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Brainfuck with while 0 loops
This page is proof that brainfuck with unbounded tape and while 0 loops is Turing complete, which is proven by simulation of Cyclic Tag.
Initialise data
put +>> at the start of a program then initialise program with 0 being >+> and 1 being >>. After initialising data append >+<<[<<]>>[
Simulating Cyclic Tag
Replace every ; with +>>, 0 with +>[[>>]->+>+<<[<<]->+]<[+>-<]- and 1 with +>[[>>]->>+<<[<<]->+]<[+>-<]-. End program with ]. It also proves that it is Turing-complete with nesting depth of 3
Explanation
To simulate Cyclic Tag in this version of brainfuck we will use following layout:
10 xy xy xy ... 1
x is 1 if bit is deleted and 0 otherwise. y is 1 if bit is 0 and 0 if bit is 1.