Recognizing Lexicographically Smallest Words and Computing Successors in Regular Languages

Abstract

For a formal language [Formula: see text], the problem of language enumeration asks to compute the length-lexicographically smallest word in [Formula: see text] larger than a given input [Formula: see text] (henceforth called the [Formula: see text]-successor of [Formula: see text]). We investigate this problem for regular languages from a computational complexity and state complexity perspective. We first show that if [Formula: see text] is recognized by a DFA with [Formula: see text] states, then [Formula: see text] states are (in general) necessary and sufficient for an unambiguous finite-state transducer to compute [Formula: see text]-successors. As a byproduct, we obtain that if [Formula: see text] is recognized by a DFA with [Formula: see text] states, then [Formula: see text] states are sufficient for a DFA to recognize the subset [Formula: see text] of [Formula: see text] composed of its lexicographically smallest words. We give a matching lower bound that holds even if [Formula: see text] is represented as an NFA. It has been known that [Formula: see text]-successors can be computed in polynomial time, even if the regular language is given as part of the input (assuming a suitable representation of the language, such as a DFA). In this paper, we refine this result in multiple directions. We show that if the regular language is given as part of the input and encoded as a DFA, the problem is in [Formula: see text]. If the regular language [Formula: see text] is fixed, we prove that the enumeration problem of the language is reducible to deciding membership to the Myhill-Nerode equivalence classes of [Formula: see text] under [Formula: see text]-uniform [Formula: see text] reductions. In particular, this implies that fixed star-free languages can be enumerated in [Formula: see text], arbitrary fixed regular languages can be enumerated in [Formula: see text] and that there exist regular languages for which the problem is [Formula: see text]-complete.