Generation of invertible functions

Reversible computing is both forward and backward deterministic. This means that a uniquely determined step exists from the previous computational conguration (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 eective. We also extend RPRF to obtain a new class of functions which are eective and Turing-complete, and represent all Kleene's µ-recursive functions. Finally, we consider reversible recursion schemes that lead outside the reversible endo-functions.

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“…A language equivalent to µRPF is in [14]. It provides invertible partial recursive functions on natural numbers and not on integers.…”

Section: Discussionmentioning

confidence: 99%

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

2018

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“…Concerning functional models of Reversible Computation, we are aware of [6] which introduces the class of reversible functions RI, which is as expressive as PR. So, RI is stronger than RPP, however we see RI as less abstract than RPP for two reasons: (i) the primitive functions of RI depend on a given specific binary representation of natural numbers; (ii) unlike RPP, which we can see as PRF in a reversible setting, it is not evident to us that RI can be considered the natural extension of a total class analogous to RPP.…”

Section: Related Workmentioning

confidence: 99%

Certifying algorithms and relevant properties of Reversible Primitive Permutations with Lean

Maletto,

Roversi

2022

Preprint

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Reversible Primitive Permutations (RPP) are recursively defined functions designed to model Reversible Computation. We illustrate a proof, fully developed with the proof-assistant Lean, certifying that: "RPP can encode every Primitive Recursive Function". Our reworking of the original proof of that statement is conceptually simpler, fixes some bugs, suggests a new more primitive reversible iteration scheme for RPP, and, in order to keep formalization and semi-automatic proofs simple, led us to identify a single pattern that can generate some useful reversible algorithms in RPP: Cantor Pairing, Quotient/Reminder of integer division, truncated Square Root. Our Lean source code is available for experiments on Reversible Computation whose properties can be certified.

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“…A forcefully non-exhaustive list of references follows. Reversible Turing machines are defined in [1,3,13,24] while [1,2,12,25] study some of their computational theoretic properties. Many research efforts have been focused on boolean functions and cellular automata as models of the reversible computation [33,48].…”

Section: Introductionmentioning

confidence: 99%

“…• Finally we discuss [12]. It introduces the class of reversible functions RI which is as expressive as Kleene's partial recursive functions [9,36].…”

Section: Introductionmentioning

confidence: 99%