Separating Regular Languages with First-Order Logic

Abstract

Abstract. Given two languages, a separator is a third language that contains the first one and is disjoint from the second one. We investigate the following decision problem, called separation: given two regular languages of finite words, decide whether there exists a firstorder definable separator. A more general problem was solved in an algebraic framework by Henckell in 1988, although the connection with separation was pointed out only in 1996, by Almeida. The result was then generalized by Henckell, Steinberg and Rhodes in 2010. In this paper, we present a new, self-contained and elementary proof of it, which actually covers the original result of Henckell.We prove that in order to answer this question, sufficient information can be extracted from semigroups recognizing the input languages, using a fixpoint computation, similar to that originally proposed by Henckell. Given as input a morphism recognizing both languages to be separated, this yields an Exptime algorithm for checking first-order separability. Moreover, the correctness proof of this algorithm yields a stronger result, namely a description of a possible separator. More precisely, one can compute a bound on the quantifier rank of potential separators, as well as a first-order formula that describes a separator, if there exists one. Finally, we prove that this technique can be generalized to answer the same question for regular languages of infinite words.

Deciding Classes of Regular Languages: The Covering Approach

“…The covering problem generalizes membership. It was first considered implicitly in [26,27] and was later formalized in [31] (along with a detailed framework designed for handling it). At the time, its introduction was motivated by two reasons.…”

“…The input consists in two objects: a regular language L and a finite set of regular languages L. One must decide whether there exists a C-cover K of L (a finite set of languages in C whose union includes L) such that no language in K intersects all languages in L. Naturally, this definition is more involved than the one of membership and it is more difficult to find an algorithm for C-covering than for Cmembership. Yet, covering was recently shown to be decidable for many natural classes (see for example [6,24,25,30,34,35]) including the star-free languages [29].…”

The Dot-Depth Hierarchy, 45 Years Later

“…The separation problem for star-free languages was first solved by Henckell [39] in its semigroup form. Successive improvements can be found in [41,79,82].…”

“…The separation problem for star-free languages was first solved by Henckell [39] in its semigroup form. Successive improvements can be found in [41,79,82].A major result of Place and Zeitoun [78] is the following much stronger result. For the signature {<, S, min, max}, the decidability of Σ n and Π n , for n 4 and that of BΣ n , for n 2, follows from Theorem 6.8.…”

Separating Many Words by Counting Occurrences of Factors

“…More generally, we could try to separate two disjoint languages A and B by providing a language X from some specified family of languages such that A ⊆ X and B ∩ X = ∅. As an example related to logic, see [17]. Alternatively, we could try to separate many words w 1 , .…”