Reversible pushdown transducers

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Abstract

Deterministic pushdown transducers are studied with respect to their ability to compute reversible transductions, that is, to transform inputs into outputs in a reversible way. This means that the transducers are also backward deterministic and thus are able to uniquely step the computation back and forth. The families of transductions computed are classified with regard to four types of length-preserving transductions as well as to the property of working reversibly. It turns out that accurate to one case separating witness transductions can be provided. For the remaining case it is possible to establish the equivalence of both families by proving that stationary moves can always be removed in length-preserving reversible pushdown transductions.

Introduction

Reversible computational models have earned much attention recently, where one incentive for the study of computational devices performing logically reversible computations is probably the question posed by Landauer of whether logical irreversibility is an unavoidable feature of useful computers. Landauer has demonstrated the physical and philosophical importance of this question by showing that whenever a physical computer throws away information about its previous state it must generate a corresponding amount of entropy that results in heat dissipation (see [2] for further details and references). As one of the first models Turing machines have been investigated towards their ability to work reversibly in the work of Lecerf [3] and Bennett [2], where it is shown that for every Turing machine an equivalent reversible Turing machine can be constructed. At the other end of the Chomsky hierarchy, Angluin introduced reversible computations in deterministic finite automata (DFA) and showed that reversible DFAs are weaker than DFAs in general [4]. Moreover, it is known that the general model and the reversible one coincide if the input head is two-way [5]. An algebraic characterization of languages accepted by reversible multiple-entry DFAs is obtained in [6]. Recent results on reversible regular languages are given in [7], [8], [9], where different aspects concerning the descriptional complexity and the minimality of reversible (one-way) DFAs are studied. For deterministic pushdown automata, it has been shown that the reversible variant is weaker than the general one [10].

Computational models are not only interesting from the viewpoint of accepting some input, but also from the more applied perspective of transforming some input into some output. For example, a parser for a programming language should not only return the information whether or not the input word can be parsed, but also the parse tree in the positive case. Transductions that are computed by different variants of transducers are studied in detail in the book of Berstel [11]. Deterministic and nondeterministic pushdown transducers are investigated in [12], where also characterizations of pushdown transductions as well as applications to the parsing of context-free languages are given. More recently, iterated finite state transducers have been introduced and studied in [13], [14], transducing variants of stack automata have been considered in [15], two-way transducers with auxiliary memory structures have been investigated in [16] with respect to their decidable problems, and the parallel model of cellular automata has been studied in [17] towards its transducing capabilities.

Reversibility in transducing devices has been investigated recently in [18], [19] for deterministic finite state transducers (DFSTs). In the former paper, reversible variants of (two-way) DFSTs are introduced where the reversibility of the transition function depends on the input only. Based on this definition the authors present constructions for the composition and uniformization of two-way DFSTs. In the latter paper, reversible DFSTs are defined whose transition function depends on the input and the output, since reversibility here is meant to preserve information both on the input and the output side. Hence, reversible DFSTs may be considered as reversible Turing machines (see, for example, [20], [2]) with a one-way input tape and a one-way output tape. The main results of [19] concern reversible DFSTs with length-preserving transductions where it is differentiated between different modes that basically differ by the fact whether or not input and output head have to move synchronously.

In this paper, we complement the investigation of reversible models by introducing reversible deterministic pushdown transducers for which we study length-preserving transductions in Mealy mode, strong mode, weak mode, and bounded delay mode. The definition of the model and of the different modes, which basically differ in possible distances between the input and output head, are given in Section 2. In Section 3, we study the computational capacity in detail and we can draw a complete picture. It can be shown by a detailed construction that the Mealy mode and the strong mode coincide. However, this holds in the reversible case only and is no longer true for possibly irreversible deterministic pushdown transducers. Apart from this equivalence for all remaining inclusions we present suitable witness transductions that show their properness both in the reversible and the general case. Finally, reversible transductions can be separated from general transductions in every of the four modes. Moreover, we can prove incomparability results in all cases where inclusion cannot be obtained.

Section snippets

Preliminaries

We denote the non-negative integers {0,1,2,} by N. Let Σ denote the set of all words over the finite alphabet Σ, and Σk denote its restriction to words of length at most k, for any k0. The empty word is denoted by λ. The length of a word w is denoted by |w| and its reversal is denoted by wR. For the number of occurrences of a symbol a in w we use the notation |w|a. Inclusions are denoted by ⊆. For convenience, we use Sx to denote S{x}, where S is a set and x is an element not belonging to S

Computational capacity

We turn to consider the computational capacity of reversible DPDTs. In particular, whenever two types of devices have different language acceptance power, then trivial transductions applied to a language from their symmetric difference would be a witness for separating also the power of the transducers. However, in the following we consider transductions of languages that are accepted by both types of devices in question. In this way, we are in fact separating the capabilities of computing

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the European COST Action IC 1405: Reversible Computation – Extending Horizons of Computing.

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    A preliminary version of this work was presented at the 22nd International Conference on Developments in Language Theory (DLT 2018) and it is published in [1].

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