On a Class of Reversible Primitive Recursive Functions and Its Turing-Complete Extensions

Reversible computing is both forward and backward deterministic. This means that a uniquely determined step exists from the previous computational configuration (backward determinism) to the next one (forward determinism) and vice versa. We present the reversible primitive recursive functions (RPRF), a class of reversible (endo-)functions over natural numbers which allows to capture interesting extensional aspects of reversible computation in a formalism quite close to that of classical primitive recursive functions. The class RPRF can express bijections over integers (not only natural numbers), is expressive enough to admit an embedding of the primitive recursive functions and, of course, its evaluation is effective. We also extend RPRF to obtain a new class of functions which are effective and Turing complete, and represent all Kleene’s -recursive functions. Finally, we consider reversible recursion schemes that lead outside the reversible endo-functions.