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.

Tile systems offer a general paradigm for modular descriptions of concurrent systems, based on a set of rewriting rules with side-effects. Monoidal double categories are a natural semantic framework for tile systems, because the mathematical structures describing system states and synchronizing actions (called configurations and observations, respectively, in our terminology) are monoidal categories having the same objects (the interfaces of the system). In particular, configurations and observations based on net-process-like and term structures are usually described in terms of symmetric monoidal and cartesian categories, where the auxiliary structures for the rearrangement of interfaces correspond to suitable natural transformations. In this paper we discuss the lifting of these auxiliary structures to double categories. We notice that the internal construction of double categories produces a pathological asymmetric notion of natural transformation, which is fully exploited in one dimension only (for example, for configurations or for observations, but not for both). Following Ehresmann (1963), we overcome this biased definition, introducing the notion of generalized natural transformation between four double functors (rather than two). As a consequence, the concepts of symmetric monoidal and cartesian (with consistently chosen products) double categories arise in a natural way from the corresponding ordinary versions, giving a very good relationship between the auxiliary structures of configurations and observations. Moreover, the Kelly–Mac Lane coherence axioms can be lifted to our setting without effort, thanks to the characterization of two suitable diagonal categories that are always present in a double category. Then, symmetric monoidal and cartesian double categories are shown to offer an adequate semantic setting for process and term tile systems.

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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 introduces coalgebraic monads as a unified model of term algebras covering fundamental examples such as initial algebras, final coalgebras, rational terms and term graphs. We develop a general method for obtaining finitary coalgebraic monads which allows us to generalise the notion of rational term and term graph to categories other than Set. As an application we sketch part of the correctness of the the term graph implementation of functional programming languages.

this paper we discuss the lifting of these auxiliary structures to double categories. We notice that the internal construction of double categories produces a pathological asymmetric notion of natural transformation, which is fully exploited in one dimension only (e.g., for configurations or for effects, but not for both). Following Ehresmann (1963), we overcome this biased definition, introducing the notion of generalized natural transformation between four

We present a categorical formulation of the rewriting of possibly cyclic term graphs, and the proof that this presentation is equivalent to the well-accepted operational definition proposed in [3] -- but for the case of circular redexes, for which we propose (and justify formally) a different treatment. The categorical framework, based on suitable 2-categories, allows to model also automatic garbage collection and rules for sharing/unsharing and folding/unfolding of structures. Furthermore, it can be used for defining various extensions of term graph rewriting, and for relating it to other rewriting formalisms.

. 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...

Graph transformation systems have been introduced for the formal specication of software systems. States are thereby modeled as graphs, and computations as graph derivations according to the rules of the specication. Operations on graph derivations provide means to reason about the distribution and composition of computations. In this paper we discuss the development of an algebra of graph derivations as a descriptive model of graph transformation systems. For that purpose we use a categorical three level approach for the construction of models of computations based on structured transition systems. Categorically the algebra of graph derivations can then be characterized as a free double category with nite horizontal colimits.

Recent years have seen 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 the application to the `distributed and concurrent systems' field, but an exhaustive comparison between them is sometimes difficult, because their presentations can be quite dissimilar. This work is a first step towards a unified view: Focusing on the primitive ingredients of distributed spaces (namely interfaces, links and basic modules), we introduce a general schema for describing a normal form presentation of many algebraic formalisms, and show that those normal forms can be thought of as arrows of suitable monoidal categories.