Thomas Place

Separating Regular Languages by Piecewise Testable and Unambiguous Languages

Separation is a classical problem asking whether, given two sets belonging to some class, it is possible to separate them by a set from another class. We discuss the separation problem for regular languages. We give a Ptime algorithm to check whether two given regular languages are separable by a piecewise testable language, that is, whether a \(\mathcal{B}\Sigma_1(<)\) sentence can witness that the languages are disjoint. The proof refines an algebraic argument from Almeida and the third author. When separation is possible, we also express a separator by saturating one of the original languages by a suitable congruence. Following the same line, we show that one can as well decide whether two regular languages can be separated by an unambiguous language, albeit with a higher complexity.

Separation and the Successor Relation

We investigate two problems for a class C of regular word languages. The C-membership problem asks for an algorithm to decide whether an input language belongs to C. The C-separation problem asks for an algorithm that, given as input two regular languages, decides whether there exists a third language in C containing the first language, while being disjoint from the second. These problems are considered as means to obtain a deep understanding of the class C. It is usual for such classes to be defined by logical formalisms. Logics are often built on top of each other, by adding new predicates. A natural construction is to enrich a logic with the successor relation. In this paper, we obtain new and simple proofs of two transfer results: we show that for suitable logically defined classes, the membership, resp. the separation problem for a class enriched with the successor relation reduces to the same problem for the original class. Our reductions work both for languages of finite words and infinite words. The proofs are mostly self-contained, and only require a basic background on regular languages. This paper therefore gives simple proofs of results that were considered as difficult, such as the decidability of the membership problem for the levels 1, 3/2, 2 and 5/2 of the dot-depth hierarchy.

Separating Regular Languages by Locally Testable and Locally Threshold Testable Languages

A separator for two languages is a third language containing the first one and disjoint from the second one. We investigate the following decision problem: given two regular input languages, decide whether there exists a locally testable (resp. a locally threshold testable) separator. In both cases, we design a decision procedure based on the occurrence of special patterns in automata accepting the input languages. We prove that the problem is computationally harder than deciding membership. The correctness proof of the algorithm yields a stronger result, namely a description of a possible separator. Finally, we discuss the same problem for context-free input languages.

Regular languages of infinite trees that are boolean combinations of open sets

In this paper, we study boolean (not necessarily positive) combinations of open sets. In other words, we study positive boolean combinations of safety and reachability conditions. We give an algorithm, which inputs a regular language of infinite trees, and decides if the language is a boolean combination of open sets.

A Decidable Characterization of Locally Testable Tree Languages

A regular tree language L is locally testable if the membership of a tree into L depends only on the presence or absence of some neighborhoods in the tree. In this paper we show that it is decidable whether a regular tree language is locally testable.

Characterization of Logics over Ranked Tree Languages

We study the expressive power of the logics EF+ Fi¾? 1, Δ 2 and boolean combinations of Σ 1 over ranked trees. In particular, we provide effective characterizations of those three logics using algebraic identities. Characterizations had already been obtained for those logics over unranked trees, but both the algebra and the proofs were dependant on the properties of the unranked structure and the problem remained open for ranked trees.

Toward model theory with data values

We define a variant of first-order logic that deals with data words, data trees, data graphs etc. The definition of the logic is based on Fraenkel-Mostowski sets (FM sets, also known as nominal sets). The key idea is that we allow infinite disjunction (and conjunction), as long as the set of disjuncts (conjunct) is finite modulo renaming of data values. We study model theory for this logic; in particular we prove that the infinite disjunction can be eliminated from formulas.

Concatenation Hierarchies: New Bottle, Old Wine

We survey progress made in the understanding of concatenation hierarchies of regular languages during the last decades. This paper is an extended abstract meant to serve as a precursor of a forthcoming long version.

On Separation by Locally Testable and Locally Threshold Testable Languages

A separator for two languages is a third language containing the first one and disjoint from the second one. We investigate the following decision problem: given two regular input languages, decide whether there exists a locally testable (resp. a locally threshold testable) separator. In both cases, we design a decision procedure based on the occurrence of special patterns in automata accepting the input languages. We prove that the problem is computationally harder than deciding membership. The correctness proof of the algorithm yields a stronger result, namely a description of a possible separator. Finally, we discuss the same problem for context-free input languages.

A decidable characterization of locally testable tree languages

A regular tree language L is locally testable if membership of a tree in L depends only on the presence or absence of some fix set of neighborhoods in the tree. In this paper we show that it is decidable whether a regular tree language is locally testable. The decidability is shown for ranked trees and for unranked unordered trees.