Duality and Equational Theory of Regular Languages

Gehrke, Mai; Grigorieff, Serge; Pin, Jean-Éric

doi:10.1007/978-3-540-70583-3_21

Cited by 104 publications

(161 citation statements)

References 21 publications

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“…The language A* is called the full language. The c lass N :J) of regular no ndense or full languages was first considered in [13]. …”

Section: Languages With Zero and Nondense Languagesmentioning

confidence: 99%

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Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited

Ballester‐Bolinches

Soler–Escrivà

2012

Forum Mathematicum

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Abstract. We present an extension of Eilenberg's variety theorem, a well-known result connecting algebra to formal languages. We prove that there is a bijective correspondence between formations of finite monoids and certain classes of languages, the formations of languages. Our result permits to treat classes of finite monoids which are not necessarily closed under taking submonoids, contrary to the original theory. We also prove a similar result for ordered monoids. This paper is the first step of a programme aiming at exploring the connections between the formations of finite groups and regular languages. The starting point is Eilenberg's variety theorem [10], a celebrated result of the 1970's which underscores the importance of varieties of finite monoids (also called pseudovarieties) in the study of formal languages. Since varieties of finite groups are special cases of varieties of finite monoids, varieties seems to be a natural structure to study languages recognized by finite groups. However, in finite group theory, varieties are challenged by another notion. Although varieties are incontestably a central notion, many results are better formulated in the setting of formations . This raised the question whether Eilenberg's variety theorem could be extended to a "formation theorem".The aim of this paper is to give a positive answer to this question. To our surprise, the resulting theorem holds not only for group formations but also for fo rmations of finite monoids. We also prove a similar result for formations of ordered finite monoids, extending in this way a theorem of [17] .Before stating these results more precisely, let us say a word on the aforementioned research programme and give a brief overview of the existing literature.

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“…The language A* is called the full language. The c lass N :J) of regular no ndense or full languages was first considered in [13]. …”

Section: Languages With Zero and Nondense Languagesmentioning

confidence: 99%

“…The resulting extension of Eilenberg's variety theory permits to treat classes of languages that are not necessarily closed under complement, contrary to the original theory. Other extensions were developed independently by Straubing [27] and Esik and lto [ 11] and more recently by Gehrke, Grigorieff and Pin [ 13].…”

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confidence: 99%

Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited

Ballester‐Bolinches

Soler–Escrivà

2012

Forum Mathematicum

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“…One may specifically mention Scott's model of the λ-calculus, which is a dual space, Esakia's duality [4] for Heyting algebras and the corresponding frame semantics for intuitionist logics, Goldblatt's paper [8] identifying extended Stone duality as the theory for completeness issues for Kripke semantics in modal logic, and Abramsky's path-breaking paper [1] linking program logic and domain theory. Our work with Grigorieff and Pin [7,9,6], with Pippenger [10] as a precursor, shows that the connection between regular languages and monoids also is a case of Stone duality.…”

Section: Stone Dualitymentioning

confidence: 78%

“…With Grigorieff and Pin, we gave a general and modular Eilenberg-Reiterman theorem based on duality theory [7]. The idea is the following.…”

Section: The Inverse Limit System Fmentioning

confidence: 99%

Duality and Recognition

Gehrke

2011

Mathematical Foundations of Computer Science 2011

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Abstract. The fact that one can associate a finite monoid with universal properties to each language recognised by an automaton is central to the solution of many practical and theoretical problems in automata theory. It is particularly useful, via the advanced theory initiated by Eilenberg and Reiterman, in separating various complexity classes and, in some cases it leads to decidability of such classes. In joint work with Jean-Éric Pin and Serge Grigorieff we have shown that this theory may be seen as a special case of Stone duality for Boolean algebras extended to a duality between Boolean algebras with additional operations and Stone spaces equipped with Kripke style relations. This is a duality which also plays a fundamental role in other parts of the foundations of computer science, including in modal logic and in domain theory. In this talk I will give a general introduction to Stone duality and explain what this has to do with the connection between regular languages and monoids. Stone dualityStone type dualities is the fundamental tool for moving between linguistic specification and spatial dynamics or transitional unfolding. As such, it should come as no surprise that it is a theory of central importance in the foundations of computer science where one necessarily is dealing with syntactic specifications and their effect on physical computing systems. In 1936, M. H. Stone initiated duality theory by presenting what, in modern terms, is a dual equivalence between the category of Boolean algebras and the category of compact Hausdorff spaces having a basis of clopen sets, so-called Boolean spaces [13]. The points of the space corresponding to a given Boolean algebra are not in general elements of the algebra -just like states of a system are not in general available as entities in a specification language but are of an entirely different sort. In models of computation these two different sorts, specification expressions and states, are given a priori but in unrelated settings. Via Stone duality, the points of the space may be obtained from the algebra as homomorphisms into the two-element Boolean algebra or equivalently as ultrafilters of the algebra. In logical terms these are valuations or models of the Boolean algebra. In computational terms they are possible states of the system. Each element of the Boolean algebra corresponds to the set of all models in which it is true, or all states in which it holds, and the topology of the space is generated by these sets. A main insight of Stone is that one may recover the original algebra as the Boolean algebra of clopen subsets of the resulting space.

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“…We recall here two versions of the variety theorem. Extended versions were also obtained in [23,9] and a unified version is proposed in [11].…”

Section: Varieties Of Finite Monoidsmentioning

confidence: 99%

A Robust Class of Regular Languages

Gómez

Lecture Notes in Computer Science

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Abstract. In this survey paper, we present known results and open questions on a proper subclass of the class of regular languages. This class, denoted by W, is especially robust: it is closed under union, intersection, product, shuffle, left and right quotients, inverse of morphisms, length preserving morphisms and commutative closure. It can be defined as the largest positive variety of languages not containing the language (ab) * . It admits a nontrivial algebraic characterization in terms of finite ordered monoids, which implies that W is decidable: given a regular language, one can effectively decide whether or not it belongs to W. We propose as a challenge to find a constructive description and a logical characterization of W.Warning. In this paper, square brackets are used as a substitute to "respectively" to gather several definitions [properties] into a single one.The search for robust classes of regular languages is an old problem of automata theory, which occurs in particular in the study of regular model checking [3]. In this survey paper, we present known results and open questions on a proper subclass of the class of regular languages, introduced a few years ago by the authors in connection with the study of the shuffle product [6,7]. This class, denoted by W, is especially robust: it is closed under union, intersection, product, shuffle, left and right quotients, inverse of morphisms, length preserving morphisms and commutative closure. Furthermore, this class is decidable: there is an algorithm to decide whether a given regular language belongs to W or not. As such, it might offer an appropriate framework for modeling certain problems arising in the verification of concurrent systems.The class W is also interesting on its own and appears in the study of three operations on languages: length preserving morphisms, inverse of substitutions and shuffle product. More specifically, W is the largest proper positive variety of languages closed under one of these operations. It is also the largest positive variety of languages not containing the language (ab) * .⋆ The authors acknowledge support from the AutoMathA programme of the European Science Foundation.

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Duality and Equational Theory of Regular Languages

Cited by 104 publications

References 21 publications

Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited

Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited

Duality and Recognition

A Robust Class of Regular Languages

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