Time-symmetric Turing machines for computable involutions

• Published in: Science of computer programming Vol. 215; p. 102748
• Main Author: Nakano, Keisuke
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.03.2022
• Subjects: Bidirectional transformation language; Computable involution; Reversible Turing machine; Self-inverse function; Time-symmetric machine; Computable involution; Reversible Turing machine; Time-symmetric machine; Self-inverse function; Bidirectional transformation language
• ISSN: 0167-6423; 1872-7964
• DOI: 10.1016/j.scico.2021.102748

•A time-symmetric Turing machine always computes an involution (also called an involutory function or a self-inverse function).•A time-symmetric Turing machine is shown to be expressive enough, i.e., it can define arbitrary computable involutions.•Every multitape time-symmetric Turing machine can be simulated by a single-tape time-symmetric Turing machine.•The existence of a universal time-symmetric Turing machine is shown in terms of an appropriate definition of universality.•A time-symmetric Turing machine is motivated by characterizing bidirectional transformation languages. A reversible Turing machine is a forward and backward deterministic Turing machine, which has been an expressive model of reversible computation. It is obvious that every reversible Turing machine computes an injective function under a function semantics in which the initial and the final configuration of a run corresponds to an input and an output of the function. Axelsen and Glück showed the opposite direction that every injective computable function can be computed by a reversible Turing machine. This paper provides a similar result on involutions instead of injective functions. An involution, also called a self-inverse function, is a function f that is its own inverse, i.e., f(f(x))=x holds whenever f(x) is defined. The paper presents a computational model of involution as a variant of Turing machines, called a time-symmetric Turing machine. The computational model is shown to be expressive in the sense that not only does a time-symmetric Turing machine always compute an involution but also every computable involution can be computed by a time-symmetric Turing machine. As any involution is injective (hence reversible), any time-symmetric Turing machine is a reversible Turing machine. Furthermore, the existence of a universal time-symmetric Turing machine is shown under an appropriate redefinition of universality introduced by Axelsen and Glück for reversible Turing machines.