Finite Semigroups and Recognizable Languages: An Introduction

Abstract

This paper is an attempt to share with a larger audience some modern developments in the theory of finite automata. It is written for the mathematician who has a background in semigroup theory but knows next to nothing on automata and languages. No proofs are given, but the main results are illustrated by several examples and counterexamples.What is the topic of this theory? It deals with languages, automata and semigroups, although recent developments have shown interesting connections with model theory in logic, symbolic dynamics and topology. Historically, in their attempt to formalize natural languages, linguists such as Chomsky gave a mathematical definition of natural concepts such as words, languages or grammars: given a finite set A, a word on A is simply an element of the free monoid on A, and a language is a set of words. But since scientists are fond of classifications of all sorts, language theory didn't escape to this mania. Chomsky established a first hierarchy, based on his formal grammars. In this paper, we are interested in the recognizable languages, which form the lower level of the Chomsky hierarchy. A recognizable language can be described in terms of finite automata while, for the higher levels, more powerful machines, ranging from pushdown automata to Turing machines, are required. For this reason, problems on finite automata are often under-estimated, according to the vague -but totally erroneous -feeling that "if a problem has been reduced to a question about finite automata, then it should be easy to solve".Kleene's theorem [23] is usually considered as the foundation of the theory. It shows that the class of recognizable languages (i.e. recognized by finite automata), coincides with the class of rational languages, which are given by rational expressions. Rational expressions can be thought of as a generalization of polynomials involving three operations: union (which plays the role of addition), product and star operation. An important corollary of Kleene's theorem is that rational languages are closed under complement. In the sixties, several classification schemes for the rational languages were proposed, based on the * From 1st Sept 1993, LITP, Université Paris VI, Tour 55-65, 4 Place Jussieu, 75252 Paris Cedex 05, FRANCE.

Correct Passive Testing Algorithms and Complete Fault Coverage

“…This could be done, for example, by characterizing the structure of formulas in temporal logic that yield this closure condition, as is done in [16] for safety and liveness properties. In the context of specifying the desired behavior operationally as an automaton, one could look for an algebraic characterization of prefix-and suffix-closed languages as suitable varieties of semigroups, in the style of [12]. Finally, in addition to fault detection, an important part of fault management is fault location which identifies the faulty local site which is responsible for the observed faulty global execution, and it would be interesting to examine systematic formal approaches to the fault location problem.…”

On the word problem for syntactic monoids of piecewise testable languages

“…Hence,ŵ is a shortest possible representant in its ∼ 3 class. Listing the triples occurring in C 3 (w) takes O(|w|3 ) time and all other steps are linear in |C 3 (w)|. Hence, finding and multiplying normal forms takes O(|w| 3 ) time, as expected.…”

Expressive power of existential first-order sentences of Büchi's sequential calculus

“…It is a well-known fact that in a finite semigroup, the subsemigroup generated by an element x contains a unique idempotent, denoted by x . See [1,9,16] for references on semigroup theory.…”