Minimal Reversible Deterministic Finite Automata

Abstract

We study reversible deterministic finite automata (REV-DFAs), that are partial deterministic finite automata whose transition function induces an injective mapping on the state set for every letter of the input alphabet. We give a structural characterization of regular languages that can be accepted by REV-DFAs. This characterization is based on the absence of a forbidden pattern in the (minimal) deterministic state graph. Again with a forbidden pattern approach, we also show that the minimality of REV-DFAs among all equivalent REV-DFAs can be decided. Both forbidden pattern characterizations give rise to [Formula: see text]-complete decision algorithms. In fact, our techniques allow us to construct the minimal REV-DFA for a given minimal DFA. These considerations lead to asymptotic upper and lower bounds on the conversion from DFAs to REV-DFAs. Thus, almost all problems that concern uniqueness and the size of minimal REV-DFAs are solved.

Minimal and Reduced Reversible Automata

“…An algorithm for converting a minimum irreversible dfa M into an equivalent minimal rev-dfa A, when possible, has been obtained in [HJK15]. We present an outline of it.…”

“…(See Algorithm 1.) As pointed out in [HJK15], in some steps of the algorithm different choices are possible (see lines 3 and 6). As a consequence, different minimal dfas equivalent to M could be produced.…”

“…An extension which allows to consider nondeterministic transitions, without changing the class of accepted languages, has been considered by Lombardy [Lom02], introducing and investigating quasi reversible automata. Classical automata, namely automata with a single initial state and a set of final states, have been considered in the works by Holzer, Jakobi, and Kutrib [Kut14,HJK15,Kut15]. In particular, in [HJK15] the authors gave a characterization of regular languages which are accepted by reversible automata.…”

“…Classical automata, namely automata with a single initial state and a set of final states, have been considered in the works by Holzer, Jakobi, and Kutrib [Kut14,HJK15,Kut15]. In particular, in [HJK15] the authors gave a characterization of regular languages which are accepted by reversible automata. This characterization is given in terms of the structure of the minimum deterministic automaton.…”

“…Investigating this phenomenon more into details, we are able to find a language for which there exist arbitrarily large, and hence infinitely many reduced reversible automata. Indeed, we present a general construction that allows to obtain arbitrarily large reversible automata for all languages whose minimum deterministic automata satisfy a further structural condition besides the one given in [HJK15]. As a consequence of such condition, for each reversible language whose minimum deterministic automaton contains a loop in the "irreversible part", it is possible to construct infinitely many arbitrarily large reduced reversible automata.…”