The study of graph structure has advanced in recent years with great strides: finite graphs can be described algebraically, enabling them to be constructed out of more basic elements. Separately the properties of graphs can be studied in a logical language called monadic second-order logic. In this book, these two features of graph structure are brought together for the first time in a presentation that unifies and synthesizes research over the last 25 years. The author not only provides a thorough description of the theory, but also details its applications, on the one hand to the construction of graph algorithms, and, on the other to the extension of formal language theory to finite graphs. Consequently the book will be of interest to graduate students and researchers in graph theory, finite model theory, formal language theory, and complexity theory.

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Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach

Discoveries in finite semigroups have influenced several mathematical fields, including theoretical computer science, tropical algebra via matrix theory with coefficients in semirings, and other areas of modern algebra. This comprehensive, encyclopedic text will provide the reader - from the graduate student to the researcher/practitioner with a detailed understanding of modern finite semigroup theory, focusing in particular on advanced topics on the cutting edge of research. Key features: (1) Develops q-theory, a new theory that provides a unifying approach to finite semigroup theory via quantization; (2) Contains the only contemporary exposition of the complete theory of the complexity of finite semigroups; (3) Introduces spectral theory into finite semigroup theory; (4) Develops the theory of profinite semigroups from first principles, making connections with spectra of Boolean algebras of regular languages; (5) Presents over 70 research problems, most new, and hundreds of exercises. Additional features: (1) For newcomers, an appendix on elementary finite semigroup theory; (2) Extensive bibliography and index. The q-theory of Finite Semigroups presents important techniques and results, many for the first time in book form, and thereby updates and modernizes the literature of semigroup theory.

The q-theory of Finite Semigroups

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Syntactic semigroups Jean-Eric Pin

Syntactic semigroups

This article is a contribution to the algebraic theory of automata, but it also contains an application to Buchi’s sequential calculus. The polynomial closure of a class of languagesC is the set of languages that are finite unions of languages of the formL
 0
 a
 1
 L
 1 ···a
 nLn, where thea
 i’s are letters and theL
 i’s are elements ofC. Our main result is an algebraic characterization, via the syntactic monoid, of the polynomial closure of a variety of languages. We show that the algebraic operation corresponding to the polynomial closure is a certain Mal’cev product of varieties. This result has several consequences. We first study the concatenation hierarchies similar to the dot-depth hierarchy, obtained by counting the number of alternations between boolean operations and concatenation. For instance, we show that level 3/2 of the Straubing hierarchy is decidable and we give a simplified proof of the partial result of Cowan on level 2. We propose a general conjecture for these hierarchies. We also show that if a language and its complement are in the polynomial closure of a variety of languages, then this language can be written as a disjoint union of marked unambiguous products of languages of the variety. This allows us to extend the results of Thomas on quantifier hierarchies of first-order logic.

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Polynomial closure and unambiguous product

Introduction to semigroup theory

We relate the problem of computing the closure of a finitely generated subgroup of the free group in the pro-V topology, where V is a pseudovariety of finite groups, with an extension problem for inverse automata which can be stated as follows: given partial one-to-one maps on a finite set, can they be extended into permutations generating a group in V? The two problems are equivalent when V is extension-closed. Turning to practical computations, we modify Ribes and Zalesskiĭ's algorithm to compute the pro-p closure of a finitely generated subgroup of the free group in polynomial time, and to effectively compute its pro-nilpotent closure. Finally, we apply our results to a problem in finite monoid theory, the membership problem in pseudovarieties of inverse monoids which are Mal'cev products of semilattices and a pseudovariety of groups. Resume: Nous etablissons un lien entre le probleme du calcul de l'adheerence d'un sous-groupe finiment engendre du groupe libre dans la topologie pro-V, ou V est une pseudovariete de groupes finis, et un probleme d'extension pour les automates inversifs qui peut etre enonce de la fac con suivante: etant donnees des transformations partielles injectives d'un ensemble fini, peuvent-elles etre etendues en des permutations qui engendrent un groupe dans V? Les deux problemes sont equivalents si V est fermee par extensions. Nous interessant ensuite aux calculs pratiques, nous modifions l'algorithme de Ribes et Zalesskiĭ pour calculer l'adherence pro-p d'un sous-groupe finiment engendre du groupe libre en temps polynomial et pour calculer effectivement sa cloture pro-nilpotente. Enfin nous appliquons nos resultats a un probleme de theorie des monoides finis, celui de de l'appartenance dans les pseudovarietes de monoides inversifs qui sont des produits de Mal'cev de demi-treillis et d'une pseudovariete de groupes.

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Closed subgroups in pro-v topologies and the extension problem for inverse automata

PSPACE-complete problems for subgroups of free groups and inverse finite automata

Many recent results in finite semigroup theory make use of profinite methods, that is, they rely on the study of certain infinite, compact semigroups which arise as projective limits of finite semigroups. These ideas were introduced in semigroup theory in the 1980s, first to describe pseudovarieties in terms of so-called pseudo-identities: this is Reiterman's theorem, which can be viewed as the (much more complex) finite algebra analogue of Birkhoff's variety theorem. Soon, these methods were used in conjunction with virtually all the other approaches of finite semigroups, notably to study the decidability of product pseudovarieties. This paper surveys the contribution of profinite methods and the way they enriched and modified finite semigroup theory.

Profinite methods in semigroup theory

We consider the two-variable fragment FO^2[<] of first-order logic over finite words. Numerous characterizations of this class are known. Th\'erien and Wilke have shown that it is decidable whether a given regular language is definable in FO^2[<]. From a practical point of view, as shown by Weis, FO^2[<] is interesting since its satisfiability problem is in NP. Restricting the number of quantifier alternations yields an infinite hierarchy inside the class of FO^2[<]-definable languages. We show that each level of this hierarchy is decidable. For this purpose, we relate each level of the hierarchy with a decidable variety of finite monoids. Our result implies that there are many different ways of climbing up the FO^2[<]-quantifier alternation hierarchy: deterministic and co-deterministic products, Mal'cev products with definite and reverse definite semigroups, iterated block products with J-trivial monoids, and some inductively defined omega-term identities. A combinatorial tool in the process of ascension is that of condensed rankers, a refinement of the rankers of Weis and Immerman and the turtle programs of Schwentick, Th\'erien, and Vollmer.

The FO^2 alternation hierarchy is decidable

We propose a new algebraic framework to discuss and classify recognizable tree languages, and to characterize interesting classes of such languages. Our algebraic tool, called preclones, encompasses the classical notion of syntactic Sigma-algebra or minimal tree automaton, but adds new expressivity to it. The main result in this paper is a variety theorem a la Eilenberg, but we also discuss important examples of logically defined classes of recognizable tree languages, whose characterization and decidability was established in recent papers (by Benedikt and Segoufin, and by Bojanczyk and Walukiewicz) and can be naturally formulated in terms of pseudovarieties of preclones. Finally, this paper constitutes the foundation for another paper by the same authors, where first-order definable tree languages receive an algebraic characterization.

Pascal Weil

Papers

Closed subgroups in pro-v topologies and the extension problem for inverse automata

PSPACE-complete problems for subgroups of free groups and inverse finite automata

Profinite methods in semigroup theory

The FO^2 alternation hierarchy is decidable

Algebraic recognizability of regular tree languages