The large diffusion of concurrent and distributed systems has spawned in recent years a variety of new formalisms, equipped with features for supporting an easy specification of such systems. The aim of our paper is to analyze three proposals, namely rewriting logic, action calculi and tile logic, chosen among those formalisms designed for the description of rule-based systems. For each of these logics we first try to understand their foundations, then we briefly sketch some applications. The overall goal of our work is to find out a common layout where these logics can be recast, thus allowing for a comparison and an evaluation of their specific features.

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.

We introduce a notion of bisimulation for graph rewriting systems, allowing us to prove observational equivalence for dynamically evolving graphs and networks. We use the framework of synchronized graph rewriting with mobility which we describe in two different, but operationally equivalent ways: on graphs defined as syntactic judgements and by using tile logic. One of the main results of the paper says that bisimilarity for synchronized graph rewriting is a congruence whenever the rewriting rules satisfy the basic source property. Furthermore we introduce an up-to technique simplifying bisimilarity proofs and use it in an example to show the equivalence of a communication network and its specification.

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...

. Multi-algebras allow to model 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 \Sigma as cartesian functors from the algebraic theory of \Sigma , Th(\Sigma), to Set. The functors we introduce are based on variations of the notion of theory, having a structure weaker than cartesian, and their target is Rel, the category of sets and relations. We argue that this functorial presentation provides an original abstract syntax for partial and multi-algebras. 1 Introduction Nondeterminism is a fundamental concept in Computer Science. It arises not only from the study of intrinsically nondeterministic computational models, like Turing machines and various kinds of automata, but also in the study of the behaviour of deterministic sys...

Abstract: A study of the classes of nite relations as enriched strict monoidal categories is presented in [CaS91]. The relations there are interpreted as connections in owchart schemes, hence an \angelic " theory of relations is used. Finite relations may be used to model the connections between the components of data ow networks [BeS98, BrS96], as well. The corresponding algebras are slightly di erent enriched strict monoidal categories modeling a \forward-demonic " theory of relations. In order to obtain a full model for parallel programs one needs to mix control and reactive parts, hence a richer theory of nite relations is needed. In this paper we (1) de ne a model of such mixed nite relations, (2) introduce enriched (weak) semiringal categories as an abstract algebraic model for these relations, and (3) show that the initial model of the axiomatization (it always exists) is isomorphic to the de ned one of mixed relations. Hence the axioms gives a sound and complete axiomatization for the these relations. Key Words: parallel programs; mixed relations; network algebra; (enriched) semiringal

. The aim of this paper is to investigate the relation between two models of concurrent systems: tile rewrite systems and coalgebras. Tiles are rewrite rules with side effects which are endowed with operations of parallel and sequential composition and synchronization. Their models can be described as monoidal double categories. Coalgebras can be considered, in a suitable mathematical setting, as dual to algebras. They can be used as models of dynamical systems with hidden states in order to study concepts of observational equivalence and bisimilarity in a more general setting. In order to capture in the coalgebraic presentation the algebraic structure given by the composition operations on tiles, coalgebras have to be endowed with an algebraic structure as well. This leads to the concept of structured coalgebras, i.e., coalgebras for an endofunctor on a category of algebras. However, structured coalgebras are more restrictive than tile models. Those models which can be presented as st...

this paper we consider bisimulation equivalences [33, 37] (with bisimilarity meaning the maximal bisimulation), where the entire branching structure of the transition system is accounted for: informally, two states are equivalent if whatever transition one can perform, the other can simulate it via a transition with the same observation, still ending in equivalent states.