The Temporal Logic of Reactive and Concurrent Systems: Specification

General Topology-I

Publisher Summary This chapter focuses on finite automata on infinite sequences and infinite trees. The chapter discusses the complexity of the complementation process and the equivalence test. Deterministic Muller automata and nondeterministic Buchi automata are equivalent in recognition power. Any nonempty Rabin recognizable set contains a regular tree and shows that the emptiness problem for Rabin tree automata is decidable. The chapter discusses the formulation of two interesting generalizations of Rabin's Tree Theorem and presents some remarks on the undecidable extensions of the monadic theory of the binary tree. A short overview of the work that studies the fine structure of the class of Rabin recognizable sets of trees is also presented in the chapter. Depending on the formalism in which tree properties are classified, the results fall in three categories: monadic second-order logic, tree automata, and fixed-point calculi.

Automata on infinite objects

The subject of this chapter is the study of formal languages (mostly languages recognizable by finite automata) in the framework of mathematical logic.

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Languages, automata, and logic

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

Varieties of Formal Languages

Infinite Words is an important theory in both Mathematics and Computer Sciences. Many new developments have been made in the field, encouraged by its application to problems in computer science. Infinite Words is the first manual devoted to this topic. Infinite Words explores all aspects of the theory, including Automata, Semigroups, Topology, Games, Logic, Bi-infinite Words, Infinite Trees and Finite Words. The book also looks at the early pioneering work of B chi, McNaughton and Sch tzenberger. Serves as both an introduction to the field and as a reference book. Contains numerous exercises desgined to aid students and readers. Self-contained chapters provide helpful guidance for lectures.

Infinite Words: Automata, Semigroups, Logic and Games

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

We present some partial results on the following conjectures arising from automata theory. The first conjecture is the triangle conjecture due to Perrin and Schiitzenberger. Let A ={a, b } be a two-letter alphabet, d a positive integer and let B d ={ a i ba j |0 ≤ i + j ≤ d }. If X C B d is a code, then | X ≤ d + 1. The second conjecture is due to Cerny and the author. Let be an automaton with n states. If there exists a word of rank n - k in , there exists such a word of length ≤ k 2 .

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On two Combinatorial Problems Arising from Automata Theory

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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Jean-Eric Pin

Papers

Varieties of Formal Languages

Infinite Words: Automata, Semigroups, Logic and Games

Syntactic semigroups

On two Combinatorial Problems Arising from Automata Theory

Polynomial closure and unambiguous product