this paper were first presented at the "IEEE Symposium on Logic in Computer Science," Ithaca, New York, June 1987

Abstract: More than sixteen years after the beginning of a linear theory for certain discrete event systems in which max-plus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison with classical linear system theory, there are areas which are practically untouched, mostly because the corresponding mathematical tools are yet to be fabricated. This is the case of the geometric approach of systems which is known, in the classical theory, to provide another important insight to system-theoretic and control-synthesis problems, beside the algebraic machinery. A preliminary discussion of geometric aspects in the max-plus algebra and their use for system theory is proposed in the last part of the paper. Résumé: Plus de seize ans après le début d’une théorie linéaire de certains systèmes à événements discrets dans laquelle l’algèbre max-plus et autres outils algébriques assimilés jouent un rôle central, ce papier cherche àdécrire quelques uns des principaux résultats obtenus de façon informelle, en s’appuyant sur des exemples. Par comparaison avec la théorie classique des systèmes linéaires, il existe des domaines pratiquement vierges, surtout en raison du fait que les outils mathématiques correspondants restent à forger. C’est en particulier le cas de l’approche géométrique des systèmes qui, dans la théorie classique, est connue pour apporter un autre regard important sur les questions de théorie des systèmes et de synthèse de lois de commandes àcôté de la machinerie purement algébrique. Une discussion préliminaire sur les aspects géométriques de l’algèbre max-plus et leur utilité pour la théorie des systèmes est proposée dans la dernière partie du papier.

This paper is a contribution to the algebraic theory of recognizable languages. The main topic of this paper is the polynomial closure, an operation that mixes together the operations of union and concatenation. Formally, the polynomial closure of a class of languages L of A ∗ is the set of languages

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

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In this paper, we present a theory of synchronous data-flow languages. Our theory is supported by both some heuristic analysis of applications and some theoretical investigation of the data-flow paradigm. Our model covers both behavioural and operational aspects, and allows both synchronous and asynchronous styles of implementation for synchronous programs. This model served as a basis to establish the gc common format for synchronous data-flow languages.

This paper extends to pomsets without auto-concurrency the fundamental notion of asynchronous cellular automata (ACA) which was originally introduced for traces by Zielonka. We generalize to pomsets the notion of asynchronous mapping introduced by Cori, Métivier and Zielonka and we show how to construct a deterministic ACA from an asynchronous mapping. Then we investigate the relation between the expressiveness of monadic second order logic, nondeterministic ACAs and deterministic ACAs. We can generalize Büchi’s theorem for finite words to a class of pomsets without auto-concurrency which satisfy a natural axiom. This axiom ensures that an asynchronous cellular automaton works on the pomset as a concurrent read and exclusive owner write machine. More precisely, in this class non-deterministic ACAs, deterministic ACAs and monadic second order logic have the same expressive power. Then we consider a class where deterministic ACAs are strictly weaker than nondeterministic ones. But in this class nondeterministic ACAs still capture monadic second order logic. Finally it is shown that even this equivalence does not hold in the class of all pomsets since there the class of recognizable pomset languages is not closed under complementation.

this paper. The extended bibliography given below shows other important contributions by Azevedo, Costa, Delgado, Pin, Teixeira, Volkov, Weil and Zeitoun.

It is well known that automatic groups can be characterized using geometric properties of their Cayley graphs. Along the same line of thought, we provide a geometric characterization of automatic monoids. This involves working with a slightly strengthened definition of an automatic monoid which is still a proper generalization of the concept of an automatic group. The two definitions coincide in the case of right cancellative monoids for which a particularly simple characterization is obtained. 1.

. We introduce frames as basic objects for the construction of transition systems, process graphs or automata. We provide an algebraic notation for frames, and display some theoretical results. Key words & Phrases: Frame, Process, Decidability. 1987 CR Categories: F.1.1, F.4.3, I.1.1. 1 Introduction In this paper we propose a very simple, mathematical structure which we think to be of the type that underlies many structures modelling (concurrent) behaviour. The objects in this structure are called frames. We provide an axiomatic, algebraic approach to reasoning about equality between frames. Frames can be characterized as a particular kind of graphs; in fact, all frames in the present paper are labeled, directed graphs. The axioms however will allow other models. The main source of such models is the introduction of various degrees of locality (invisibility, hiding) for states. If no hiding mechanism on states is present, all states are called sharp. In this paper we will restrict a...