This chapter presents that part of the theory of the \(\mu\)-calculus that is relevant to the model-checking problem as broadly understood. The \(\mu\)-calculus is one of the most important logics in model checking. It is a logic with an exceptional balance between expressiveness and algorithmic properties.

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The mu-calculus and Model Checking

We investigate the quantifier alternation hierarchy in first-order logic on finite words. Levels in this hierarchy are defined by counting the number of quantifier alternations in formulas. We prove that one can decide membership of a regular language to the levels \(\mathcal{B}\Sigma_{2}\) (boolean combination of formulas having only 1 alternation) and Σ3 (formulas having only 2 alternations beginning with an existential block). Our proof works by considering a deeper problem, called separation, which, once solved for lower levels, allows us to solve membership for higher levels.

Going Higher in the First-Order Quantifier Alternation Hierarchy on Words

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.

Separating Regular Languages by Piecewise Testable and Unambiguous Languages

On the index of Simon's congruence for piecewise testability

In 1970, R. S. Cohen and Janusz A. Brzozowski introduced a hierarchy of star-free languages called the dot-depth hierarchy. This hierarchy and its generalisations, together with the problems attached to them, had a long-lasting influence on the development of automata theory. This survey article reports on the numerous results and conjectures attached to this hierarchy. This paper is a follow-up of the survey article Open problems about regular languages, 35 years later [57]. The dot-depth hierarchy, also known as Brzozowski hierarchy, is a hierarchy of star-free languages first introduced by Cohen and Brzozowski [25] in 1971. It immediately gave rise to many interesting questions and an account of the early results can be found in Brzozowski’s survey [20] from 1976. 1 Terminology, notation and background Most of the terminology used in this paper was introduced in [57]. We just complete these definitions by giving the ordered versions of the notions of syntactic monoid and variety of finite monoids. 1.1 Syntactic order and positive varieties An ordered monoid is a monoid equipped with an order 6 compatible with the multiplication: x 6 y implies zx 6 zy and xz 6 yz. The syntactic preorder of a language L of A is the relation 6L defined on A by u 6L v if and only if, for every x, y ∈ A ,

The Dot-Depth Hierarchy, 45 Years Later.

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.

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Separation and the Successor Relation

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: given two regular input languages of finite words, decide whether there exists a first-order definable separator. We prove that in order to answer this question, sufficient information can be extracted from semigroups recognizing the input languages, using a fixpoint computation. 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. Finally, we prove that this technique can be generalized to answer the same question for regular languages of infinite words.

/pdf/separating-regular-languages-with-first-order-logic-193py3j0pb.pdf

Separating regular languages with first-order logic

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: given two regular input languages of finite words, decide whether
there exists a first-order definable separator. We prove that in order to
answer this question, sufficient information can be extracted from semigroups
recognizing the input languages, using a fixpoint computation. 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. Finally, we generalize this technique to
answer the same question for regular languages of infinite words.

/pdf/separating-regular-languages-with-first-order-logic-2nprdmtq3i.pdf

Thomas Place

Papers

Going Higher in the First-Order Quantifier Alternation Hierarchy on Words

Separating Regular Languages by Piecewise Testable and Unambiguous Languages

Separation and the Successor Relation

Separating regular languages with first-order logic

Separating Regular Languages with First-Order Logic