Can every recursion be converted into iteration?

Question

A reddit thread brought up an apparently interesting question:

Tail recursive functions can trivially be converted into iterative functions. Other one

Answer 1

Is it always possible to write a non-recursive form for every recursive function?

Yes. A simple formal proof is to show that both µ recursion and a non-recursive calculus such as GOTO are both Turing complete. Since all Turing complete calculi are strictly equivalent in their expressive power, all recursive functions can be implemented by the non-recursive Turing-complete calculus.

Unfortunately, I’m unable to find a good, formal definition of GOTO online so here’s one:

A GOTO program is a sequence of commands P executed on a register machine such that P is one of the following:

• HALT, which halts execution
• r = r + 1 where r is any register
• r = r – 1 where r is any register
• GOTO x where x is a label
• IF r ≠ 0 GOTO x where r is any register and x is a label
• A label, followed by any of the above commands.

However, the conversions between recursive and non-recursive functions isn’t always trivial (except by mindless manual re-implementation of the call stack).

For further information see this answer.