. This paper surveys the work of many researchers on rewriting logic since it was first introduced in 1990. The main emphasis is on the use of rewriting logic as a semantic framework for concurrency. The goal in this regard is to express as faithfully as possible a very wide range of concurrency models, each on its own terms, avoiding any encodings or translations. Bringing very different models under a common semantic framework makes easier to understand what different models have in common and how they differ, to find deep connections between them, and to reason across their different formalisms. It becomes also much easier to achieve in a rigorous way the integration and interoperation of different models and languages whose combination offers attractive advantages. The logic and model theory of rewriting logic are also summarized, a number of current research directions are surveyed, and some concluding remarks about future directions are made. Table of Contents 1 In...

In this paper we introduce a model for a wide class of computational systems, whose behaviour can be described by certain rewriting rules. We gathered our inspiration both from the world of term rewriting, in particular from the rewriting logic framework [Mes92], and of concurrency theory: among the others, the structured operational semantics [Plo81], the context systems [LX90] and the structured transition systems [CM92] approaches. Our model recollects many properties of these sources: first, it provides a compositional way to describe both the states and the sequences of transitions performed by a given system, stressing their distributed nature. Second, a suitable notion of typed proof allows to take into account also those formalisms relying on the notions of synchronization and side-effects to determine the actual behaviour of a system. Finally, an equivalence relation over sequences of transitions is defined, equipping the system under analysis with a concurrent semantics, ...

. In the last years there has been a growing interest towards categorical models for term rewriting systems (trs's). In our opinion, very interesting are those associating to each trs's a cat-enriched structure: a category whose hom-sets are categories. Interpreting rewriting steps as morphisms in hom-categories, these models provide rewriting systems with a concurrent semantics in a clean algebraic way. In this paper we provide a unified presentation of two models recently proposed in literature by Jos'e Meseguer [Mes90, Mes92, MOM93] and John Stell [Ste92, Ste94], respectively, pursuing a critical analysis of both of them. More precisely, we show why they are to a certain extent unsatisfactory in providing a concurrent semantics for rewriting systems. It turns out that the derivation space of Meseguer's Rewriting Logic associated with each term (i.e., the set of coinitial computations) fails in general to form a prime algebraic domain: a condition that is generally cons...

. Rational terms (possibly infinite terms with finitely many subterms) can be represented in a finite way via -terms, that is, terms over a signature extended with self-instantiation operators. For example, f ! = f(f(f(: : :))) can be represented as x :f(x) (or also as x :f(f(x)), f(x :f(x)), . . . ). Now, if we reduce a -term t to s via a rewriting rule using standard notions of the theory of Term Rewriting Systems, how are the rational terms corresponding to t and to s related? We answer to this question in a satisfactory way, resorting to the definition of infinite parallel rewriting proposed in [7]. We also provide a simple, algebraic description of -term rewriting through a variation of Meseguer's Rewriting Logic formalism. 1 Introduction Rational terms are possibly infinite terms with a finite set of subterms. They show up in a natural way in Theoretical Computer Science whenever some finite cyclic structures are of concern (for example data flow diagrams, cyclic te...

In [12] we introduced the tile model, a framework encompassing a wide class of computational systems, whose behaviour can be described by certain rewriting rules. We gathered our inspiration both from the world of term rewriting and of concurrency theory, and our formalism recollects many properties of these sources. For example, it provides a compositional way to describe both the states and the sequences of transitions performed by a given system, stressing their distributed nature. Moreover, a suitable notion of typed proof allows to take into account also those formalisms relying on the notions of synchronization and side-effects to determine the actual behaviour of a system. In this work we narrow our scope, presenting a restricted version of our tile model and focussing our attention on its expressive power. To this aim, we recall the basic definitions of the process algebras paradigm [3,24], centering the paper on the recasting of this framework in our formalism.

. In this paper we discuss a general methodology to provide a categorical semantics for a wide class of computational systems, whose behaviour can be described by a suitable set of transition steps. We open our survey presenting some results on the semantics of Petri Nets. Starting from this, we elaborate a two-steps procedure allowing for the description of all the sequences of transitions performed by a given system, and equipping them with a suitable equivalence relation. This relation provides the sistem under analisys with a concurrent semantics: equivalence classes denote families of "computationally equivalent" behaviours, corresponding to the execution of the same set of (causally) independent transition steps. 1 Introduction The latest years have seen a wide amount of different approaches to the semantics of computional sistems: a variety that, if only for the comparison between the various formalisms, calls for a unified framework. In this paper we aim to show that enriched ...

We consider (a slight variant of) the ccs calculus, and we analyze two operational semantics defined in the literature: the first exploits Proved Transition Systems (pts) and the second Rewriting Logic (rl). We show that the interleaving interpretation of both semantics agree, in that they define the same transitions and exhibit the same nondeterministic structure. In addition, we study causality in ccs computations. We recall the treatment via pts, that exhibits the notion of causality presented in the literature, and we show how to recast it in the rl semantics via suitable axioms. 1 Introduction Concurrency is maybe the basic aspect of the operational interpretation of rewriting logic. And as Jos'e Meseguer says in his lecture at concur'96 [20], . . . my main emphasis in this talk will be on rewriting logic as a semantic framework for concurrency. . . . The goal is . . . to express as faithfully as possible each model [of concurrency] on its own terms, avoiding any encodings or tr...

Abstract. The notion of path is classical in graph theory but not directly used in the term rewriting community. The main idea of this work is to raise the notion of path to the level of first-order terms, i.e. paths become part of the terms and not just meta-information about them. These paths are represented by words of integers (positive or negative) and are interpreted as relative addresses in terms. In this way, paths can also be seen as a generalization of the classical notion of position for the first-order terms and are inspired by de Bruijn indexes. In this paper, we define an original framework called Referenced Term Rewriting where paths are used to represent pointers between subterms. Using this approach, any term-graph rewriting systems can be simulated using a term rewrite-based environment. 1

Îlots formels et certification du filtrage 31 Gestion mémoire et structures de données 61 Applications 121 Extensions aux anti-patterns et aux graphes 155Présentations de Tom