B^2 Turing Completeness

This page will convert brainfuck, a known Turing complete language, into B². B² is not turing complete but is a bounded-storage machine (see "Counter Argument" for details).

Required For Turing-Completeness

Brainfuck	B2 Equivalent
+	x = add x 1 ;
-	x = sub x 1 ;
[	while ( x == 0 ) {
]	}
<	No exact match.
>	No exact match.

Not Required

Brainfuck	B2 Equivalent
.	output x ;
,	number x = input ; or decimal x = input ;

Counter-argument

If one argues that < and > could also be represented using variables, B² is still not turing complete. These variables are all of limited size, and only a limited amount of them are accessible. Even a program like this:

number cell = firstCell ;
if ( currentCell == 1 ) {
cell = secondCell ;
}
if ( currentCell == 2 ) {
cell = thirdCell ;
}

for however many variables/cells are used, that's still a limited amount of variables — and although that would be Turing-complete if variables were unbounded, they're not, and since a finite amount of variables times a finite size for each of those variables equals a finite amount of storage, while a Turing machine needs an infinite (or unbounded) amount of storage, B² isn't Turing complete. Since bounded storage machines have finite storage, but are the same as Turing machines in all other respects (just like B²), B² is not Turing-complete but is a bounded-storage machine.