Bitstring Physics

Question

Background

Yes, bitstring physics is a real thing. The idea is to construct a new theory of physics using only strings of bits that evolve under a probabilistic rule... or something. Despite reading a couple of papers about it, I'm still pretty confused. However, the bitstring universe makes for a nice little code golf.

Program Universe

Bitstring physics takes place in a so-called program universe. At each step of the evolution of the universe, there is a finite list L of bitstrings of some length k, starting with the two-element list [10,11] where k = 2. One timestep is processed as follows (in Python-like pseudocode).

A := random element of L
B := random element of L
if A == B:
    for each C in L:
        append a random bit to C
else:
    append the bitwise XOR of A and B to L

All random choices are uniformly random and independent of each other.

Example

An example evolution of 4 steps might look like the following. Start with the initial list L:

10
11

We randomly choose A := 10 and B := 10, which are the same row, which means we need to extend each string in L with a random bit:

101
110

Next, we choose A := 101 and B := 110, and since they are not equal, we add their XOR to L:

101
110
011

Then, we choose A := 011 and B := 110, and again append their XOR:

101
110
011
101

Finally, we choose A := 101 (last row) and B := 101 (first row), which are equal, so we extend with random bits:

1010
1100
0111
1010

The Task

Your task is to take a nonnegative integer t as input, simulate the program universe for t timesteps, and return or print the resulting list L. Note that t = 0 results in the initial list [10,11]. You can output L as a list of lists of integers, list of lists of boolean values or a list of strings; if output goes to STDOUT, you may also print the bitstrings one per line in some reasonable format. The order of the bitstrings is significant; in particular, the initial list cannot be [11,10], [01,11] or anything like that. Both functions and full programs are acceptable, standard loopholes are disallowed, and the lowest byte count wins.

Answer 1

Pyth, 27 26 bytes

u?+RO2GqFKmOG2aGxVFKQ*]1U2

Try it online: Demonstration

Explanation:

                              implicit: Q = input number
                     *]1U2    the initial list [[1,0], [1,1]]
u                   Q         reduce, apply the following expression Q times to G = ^
          mOG2                  take two random elements of G
         K                      store in K
       qF                       check if they are equal
 ?                              if they are equal:
  +RO2G                           append randomly a 0 or 1 to each element of G
                                else:
              aG                  append to G
                xVFK              the xor of the elements in K

Answer 2

CJam, 42 40 38 37 bytes

1 byte saved by Sp3000.

B2b2/q~{:L_]:mR_~#L@~.^a+L{2mr+}%?}*p

Explanation

Create the initial state as a base-2 number:

B2b e# Push the the binary representation of 11: [1 0 1 1]
2/  e# Split into chunks of 2 to get [[1 0] [1 1]]

And then perform the main loop and pretty-print the result at the end:

q~       e# Read and eval input t.
{        e# Run this block t times.
  :L     e#   Store the current universe in L.
  _]     e#   Copy it and wrap both copies in an array.
  :mR    e#   Pick a random element from each copy.
  _~     e#   Duplicate those two elements, and unwrap them.
  #      e#   Find the second element in the first. If they are equal, it will be found at
         e#   index 0, being falsy. If they are unequal, it will not be found, giving
         e#   -1, which is truthy.

         e#   We'll now compute both possible universes for the next step and then select
         e#   the right one based on this index. First, we'll build the one where they were
         e#   not equal.

  L@~    e#   Push L, pull up the other copy of the selected elements and unwrap it.
  .^     e#   Take the bitwise XOR.
  a+     e#   Append this element to L.

  L      e#   Push L again.
  {      e#   Map this block onto the elements in L.
    2mr+ e#     Append 0 or 1 at random. 
  }%     
  ?      e#   Select the correct follow-up universe.
}*
p        e# Pretty-print the final universe.

Test it here.

Answer 3

Julia, 141 129 bytes

t->(L=Any[[1,0],[1,1]];for i=1:t r=1:length(L);A=L[rand(r)];B=L[rand(r)];A==B?for j=r L[j]=[L[j],rand(0:1)]end:push!(L,A$B)end;L)

Nothing clever. Creates an unnamed function that accepts an integer as input and returns an array of arrays. To call it, give it a name, e.g. f=t->....

Ungolfed + explanation:

function f(t)
    # Start with L0
    L = Any[[1,0], [1,1]]

    # Repeat for t steps
    for i = 1:t
        # Store the range of the indices of L
        r = 1:length(L)

        # Select 2 random elements
        A = L[rand(r)]
        B = L[rand(r)]

        if A == B
            # Append a random bit to each element of L
            for j = r
                L[j] = [L[j], rand(0:1)]
            end
        else
            # Append the XOR of A and B to L
            push!(L, A $ B)
        end
    end

    # Return the updated list
    L
end

Examples:

julia> f(4)
4-element Array{Any,1}:
 [1,0,1,0]
 [1,1,1,1]
 [0,1,1,0]
 [0,1,0,0]

julia> f(3)
3-element Array{Any,1}:
 [1,0,1,1]
 [1,1,1,0]
 [0,1,0,1]

Saved 12 bytes thanks to M L!

Answer 4

Python 2, 141

I tried a few different methods, but the best I could get was relatively straightforward. Thanks to @Sp3000 for 15 chars or so (and for teaching me about the existence of int.__xor__).

from random import*
L=[[1,0],[1,1]];C=choice
exec"A=C(L);B=C(L);L=[L+[map(int.__xor__,A,B)],[x+[C([1,0])]for x in L]][A==B];"*input()
print L

Answer 5

Python 2, 127 122

Assuming python bit strings of the form '0b1' etc. are OK:

from random import*
C=choice
L=[2,3]
exec"x=C(L)^C(L);L=L+[x]if x else[a*2+C([0,1])for a in L];"*input()
print map(bin,L)

Only mild nuance here is use of the fact that XOR(A,B)=0 iff A=B.

Thanks to @Sp300 for shortening the enclosing for loop

Answer 6

Pyth, 34

u?+RO`TGqJOGKOG+Gsm`s!dqVJKQ[`T`11

Uses reduce to apply each iteration. I'll explain when I'm done golfing.

Try it here

Answer 7

K, 46 53 46 bytes

{x{:[~/t:2?x;{x,*1?2}'x;x,,,/~=/t]}/(1 0;1 1)}

A good chunk of the size (about 7 bytes) of this is do to the fact that K has no xor operator, so I had to implement one myself. Originally, I used a list of strings, followed by realizing that was insanely stupid. So now I cut off the 7 bytes again!

Before:

{x{:[~/t:2?x;{x,*$1?2}'x;x,,,/$~=/(0$')'t]}/$:'10 11}

@JohnE pointed out in the comments that the initial state was supposed to be hardcoded, which cost 7 extra bytes. :/

Answer 8

JavaScript (ES6) 152

A function, using strings (with numbers it should be shorter, but in javascript bit operations are limited to 32 bit integers).

Test in Firefox using the snippet below.

F=(t,L=['10','11'],l=2,R=n=>Math.random()*n|0,a=L[R(l)],b=L[R(l)])=>
   t--?a==b
     ?F(t,L.map(x=>x+R(2)),l)
     :F(t,L,L.push([...a].map((x,p)=>x^b[p]).join('')))
  :L
  
test=_=>O.innerHTML=F(+I.value).join('\n')

#I{width:3em}

<input id=I value=10><button onclick=test()>-></button><pre id=O></pre>

Answer 9

Javascript, 241 233 bytes

That's kind of long.

a=[[1,0],[1,1]];for(b=prompt();b--;)if(c=a.length,d=a[c*Math.random()|0],e=a[c*Math.random()|0],d+""==e+"")for(f=0;f<c;f++)a[f].push(2*Math.random()|0);else{g=[];for(h=0;h<d.length;h++)g.push(d[h]^e[h]);a.push(g)}alert(a.join("\n"));

Answer 10

K, 45 41 38 bytes

{x{(x,,~=/t;{x,1?2}'x)@~/t:2?x}/1,'!2}

The structure of my answer is quite similar to that of @kirbyfan64sos, but instead of strings I used 1/0 vectors and I avoid the need for a conditional (:[ ; ; ]) by instead indexing into a list.

A few runs:

  {x{(x,,~=/t;{x,1?2}'x)@~/t:2?x}/1,'!2}3
(1 0 0 0
 1 1 1 1
 0 1 1 1)

  {x{(x,,~=/t;{x,1?2}'x)@~/t:2?x}/1,'!2}3
(1 0 0
 1 1 0
 0 1 0
 1 0 0)

  {x{(x,,~=/t;{x,1?2}'x)@~/t:2?x}/1,'!2}3
(1 0 0
 1 1 0
 0 1 0
 1 1 0)

Edit:

Saved four bytes with a more compact way to build the initial universe:

1,'!2     / new
(1 0;1 1) / old

Edit2:

I forgot that "choose" can take a list as its right argument:

  2?"abcd"
"dc"
  2?"abcd"
"cc"
  2?"abcd"
"ca"

So I can simplify part of this. Credit where it's due, Kirby got this trick before I did.

2?x    / new
x@2?#x / old

Answer 11

T-SQL (2012+), 1019

I'm really sorry this is nowhere near competitive, but to be honest I didn't think I could get this to work and had to post it once I did. I did try to golf it a bit :)

To handle the binary/integer conversions I had to create a couple of scalar functions (513 of the bytes). A goes from integer to a bit string. B does the reverse.

CREATE FUNCTION A(@ BIGINT)RETURNS VARCHAR(MAX)AS
BEGIN
DECLARE @S VARCHAR(MAX);
WITH R AS(SELECT @/2D,CAST(@%2 AS VARCHAR(MAX))M
UNION ALL
SELECT D/2,CAST(D%2 AS VARCHAR(MAX))+M
FROM R
WHERE D>0)SELECT @S=M FROM R WHERE D=0
RETURN @S
END
CREATE FUNCTION B(@ VARCHAR(MAX))RETURNS BIGINT AS
BEGIN
DECLARE @I BIGINT;
WITH R AS(SELECT CAST(RIGHT(@,1)AS BIGINT)I,1N,LEFT(@,LEN(@)-1)S
UNION ALL 
SELECT CAST(RIGHT(S,1)AS BIGINT)*POWER(2,N),N+1,LEFT(S,LEN(S)-1)
FROM R
WHERE S<>''
)SELECT @I=SUM(I)FROM R
RETURN @I
END

Then there is the procedure. @C is the number of steps

DECLARE @C INT=9
DECLARE @ TABLE(S VARCHAR(MAX))
DECLARE @R VARCHAR(MAX)
INSERT @ VALUES('10'),('11')
WHILE(@C>=0)
BEGIN
SET @C-=1
SELECT @R=CASE WHEN MAX(S)=MIN(S)THEN''ELSE RIGHT(REPLICATE('0',99)+dbo.A(dbo.B(MAX(S))^dbo.B(MIN(S))),LEN(MAX(S)))END
FROM(SELECT TOP 2S,ROW_NUMBER()OVER(ORDER BY(SELECT\))N FROM @,(VALUES(1),(1),(1))D(D)ORDER BY RAND(CAST(NEWID()AS VARBINARY(50))))A
IF @R=''UPDATE @ SET S=CONCAT(S,ROUND(RAND(CAST(NEWID() AS VARBINARY(50))),0))
ELSE INSERT @ VALUES(@R)
END
SELECT * FROM @

Ten thousand iterations took around 2 minutes and returned 9991 rows

1001001100110
1101001001110
0111100100101
1111100001011
1111001010011
0110101001101
...
1110101000100
1111011101100
1100001100010
0110010001001
1110100010100

Answer 12

Pyth - 37 bytes

Obvious, just follows psuedocode. Probably can golf a lot.

KPP^,1Z2VQ=K?m+dO1KqFJ,OKOKaKxVhJeJ;K

Try it here online.

Answer 13

Mathematica, 106 bytes

Nest[If[Equal@@#,Map[#~Append~RandomInteger[]&],Append[BitXor@@#]]&[#~RandomChoice~2]@#&,{{1,0},{1,1}},#]&

Answer 14

Perl, 102

#!perl -p
@l=qw(10 11);$_=$l[rand@l]^$l[rand@l],$_|=y//0/cr,@l=/1/?(@l,$_):map{$_.~~rand 2}@l for 1..$_;$_="@l"

Try me.

Answer 15

R, 186

L=list(0:1,c(1,1))
if(t>0)for(t in 1:t){A=sample(L,1)[[1]]
B=sample(L,1)[[1]]
if(all(A==B)){L=lapply(L,append,sample(0:1, 1))}else{L=c(L,list(as.numeric(xor(A,B))))}}
L

Nothing magical here. Enter the value for t in the R console and run the script. It's hard to "golf" R code but here's a more readable version:

L <- list(0:1, c(1, 1))
if(t > 0) {
  for(t in 1:t) {
    A <- sample(L, 1)[[1]]
    B <- sample(L, 1)[[1]]
    if (all(A == B)) {
      L <- lapply(L, append, sample(0:1, 1))
    } else {
      L <- c(L,list(as.numeric(xor(A, B))))
    }
  }
}
L

Answer 16

Ruby, 82

Pretty much straight-forward. Compared with other non-golfing languages, ruby seems to do well with its large standard library.

->t{l=[2,3]
t.times{a,b=l.sample 2
a.equal?(b)?l.map!{|x|x*2+rand(2)}:l<<(a^b)}
l}

Sample output for t=101010 :

[9, 15, 6, 13, 5, 12, 10, 11, 5, 4, 15, 13, 2, 7, 11, 9, 3, 3, 8, 6, 3, 13, 13, 12, 10, 9, 2, 4, 14, 9, 9, 14, 15, 7, 10, 4, 10, 14, 13, 7, 15, 7]