Minimal and Reduced Reversible Automata

Abstract

A condition characterizing the class of regular languages which have several nonisomorphic minimal reversible automata is presented. The condition concerns the structure of the minimum automaton accepting the language under consideration. It is also observed that there exist reduced reversible automata which are not minimal, in the sense that all the automata obtained by merging some of their equivalent states are irreversible. Furthermore, a sufficient condition for the existence of infinitely many reduced reversible automata accepting a same language is given. It is also proved that, when the language is accepted by a unique minimal reversible automaton (that does not necessarily coincide with the minimum deterministic automaton), then no other reduced reversible automata accepting it can exist.Comment: Preliminary version presented at DCFS 2016 --- Descriptional Complexity of Formal Systems, Bucharest, Romania, Jul 5-8, 201

Reversible Languages Having Finitely Many Reduced Automata

“…Contrary to the general case of regular languages, there can be more than one reduced reversible automata, up to isomorphism, recognizing the same (reversible) language. For example, see Figure 2 of [16]. In that example the minimal automaton (depicted on Subfigure (a)) is not reversible and there are two nonisomorphic reversible reduced automata recognizing the same language (with four states).…”

“…Several variants of reversible automata were defined and studied [21,1,17]. The variant we work with (partial deterministic automata with a single initial state and and arbitrary set of final states) have been treated in [9,10,6,16,15,14]. In particular, in [6] the class of reversible languages is characterized by means of a forbidden pattern in the minimal automaton of the language in question, and an algorithm is provided to compute a minimal reversible automaton for a reversible language, given its minimal automaton.…”

“…Here "minimal" means minimizing the number of states, and the minimal reversible automaton is shown to be not necessarily unique. In [16], the notion of reduced reversible automata is introduced: a reversible automaton is reduced if it is trim (all of its states are accessible and coaccessible), and none of its nontrivial factor automata is reversible. The authors characterize the class of those reversible languages (again, by developing a forbidden pattern) having a unique reduced reversible automaton (up to isomorphism), and leave open the problem to find a characterization of the class of those reversible languages having finitely many reduced reversible automata (up to isomorphism).…”

“…In this paper we solve this open problem of [16], by also developing a forbidden pattern characterization which allows an NL algorithm. edges, not even loops), and nontrivial otherwise.…”

Weakly and Strongly Irreversible Regular Languages

“…The first works on this topic already appeared half a century ago and are due to Landauer and Bennet [9,2]. More recently, several papers presenting investigations on reversibility in space bounded Turing machines, finite automata, and other devices appeared in the literature (see, e.g., [1,15,6,10,13,3,11]).…”