cells}};    \$this->{cells} = [        @{\$this->{rule}}[        map {          4*\$previous[(\$_ - 1) % @previous]        + 2*\$previous[\$_]        +   \$previous[(\$_ + 1) % @previous]        } 0 .. @previous - 1        ]    ];    return \$this;    }    use overload    q{""} => sub {    my \$this = shift;    join '', map { \$_ ? '#' : ' ' } @{\$this->{cells}}    };} my \$a = Automaton->new(30, 1, map 0, 1 .. 100); for my \$n (1 .. 10) {    my \$sum = 0;    for my \$b (1 .. 8) {	\$sum = \$sum * 2 + \$a->{cells}[0];	\$a->next;    }    print \$sum, \$n == 10 ? "\n" : " ";}`
Output:
`220 197 147 174 117 97 149 171 240 241`

## Perl 6

`class Automaton {    has \$.rule;    has @.cells;    has @.code = \$!rule.fmt('%08b').flip.comb».Int;     method gist { "|{ @!cells.map({+\$_ ?? '#' !! ' '}).join }|" }     method succ {        self.new: :\$!rule, :@!code, :cells(             @!code[                    4 «*« @!cells.rotate(-1)                »+« 2 «*« @!cells                »+«       @!cells.rotate(1)            ]        )    }} my Automaton \$a .= new: :rule(30), :cells( flat 1, 0 xx 100 ); say :2[\$a++.cells[0] xx 8] xx 10;`
Output:
`220 197 147 174 117 97 149 171 240 241`

## Python

### Python: With zero padded ends

`from elementary_cellular_automaton import eca, eca_wrap def rule30bytes(lencells=100):    cells = '1' + '0' * (lencells - 1)    gen = eca(cells, 30)    while True:        yield int(''.join(next(gen)[0] for i in range(8)), 2) if __name__ == '__main__':    print([b for i,b in zip(range(10), rule30bytes())])`
Output:
`[255, 255, 255, 255, 255, 255, 255, 255, 255, 255]`

!

### Python: With wrapping of end cells

`def rule30bytes(lencells=100):    cells = '1' + '0' * (lencells - 1)    gen = eca_wrap(cells, 30)    while True:        yield int(''.join(next(gen)[0] for i in range(8)), 2))`
Output:
`[220, 197, 147, 174, 117, 97, 149, 171, 240, 241]`

## Racket

Implementation of Elementary cellular automaton is saved in "Elementary_cellular_automata.rkt"

`#lang racket;; below is the code from the parent task(require "Elementary_cellular_automata.rkt")(require racket/fixnum) ;; This is the RNG automaton(define (CA30-random-generator         #:rule [rule 30] ; rule 30 is random, maybe you're interested in using others         ;; width of the CA... this is implemented as a number of words plus,         ;; maybe, another word containing the spare bits         #:bits [bits 256])  (define-values [full-words more-bits]    (quotient/remainder bits usable-bits/fixnum))  (define wrap-rule    (and (positive? more-bits) (wrap-rule-truncate-left-word more-bits)))  (define next-gen (CA-next-generation 30 #:wrap-rule wrap-rule))  (define v (make-fxvector (+ full-words (if more-bits 1 0))))  (fxvector-set! v 0 1) ; this bit will always have significance   (define (next-word)    (define-values [v+ o] (next-gen v 0))    (begin0 (fxvector-ref v 0) (set! v v+)))   (lambda (bits)    (for/fold ([acc 0]) ([_ (in-range bits)])      ;; the CA is fixnum, but this function returns integers of arbitrary width      (bitwise-ior (arithmetic-shift acc 1) (bitwise-and (next-word) 1))))) (module+ main  ;; To match the other examples on this page, the automaton is 30+30+4 bits long  ;; (i.e. 64 bits)  (define C30-rand-64 (CA30-random-generator #:bits 64))  ;; this should be the list from "C"  (for/list ([i 10]) (C30-rand-64 8))   ; we also do big numbers...  (number->string (C30-rand-64 256) 16)  (number->string (C30-rand-64 256) 16)  (number->string (C30-rand-64 256) 16)  (number->string (C30-rand-64 256) 16))`
Output:
```(220 197 147 174 117 97 149 171 100 151)
"ecd9fbcdcc34604d833950deb58447124b98706e74ccc74d9337cb4e53f38c5e"
"fc6fd302f5ea8f2fba6f476cfe9d090dc877dbd558e5afba49044d05b14d258"```

## Ruby

`size = 100eca = ElemCellAutomat.new("1"+"0"*(size-1), 30)eca.take(80).map{|line| line[0]}.each_slice(8){|bin| p bin.join.to_i(2)}`
Output:
```220
197
147
174
117
97
149
171
240
241
```

## Rust

` //Assuming the code from the Elementary cellular automaton task is in the namespace.fn main() {    struct WolfGen(ElementaryCA);    impl WolfGen {        fn new() -> WolfGen {            let (_, ca) = ElementaryCA::new(30);            WolfGen(ca)        }        fn next(&mut self) -> u8 {            let mut out = 0;            for i in 0..8 {                out |= ((1 & self.0.next())<<i)as u8;            }            out        }    }    let mut gen = WolfGen::new();    for _ in 0..10 {        print!("{} ", gen.next());    }} `
Output:
```157 209 228 58 87 195 212 106 147 244
```

## Scheme

` ; uses SRFI-1 library http://srfi.schemers.org/srfi-1/srfi-1.html (define (random-r30 n)  (let ((r30 (vector 0 1 1 1 1 0 0 0)))    (fold      (lambda (x y ls)	(if (= x 1)	  (cons (* x y) ls)	  (cons (+ (car ls) (* x y)) (cdr ls))))      '()      (circular-list 1 2 4 8 16 32 64 128)      (unfold-right	(lambda (x) (zero? (car x)))	cadr	(lambda (x) (cons (- (car x) 1)			  (evolve (cdr x) r30)))	(cons (* 8 n) (cons 1 (make-list 79 0))))))) ; list (random-r30 10) `
Output:
```(220 197 147 174 117 97 149 171 240 241)
```

## Sidef

`var auto = Automaton(30, [1] + 100.of(0)); 10.times {    var sum = 0;    8.times {        sum = (2*sum + auto.cells[0]);        auto.next;    };    say sum;};`
Output:
```220
197
147
174
117
97
149
171
240
241
```

## Tcl

Works with: Tcl version 8.6
`oo::class create RandomGenerator {    superclass ElementaryAutomaton    variable s    constructor {stateLength} {	next 30	set s [split 1[string repeat 0 \$stateLength] ""]    }     method rand {} {	set bits {}	while {[llength \$bits] < 8} {	    lappend bits [lindex \$s 0]	    set s [my evolve \$s]	}	return [scan [join \$bits ""] %b]    }}`

Demonstrating:

`set rng [RandomGenerator new 31]for {set r {}} {[llength \$r]<10} {} {    lappend r [\$rng rand]}puts [join \$r ,]`
Output:
```220,197,147,174,241,126,135,130,143,234
```

Note that as the number of state bits is increased (the parameter to the constructor), the sequence tends to a limit of ${\displaystyle 220,}$ ${\displaystyle 197,}$ ${\displaystyle 147,}$ ${\displaystyle 174,}$ ${\displaystyle 117,}$ ${\displaystyle 97,}$ ${\displaystyle 149,}$ ${\displaystyle 171,}$ ${\displaystyle 240,}$ ${\displaystyle 241,}$ ${\displaystyle \ldots }$ and that deviations from this are due to interactions between the state modification “wavefront” as the automaton wraps round.

## zkl

No attempts at extra credit and not fast.

`fcn rule(n){ n=n.toString(2); "00000000"[n.len() - 8,*] + n }fcn applyRule(rule,cells){   cells=String(cells[-1],cells,cells[0]);  // wrap edges   (cells.len() - 2).pump(String,'wrap(n){ rule[7 - cells[n,3].toInt(2)] })}fcn rand30{   var r30=rule(30), cells="0"*63 + 1; // 64 bits (8 bytes), arbitrary   n:=0;   do(8){      n=n*2 + cells[-1];          // append bit 0      cells=applyRule(r30,cells); // next state   }   n}`

Note that "var" in a function is "static" in C, ie function local variables, initialized once.

`do(10){ rand30().print(","); }`
Output:
`220,197,147,174,117,97,149,171,100,151,`