|Six of the commands are identical to brainfuck.|
||Increment the current cell.|
||Decrement the current cell.|
||The cell to the right of the current cell becomes current.|
||The cell to the left of the current cell becomes current.|
||Output a character whose ASCII code is that of the current cell.|
|| If the current cell is nonzero, jump backwards to just after the matching |
|Two of the commands are changed.|
||Input a character and store its ASCII code in the current cell, if it contains a 0. If the current cell doesn't contain a 0, end the program. EOF is treated as if a 0 were read.|
|| If the current cell is nonzero, jump forwards to just after the matching |
Reversible Brainfuck uses a right-infinite tape. Each cell on the tape must be able to be decremented 255 times from zero and incremented 255 times from zero without giving a zero result on any increment. (However, it's legal to use a wrapping system where, for instance, -128 and 128 are the same number.)
Some algorithms to use in later computational class constructions.
Inspired by the notation in brainfuck algorithms, the balanced loop algorithms below don't use
<> directly. For portability, other commands are instead labeled with the cell they affect. To make this more visually reversible, the labels are put on a separate line above the commands.
Moving a value
Moving from cell
x to cell
y, initially zero, using temp flag
MOV(x,y;f): x=a>=0, y=0, f=0 --> x=0, y=a, f=0 x f x f y f y x f x f y f y [ + ] [ [ + ]+ -[ - ] ] [ - ]
Other commands can be inserted inside the main loop, e.g. to do addition:
ADD(x,y,z;f): x=a>=0, y=0, z=b, f=0 --> x=0, y=a, z=a+b, f=0 x f x f y f y z x f x f y f y [ + ] [ [ + ]+ + -[ - ] ] [ - ]
Two methods of simulating a wrapping cell with cell size
b (either with unbounded cells or simply with a larger wrapping) are given.
The first one can also be used if
b is not statically known. It uses two cells, one with the intended value
a and the other with value
b-a. In addition, increment uses a temp flag with value 1:
W1+(x,y;f): x=a, y=b-a, f=1 --> x=(a+1)%b, y=b-(a+1)%b, f=1 x f x f x f x y f y f y f y x W1+(x,y;f) = [ - ] [ [ + ]+ -[ - ] ] [ + ]+ -
Reverse to decrement:
x y f y f y f y x f x f x f x W1-(x,y;f) = + -[ - ] [ [ + ]+ -[ - ] ] [ + ]
The second method is defined recursively on
b, so requires it to be statically known, and is also more complicated except for
b=2. However, it uses one fewer cell:
W2+(x;f;b): x=a, f=0 --> x=(a+1)%b, f=0 x f x f x f x f x f W2+(x;f;2) = [ + ]+ [ -- ] [ + ] - x f x f x x f x f x f x f x f x W2+(x;f;b) = [ + ] [ - W2+(x;f;b-1) ] [ W2+(f;x;2) ] [ + ] [ - ]
b=2 is its own inverse):
x f x f x f x f x f x x f x f x W2-(x;f;b) = [ + ] [ - ] [ W2+(f;x;2) ] [ W2-(x;f;b-1) + ] [ - ]
The following table shows how each subloop in the recursive
W2+ case acts on
a=0 | 1<=a<b | a=b -------+----------+------- (0, 0) | (a, 0) | (b, 0) (0, 1) | (a, 0) | (b, 0) (0, 1) | (a, 0) | (0, 0) (0, 0) | (a, 0) | (0, 1) (1, 0) | (a+1, 0) | (0, 1) (1, 0) | (a+1, 0) | (0, 0)
Pushing / popping digits from a base
This algorithm and its reverse can be used to implement a stack, and to do multiplication/division with remainder, which will be the core of later computational class proofs.
X is a wrapping cell with size
b, simulated using either of the methods above.
PUSH(X,s,t;f): X=a, s=c>=0, t=0, f=0 --> X=0, s=0, t=a+b*c, f=0 X s f s X f t f t X s X s f s X f t f t PUSH(X,s,t;f) = [ [ + ] ] [ [ + ]+ [ - ]W-[ [ - ] ] ] [ - ]
The algorithm here is essentially the one for
MOV(X,t;f) adjusted such that both
s must be 0 to skip/exit the loop, and such that
s is decremented whenever
X is decremented from 0.
t f t f X s f s X s X t f t f X s f s X POP(X,s,t;f) = [ + ] [ [ [ + ] ]W+[ + ] -[ - ] ] [ [ - ] ]
Using the first wrapping method where
X is implemented with
y summing to
PUSH1(x,y,s,t;f) expands to:
x s f s x f t f t x s f x f x f x y f y f y f y x s f s x f t f t [ [ + ] ] [ [ + ]+ [ - - ] [ [ + ]+ -[ - ] ] [ + ]+ -[ [ - ] ] ] [ - ]
Using the second method,
f needs to be adjusted:
x s f s x f t f t x s x f x f x s f s x f t f t PUSH2(x,s,t;f;b) = [ [ + ] ] [ [ + ]+ [ - ] - W2-(x;f;b) + [ [ - ] ] ] [ - ]
To avoid the impossible deletion of information in a reversible language, the translation works by keeping a complete history of all branching.
Note that all non-brainfuck data cells will contain only values 0 or 1. Therefore this reduction can be applied to any cell size, even single bit cells. (E.g. this also proves Turing complete reversible boolfuck.)
|0||0||Left boundary data search mark|
|1||0||Left boundary history search mark|
|2||d0||First brainfuck data cell|
|3||h0||First branch history flag|
|4n||1||Search marks to the left of current brainfuck data cell|
|0||Search marks to the right of current brainfuck data cell|
|4n+1||1||Search marks to the left of current branch history flag|
|0||Search marks to the right of current branch history flag|
|4n+2||dn||Brainfuck data cell|
|4n+3||hn||Branch history flag|
|Brainfuck command||Reversible brainfuck|
[>>[<<<<]>[>>>>]<< +>>[<<<<]<[>>>>]<< ]>>[<<<<]>[>>>>]<< [>>+>> >>[<<<<]<[>>>>]<<
>>[<<<<]>[>>>>]<< +>>[<<<<]<[>>>>]<< [>>[<<<<]>[>>>>]<< ->>[<<<<]<[>>>>]<< ]>>[<<<<]>[>>>>]<< ]>>+>> >>[<<<<]<[>>>>]<<
Unbounded cells, bounded tape
The previous construction shows that Reversible Brainfuck with bounded cells and unbounded tape is Turing complete. Using the notation from the Algorithms section, two well-known constructions for Minsky machines can be adapted to give further reductions to a bounded tape with unbounded (only non-negative values used) cells, thus proving that too Turing complete.
The first construction does a reduction from an unbounded tape with size
b cells to five unbounded registers:
x- source program's current cell
f- status flag
l- left part of tape, as a base
r- right part of tape, as a base
t- temporary storage
x x x x [ ] . , are implemented as [ ] . , respectively (that is, as themselves applied at cell x). x x + W2+(x;f;b), or just + if known not to wrap. x x - W2-(x;f;b), or just - if known not to wrap. > PUSH2(x,l,t;f;b) MOV(t,l;f) POP2(x,r,t;f;b) MOV(t,r;f) < PUSH2(x,r,t;f;b) MOV(t,r;f) POP2(x,l,t;f;b) MOV(t,l;f)
The second construction translates a cell labeled program on an already bounded tape of unbounded non-negative cells, to a similar program using only four cells. I/O is not supported.
An original RBF program needs first to be put in cell labeled notation, which means each command must be applied to a statically known cell, which is equivalent to all of its loops having an equal number of
>s, aka a "balanced loop" program.
This construction encodes all the source cells as prime exponents of a single number. First, to each cell
c in the source program assign a distinct prime number
p(x)=2, p(f)=3, p(l)=5, p(r)=7, p(t)=11
The cells of the target program are:
x- used for divisibility checking, otherwise kept zero
f- status flag
n- number whose prime exponents encode the source cells
t- temporary storage
Next, translate each basic labelled command as follows:
n Start + (Initialize n to 1, or all exponents to 0.) c + PUSH2(x,n,t;f;p(c)) MOV(t,n;f) (multiplies n by p(c)) c - POP2(x,t,n;f;p(c)) MOV(t,n;f) (divides n by p(c), undefined behavior if not divisible) c x f x f x f x [ POP2(x,t,n;f;p(c)) [ + ] [ [ - ] PUSH2(x,t,n;f;p(c)) (divides n by p(c), checks the remainder to branch, then multiplies back) c x f x f x f x ] POP2(x,t,n;f;p(c)) [ + ] ] [ - ] PUSH2(x,t,n;f;p(c)) (ditto)
As a final note, the resulting labeled 4 cell program can also be interpreted as a "reversible Minsky machine", whose registers are
t and whose finite state consists of a combination of the RBF program position, and the cells
f (whose values are bounded).
This then proves the probably well-known result that a reversible machine needs only two unbounded registers to be TC, same as a non-reversible one.
weave.rb supports Reversible Brainfuck with the -r flag.