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

• Cellular automata: Is Rule 30 random? (PDF).

C

64-bits array size, cyclic borders. <lang c>#include <stdio.h>

1. include <limits.h>

typedef unsigned long long ull;

1. define N (sizeof(ull) * CHAR_BIT)
2. 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; }</lang>

Output:

 220 197 147 174 117 97 149 171 100 151

C++

We'll re-write the code of the parent task here. <lang cpp>#include <bitset>

1. include <stdio.h>

1. define SIZE 80
2. define RULE 30
3. 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;

}</lang>

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. <lang d>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;

}</lang>

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

<lang go>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)

}</lang>

Output:

220 197 147 174 117 97 149 171 100 151 

Haskell

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

<lang Haskell>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</lang>

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:

<lang Haskell>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)))</lang>

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. <lang J> 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 1@]) STATE' coclass'base'

coclass'rng' coinsert'ca' bit =: 3 :'([ next) ({. STATE)' byte =: [: #. [: , [: bit"0 (i.8)"_ coclass'base' </lang> 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

<lang scala>// 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)

}</lang>

Output:

 220 197 147 174 117 97 149 171 100 151

Pascal

Works with: Free Pascal

Use 32-Bit assembler for speed like in Delphi-forum years ago.Sometimes it helps.
Speedtest only for Next_State_Rule_30, no extraction of Bits like in D [1] with 8x2E6 calls. <lang pascal>Program Rule30; //https://www.entwickler-ecke.de/viewtopic.php?t=111812 //http://en.wikipedia.org/wiki/Next_State_Rule_30; //http://mathworld.wolfram.com/Rule30.html {$IFDEF FPC}

 {$Mode Delphi}
 {$ASMMODE INTEL}
 {$OPTIMIZATION ON,ALL}
 {$CODEALIGN proc=8}

{$ELSE}

 {$APPTYPE CONSOLE}

{$ENDIF} uses

 SysUtils;

const

 maxRounds = 2*1000*1000;
 rounds    = 10;

 CpuF = 3.7e9; // AMD Ph XII 955 3.2 Ghz  // Ryzen 5 1600 Turbo 3.7 Ghz

const

 RULE30_BITSIZE = 8*8;

 SizeOfRegister  = SizeOf(NativeUint);
 BitsPerRegister = 8*SizeOfRegister;

 Limit32Arr = RULE30_BITSIZE DIV BitsPerRegister -1;
 Limit08Arr = RULE30_BITSIZE DIV 8 -1;

type

 tLimit32 = 0..Limit32Arr+1;
 tLimit08 = 0..Limit08Arr+1;
 tArr32 = Array[tLimit32] OF Uint32;
 tArr08 = Array[tLimit08] OF BYTE;
 tpArr08 =^tArr08;

var {$ALIGN 32}

 Rule30_State : tArr32;

procedure InitRule30_State; var

 i : integer;

begin

 Rule30_State[Low(tArr32)]:= 1;
 For i := Low(tArr32)+1 to High(tArr32) do
   Rule30_State[i] := 0;

end;

function BinStr(Zahl: Uint32): String; var

 i : integer;

begin

 setlength(result,9);
 result[1] :='_';
 For i := 0 to 7 do
 begin
   result[7-i+2] := chr(Zahl AND 1+Ord('0'));
   Zahl := Zahl shr 1;
 end;

end;

procedure Out_Rule30_State; var

 i : integer;
 pArr08 :tpArr08;

begin

 pArr08 := @Rule30_State[0];
 For i := High(tLimit08)-1 Downto LOW(tLimit08) do
   write(pArr08^[i]:4,BinStr(pArr08^[i]));
 write('D',BinStr(pArr08^[High(tLimit08)-1]));
 writeln;

end;

function dummy(a:pUint32):Uint32;assembler; asm

end;

function Next_State_Rule_30(a:pUint32):Uint32;assembler; //EAX = a , EDX free to use //EBX Rule30_State[0] //ESI index 0..LimitArr-1

//EAX value //ECX value one bit to the right //EDX value one bit to the left //EDI next value asm

 push EBX; push ESI;push EDI;

 MOV   EBX,EAX
 MOV   ESI,Limit32Arr*SizeOfRegister
 ADD   ESI,EBX

 MOV   ECX,Dword Ptr [ESI];  // the highest position into previous
 MOV   EAX,Dword Ptr [EBX];                // the lowest
 MOV   Dword Ptr [ESI+SizeOfRegister],EAX; // into one behind the end

@Loop:

 MOV   EDI,Dword Ptr [EBX+SizeOfRegister]; // the next

 BT    ECX,31      // MSB of prev
 MOV   ECX,EAX
 RCL   ECX,1       // shift MSB into LSB

 BT    EDI,0       // LSB of next
 MOV   EDX,EAX
 RCR   EDX,1       // shift LSB into MSB

  OR   EDX,EAX     // POS[i] OR POS[i+1]
 XOR   ECX,EDX     // POS[i] XOR (POS[i] OR POS[i+1])
 MOV   Dword Ptr [EBX],ECX;  // save
 ADD   EBX,SizeOfRegister    // next Pos
 CMP   EBX,ESI     // MOV does not change flags
 MOV   ECX,EAX     // running to previous
 MOV   EAX,EDI     // next to running
 JBE   @Loop

 POP EDI;POP ESI;POP EBX;

end;// ['EBX','ESI','EDI'];

procedure Speedtest; var

 T1,T0 : TDateTime;
 i,j,b: NativeInt;

Begin

 writeln('Speedtest for statesize of ',RULE30_BITSIZE,' bits');
 InitRule30_State;
 T0 := time;
 For  i := 8*maxRounds-1 downto 0 do
   dummy(@Rule30_State[0]);
 T1 := time;
 writeln('Dummy calls ',FormatDateTime('HH:NN:SS.zzz',T1-T0));
 // Takte pro Durchlauf
 writeln('cycles per call : ',((T1-t0)*86400*CpuF)/(maxRounds):0:2);
 Out_Rule30_State;
 T0 := time;
 For  i := maxRounds-1 downto 0 do
 Begin
   b := 0;
   For j := 7 downto 0 do
   Begin
     b := (b+b) OR (Rule30_State[0] AND 1);
     Next_State_Rule_30(@Rule30_State[0]);
   end;
 end;
 T1 := time;
 Out_Rule30_State;
 writeln(maxRounds,' calls take ',FormatDateTime('HH:NN:SS.zzz',T1-T0));
 writeln('cycles per Byte : ',((T1-t0)*86400*CpuF)/maxRounds:0:2);
 writeln;

end;

procedure Task; var

 k,j,b: integer;

Begin

 writeln('The task ');
 InitRule30_State;
 For  k := 1 to rounds do
 Begin
   b := 0;
   For j := 7 downto 0 do
   Begin
     b := (b+b) OR (Rule30_State[0] AND 1);
     Next_State_Rule_30(@Rule30_State[0]);
   end;
   write(b:4);
 end;
 writeln;
 writeln;

end;

Begin

 SpeedTest;
 Task;
 readln;

end.</lang>

Output:

Speedtest for statesize of 64 bits
Dummy calls 00:00:00.031
cycles per call : 57.35
   0_00000000   0_00000000   0_00000000   0_00000000   0_00000000   0_00000000   0_00000000   1_00000001D_00000000
 206_11001110 118_01110110 103_01100111 126_01111110 188_10111100 192_11000000 103_01100111 206_11001110D_11001110
2000000 calls take 00:00:00.069
cycles per Byte : 127.65

The task 
 220 197 147 174 117  97 149 171 100 151

Perl

Translation of: Perl 6

<lang perl>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" : " ";

}</lang>

Output:

220 197 147 174 117 97 149 171 240 241

Perl 6

<lang perl6>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;</lang>

Output:

220 197 147 174 117 97 149 171 240 241

Phix

Making the minimum possible changes to Elementary_cellular_automaton#Phix, output matches C, D, Go, J, Kotlin, Racket, and zkl, and with the changes marked [2] C++, Haskell, Perl, Python, Ruby, Scheme, and Sidef, but completely different to Rust and Tcl. No attempt to optimise. <lang Phix>--string s = ".........#.........", --(original) string s = "...............................#"&

          "................................", 

--string s = "#"&repeat('.',100), -- [2]

      t=s, r = "........"

integer rule = 30, k, l = length(s), w = 0 for i=1 to 8 do

   r[i] = iff(mod(rule,2)?'#':'.')
   rule = floor(rule/2)

end for sequence res = {} for i=0 to 80 do

   w = w*2 + (s[32]='#')

-- w = w*2 + (s[1]='#') -- [2]

   if mod(i+1,8)=0 then res&=w w=0 end if
   for j=1 to l do
       k = (s[iff(j=1?l:j-1)]='#')*4
         + (s[          j   ]='#')*2
         + (s[iff(j=l?1:j+1)]='#')+1
       t[j] = r[k]
   end for
   s = t

end for ?res</lang>

Output:

{220,197,147,174,117,97,149,171,100,151}

Output:

with the changes marked [2]

{220,197,147,174,117,97,149,171,240,241}

Python

Python: With zero padded ends

<lang python>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())])</lang>

Output:

[255, 255, 255, 255, 255, 255, 255, 255, 255, 255]

!

Python: With wrapping of end cells

<lang python>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))</lang>

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>#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))</lang>

Output:

(220 197 147 174 117 97 149 171 100 151)
"ecd9fbcdcc34604d833950deb58447124b98706e74ccc74d9337cb4e53f38c5e"
"9c8b6471a4bc2cb3508f10b6635e4eb959ad8bbe484480695e8ddb5795f956a"
"6d85153a987dad6f013bc6159a41bf95b9d9b14af87733e17c702a3dc9052172"
"fc6fd302f5ea8f2fba6f476cfe9d090dc877dbd558e5afba49044d05b14d258"

Ruby

<lang ruby>size = 100 eca = ElemCellAutomat.new("1"+"0"*(size-1), 30) eca.take(80).map{|line| line[0]}.each_slice(8){|bin| p bin.join.to_i(2)}</lang>

Output:

220
197
147
174
117
97
149
171
240
241

Rust

<lang 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());
   }

} </lang>

Output:

157 209 228 58 87 195 212 106 147 244 

Scheme

<lang 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) </lang>

Output:

(220 197 147 174 117 97 149 171 240 241)

Sidef

<lang ruby>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;

};</lang>

Output:

220
197
147
174
117
97
149
171
240
241

Tcl

Works with: Tcl version 8.6

<lang tcl>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]

   }

}</lang> Demonstrating: <lang tcl>set rng [RandomGenerator new 31] for {set r {}} {[llength $r]<10} {} {

   lappend r [$rng rand]

} puts [join $r ,]</lang>

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 $220,$ $197,$ $147,$ $174,$ $117,$ $97,$ $149,$ $171,$ $240,$ $241,$ $\ldots$ 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. <lang zkl>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

}</lang> Note that "var" in a function is "static" in C, ie function local variables, initialized once. <lang zkl>do(10){ rand30().print(","); }</lang>

Output:

220,197,147,174,117,97,149,171,100,151,