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.

/pdf/languages-automata-and-logic-418xw4tas7.pdf

Languages, automata, and logic

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

We show that every finite semigroup is a quotient of a finite semigroup in which every right stabilizer satisfies the identities x = x^2 and xy = xyx. This result has several consequences. We first give a geometrical application : every finite transformation semigroup has a fixpoint-free covering (a transformation semigroup is fixpoint-free if every element which stabilizes a point is idempotent). Next we use our result and a result of I. Simon on congruences on paths to obtain a purely algebraic proof of a deep theorem of McNaughton on infinite words. Finally, we give an algebraic proof of a theorem of Brown on a finiteness condition for semigroups.

https://hal.archives-ouvertes.fr/hal-00019824/document

Semigroups with idempotent stabilizers and applications to automata theory

The aim of this paper is to study the concatenation hierarchy whose level 0 consists of all group languages. The union of all the levels of this hierarchy is the closure of group languages under product and boolean operations. Our first result states that this union is a decidable variety of languages. The rest of the paper is devoted to the study of level 1. This variety of languages, and the corresponding variety of monoids ◊G, appear in many different contexts. First, ◊G is exactly the variety J * G generated by all semidirect products of a J-trivial monoid by a group. Second ◊G is also the variety generated by powermonoids of groups. Finally, languages of level 1 arise in the study of the finite group topology for the free monoid. An important problem is to know whether this variety is decidable. The discussion of this problem motivated the introduction of the variety of monoids BG, which is the variety of all monoids M such that, for every idempotent e, f of M, efe = e implies e = f. Several equivalent definitions are given in the paper. In particular, we show that a monoid M is in BG if and only if the submonoid generated by all idempotents of M belongs to J. We also prove that BG is generated by all monoids that are, in some sense, extensions of a group by a monoid of J. Finally, ◊G is contained in BG but we don't know if this inclusion is strict or not.

https://hal.archives-ouvertes.fr/hal-00870714/document

Product of Group Languages

We study the variety W generated by monoids of upper-triangular boolean matrices. First, we present W as a natural extension of the variety J of finite J-trivial monoids and we give a description of the family of recognizable languages whose syntactic monoids are in W. Then we show that W can be described in terms of the generalized Schutzenberger product of finite monoids. We also show that W is generated by the power monoids of members of J. Finally we consider the membership problem for W and the connection with the dot-depth hierarchy in language theory. Although the majority of our results are purely "semigroup-theoretic" we use recognizable languages constantly in the proofs.

https://hal.archives-ouvertes.fr/hal-00870684/document

Monoids of upper triangular boolean matrices

The aim of this paper is to present algorithms to compute finite semigroups. The semigroup is given by a set of generators taken in a larger semigroup, called the "universe". This universe can be for instance the semigroup of all functions, partial functions, or relations on the set {1, ..., n}, or the semigroup of n x n matrices with entries in a given finite semiring. The algorithm produces simultaneously a presentation of the semigroup by generators and relations, a confluent rewriting system for this presentation and the Cayley graph of the semigroup. The elements of the semigroups are identified with the reduced words of the rewriting system. We also give some efficient algorithms to compute the Green relations, the local subsemigroups and the syntactic quasi-order of subsets of the semigroup.

https://hal.archives-ouvertes.fr/hal-00143949/document

Algorithms for computing finite semigroups

We give a survey on the following problem (known as the synchronization problem). Let A = (Q, X) be a finite automaton. Every word m in X* defines a function from Q into Q; the rank of m in A is the integer Card {qm ¦ q in Q }. A word of rank 1 maps all the states onto a single state and is called a synchronizing word (if such a word exists, the automaton itself is called synchronizing). Let A an automaton with n states. Cerný has conjectured that if A is synchronizing, then there exists a synchronizing word of length ≤ (n-1)^2. A generalization of this conjecture states that if there exists a word of rank ≤ k in A, then there exists such a word of length ≤ (n-k)^2.

https://hal.archives-ouvertes.fr/hal-00340776/document

Jean-Eric Pin

Papers

Semigroups with idempotent stabilizers and applications to automata theory

Product of Group Languages

Monoids of upper triangular boolean matrices

Algorithms for computing finite semigroups

Le problème de la synchronisation et la conjecture de Cerný