. We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the well-known characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particular, we show that term graphs over a signature \Sigma are one-to-one with the arrows of the free gs-monoidal category generated by \Sigma. Such a category satisfies all the axioms for Cartesian categories but for the naturality of two transformations (the discharger ! and the duplicator r), providing in this way an abstract and clear relationship between terms and term graphs. In particular, the absence of the naturality of r and ! has a precise interpretation in terms of explicit sharing and of loss of implicit garbage collection, respectively. Keywords: algebraic theories, directed acyclic graphs, gs-monoidal categories, symmetric monoidal categories, term graphs. Mathematical Subject Clas...

Visual description techniques are particularly important for the design of hybrid systems because specifications of such systems must usually be discussed between engineers from a number of different disciplines. Modularity is vital for hybrid systems not only because it allows to handle large systems, but also because hybrid systems are naturally decomposed into the system itself and its environment. Based on two different interpretations for hierarchic graphs and on a clear hybrid computation model, we develop HyCharts, two modular visual formalisms for the specification of the architecture and behavior of hybrid systems. The operators on hierarchic graphs enable us to give a surprisingly simple denotational semantics for many concepts known from statechart-like formalisms. Due to a verygeneral composition operator, HyCharts can easily be composed with description techniques from other engineering disciplines. Such heterogeneous system specifications seem to be particularly appropriate for hybrid systems because of their interdisciplinary character.

. 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...

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.

Visual description techniques are particularly important for the design of hybrid systems because specifications of such systems usually have to be discussed between engineers from a number of different disciplines. Modularity is vital for hybrid systems not only because it allows to handle large systems, but also because hybrid systems are naturally decomposed into the system itself and its environment. Based on two different interpretations for hierarchic graphs and on a clear hybrid computation model, we develop HyCharts. HyCharts consist of two modular visual formalisms, one for the specification of the architecture and one for the specification of the behavior of hybrid systems. The operators on hierarchic graphs enable us to give a surprisingly simple denotational semantics for many concepts known from statechart-like formalisms. Due to a very general composition operator, HyCharts can easily be composed with description techniques from other engineering disciplines. Such hetero...

. 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...

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.

this paper is to present a few results on mixalgebras. A mixalgebra is an algebraic structure based on functions fA : A s ! A t 1 + \Delta \Delta \Delta +A t n , with s; t 1 ; : : : ; t n products of atomic sorts. The restriction to functions with n = 1 produces classical algebras, while the dual restriction to functions with s; t 1 ; : : : ; t n atomic sorts produces coalgebras. The proper setting to speak about (co)algebras is that of a (co)algebraic theory: one needs tuples of terms when dealing with term substitution. In a similar way, the proper setting to handle mixalgebra substitution is one using tuples and cotuples of ordinary terms. Such a natural setting is provided by categories with coproducts and products such that products distribute over coproducts, known as "distributive categories". Further, distributivity requires the splitting of ordinary coalgebraic terms in "paths". A path is modeled as an extended conditional term and it is called "elementary plan". Plans are tuples and cotuples of elementary plans obeying certain conditions. They play the role the ordinary terms are playing for (co)algebras. Plans. In the construction of the plans we start with a pair of sets (O; P). The elements of O are individual or object variables, shortly called obvars. The elements in P are universal variables for elementary properties of the objects. They are shortly called propvars. Each occurrence of a variable should be fully specified as a:x, which may be read, e.g., as propvar x applied to obvar a. Notations O:x and a:P for the corresponding sets of occurrences are used, as well. To simplify the explanation we suppose the signature contains functions fA : A s1 \Theta : : : \Theta A sm ! A t 1 + \Delta \Delta \Delta + A t n , with s 1 ; : : : ; s m ; t 1 ;...