. The dynamic behavior of rule-based systems (like term rewriting systems [24], process algebras [27], and so on) can be traditionally determined in two orthogonal ways. Either operationally, in the sense that a way of embedding a rule into a state is devised, stating explicitly how the result is built: This is the role played by (the application of) a substitution in term rewriting. Or inductively, showing how to build the class of all possible reductions from a set of basic ones: For term rewriting, this is the usual definition of the rewrite relation as the minimal closure of the rewrite rules. As far as graph transformation is concerned, the operational view is by far more popular: In this paper we lay the basis for the orthogonal view. We first provide an inductive description for graphs as arrows of a freely generated dgs-monoidal category. We then apply 2-categorical techniques, already known for term and term graph rewriting [29, 7], recasting in this framework the...

The paper presents a case study on the synthesis of labelled transition systems (ltss) for process calculi, choosing as testbed Milner’s Calculus of Communicating System (ccs). The proposal is based on a graphical encoding: each ccs process is mapped into a graph equipped with suitable interfaces, such that the denotation is fully abstract with respect to the usual structural congruence. Graphs with interfaces are amenable to the synthesis mechanism based on borrowed contexts (bcs), proposed by Ehrig and König (which are an instance of relative pushouts, originally introduced by Milner and Leifer). The bc mechanism allows the effective construction of an lts that has graphs with interfaces as both states and labels, and such that the associated bisimilarity is automatically a congruence. Our paper focuses on the analysis of the lts distilled by exploiting the encoding of ccs processes: besides offering some technical contributions towards the simplification of the bc mechanism, the key result of our work is the proof that the bisimilarity on processes obtained via bcs coincides with the standard strong bisimilarity for ccs.

. In this paper we describe an approach to model the dynamics of distributed systems. For distributed systems we mean systems consisting of concurrent processes communicating via shared ports and posing certain synchronization requirements, via the ports, to the adjacent processes. The basic idea is to use graphs to represent states of such systems, and graph rewriting to represent their evolution. The kind of graph rewriting we use is based on simple context-free productions which are however combined by means of a synchronization mechanism. This allows for a good level of expressivity in the system without sacrifying full distribution. To formally model this kind of graph rewriting, however, we do not adopt the classical graph rewriting style but a more general framework, called the tile model, which allows for a clear separation between sequential rewriting and synchronization. Then, since the problem of satisfying the synchronization requirements may be a complex combinatorial pro...

. In the classical theory of Petri nets, a process is an operational description of the behaviour of a net, which takes into account the causal links between transitions in a sequence of firing steps. In the categorical framework developed in [19, 11], processes of a P/T net are modeled as arrows of a suitable monoidal category: In this paper we lay the basis of a similar characterization for contextual P/T nets, that is, P/T nets extended with read arcs, which allows a transition to check for the presence of a token in a place, without consuming it. 1 Introduction Petri nets [24] are probably the best studied and most used model for concurrent systems: Their range of applications covers a wide spectrum, from their use as a specification tool to their analysis as a suitable semantical domain. A recent extension to the classical model concerns a class of nets where transitions are able to check for the presence of a token in a place without actually consuming it. While the possibility ...

Recently there has been a growing interest towards algebraic structures that are able to express formalisms different from the standard, tree-like presentation of terms. Many of these approaches reveal a specific interest towards their application in the "distributed and concurrent systems" field, but an exhaustive comparison between them is difficult because their presentations can be quite dissimilar. This work is a first step towards a unified view, which is able to recast all those formalisms into a more general one, where they can be easily compared. We introduce a general schema for describing a characteristic normal form for many algebraic formalisms, and show that those normal forms can be thought of as arrows of suitable concrete monoidal categories.

Multi-algebras allow for the modeling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multi-algebras and partial algebras, analogous to the classical presentation of algebras over a signature as cartesian functors from the algebraic theory over to Set. We introduce two dierent notions of theory over a signature, both having a structure weaker than cartesian, and we consider functors from them to Rel or Pfn, the categories of sets and relations or partial functions, respectively. Next we discuss how the functorial presentation provides guidelines for the choice of syntactical notions for a class of algebras, and as an application we argue that the natural generalization of usual terms are \conditioned terms" for partial algebras, and \term graphs" for multi-algebras. Contents 1 Introduction 2 2 A short recap on multi-algebras 4 3...

The definition of sos formats ensuring that bisimilarity on closed terms is a congruence has received much attention in the last two decades. For dealing with open system specifications, the congruence is usually lifted from closed terms to open terms by instantiating the free variables in all possible ways; the only alternatives considered in the literature relying on Larsen and Xinxin's context systems and Rensink's conditional transition systems. We propose a different approach based on tile logic, where both closed and open terms are managed analogously. In particular, we analyze the `bisimilarity as congruence' property for several tile formats that accomplish di erent concepts of subterm sharing.

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. We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2-theories. We show that this presentation is equivalent to the well-accepted operational definition proposed by Barendregt et alii---but for the case of circular redexes, for which we propose (and justify formally) a different treatment. The categorical framework allows us to model in a concise way also automatic garbage collection and rules for sharing/unsharing and folding/unfolding of structures, and to relate term graph rewriting to other rewriting formalisms. R'esum'e. Nous pr'esentons une formulation cat'egorique de la r'e'ecriture des graphes cycliques des termes, bas'ee sur une variante de 2-theorie alg'ebrique. Nous prouvons que cette pr'esentation est 'equivalente `a la d'efinition op'erationnelle propos'ee par Barendregt et d'autres auteurs, mais pas dons le cas des radicaux circulaires, pour lesquels nous proposons (et justifions formellem...

This paper investigates some basic algebraic properties of the categories of spans and cospans (up to isomorphic supports) over the category Set of (small) sets and functions, analyzing the monoidal structures induced over both spans and cospans by the cartesian product and disjoint union of sets. Our results nd analogous counterparts in (and are partly inspired by) the theory of relational algebras, thus our paper also shed some light on the relationship between (co)spans and the categories of (multi)relations and of equivalence relations. And, since (co)spans yields an intuitive presentation in terms of dynamical system with input and output interfaces, our results introduce an expressive, two-fold algebra that can serve as a specication formalism for rewriting systems and for composing software modules and open programs. Key words: Spans, multi-relations, monoidal categories, system specications. Introduction The use of spans [1,6] (and of the dual notion of cospans) have been...