On reversible Turing machines and their function universality

Abstract

We provide a treatment of the reversible Turing machines (RTMs) under a strict function semantics. Unlike many existing reversible computation models, we distinguish strictly between computing the function λx. f (x) and computing the function λx. (x, f (x)), or other injective embeddings of f . We reinterpret and adapt a number of important foundational reversible computing results under this semantics. Unifying the results in a single model shows that, as expected (and previously claimed), the RTMs are robust and can compute exactly all injective computable functions. Because injectivity entails that the RTMs are not strictly Turing-complete w.r.t. functions, we use an appropriate alternative universality definition, and show how to derive universal RTMs (URTMs) from existing irreversible universal machines. We then proceed to construct a URTM from the ground up. This resulting machine is the first URTM which does not depend on a reversible simulation of an existing universal machine. The new construction has the advantage that the interpretive overhead of the URTM is limited to a (program dependent) constant factor. Another novelty is that the URTM can function as an inverse interpreter at no asymptotic cost. Mathematics Subject Classification 68Q05 · 68Q10 · 03D10Revised and extended version of the conference papers [4,5].

A class of Recursive Permutations which is Primitive Recursive complete

“…In the previous subsection we refer to ancillae as tools to circumvent nocloning. For remarking once more that ancillae are harmless as far as reversible computing is concerned we focus on reversible computation as formalized by means of reversible Turing-machines (RTM) [1,2,3,13]. This is like saying that a function is reversible when, and only when, some RTM exists that computes it.…”

“…What if we think of asking for the values m and n such that sum(m,n) = 5? If we had implemented sum in a prolog-like language, then we could exploit its non deterministic evaluation mechanism to list all the pairs (0, 5), (1,4), (2,3), (3,2), (4, 1) and (5, 0) every of which would well be a correct instance of (m, n). In a reversible setting we would obtain exactly the pair we started from.…”

“…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].…”

Simulating reversible computation with reaction systems

“…This result spawned extensive research interest in the field of reversible computing since it proved that reversible computing is the tool to overcome the performance constraints of traditional irreversible systems [13]. A more recent and rigorous treatment of reversibility concerning Turing machine computations can also be found in [7].…”

Foundations of Reversible Computation

“…Turing machines are traditionally viewed as language acceptors, but one can also incorporate outputs in the model so that the machine becomes a transducer that computes a (partial) function. In [12] the authors investigate RTM under the strict function semantics that requires that at the end of the computation only the output remains on the tape, and they develop a rigorous foundational theory of reversible computation of functions in this semantics, including the appropriate concept of universality and a design of a universal machine.…”