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Symmetric Monoidal and Cartesian Double Categories as a Semantic Framework for Tile Logic
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 2002
"... Tile systems offer a general paradigm for modular descriptions of concurrent systems, based on a set of rewriting rules with sideeffects. Monoidal double categories are a natural semantic framework for tile systems, because the mathematical structures describing system states and synchronizing acti ..."
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Cited by 14 (9 self)
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Tile systems offer a general paradigm for modular descriptions of concurrent systems, based on a set of rewriting rules with sideeffects. 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 netprocesslike 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.
Axioms for Contextual Net Processes
 In Automata, Languages and Programming, volume 1443 of LNCS
, 1998
"... . 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 ..."
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. 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 ...
Rewriting On Cyclic Structures: Equivalence Between The Operational And The Categorical Description
, 1999
"... . We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2theories. We show that this presentation is equivalent to the wellaccepted operational definition proposed by Barendregt et aliibut for the case of circular redexes, fo ..."
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Cited by 14 (7 self)
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. We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2theories. We show that this presentation is equivalent to the wellaccepted operational definition proposed by Barendregt et aliibut 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 2theorie 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...
Normal Forms for Partitions and Relations
 Recent Trends in Algebraic Development Techniques, volume 1589 of Lect. Notes in Comp. Science
, 1999
"... Recently there has been a growing interest towards algebraic structures that are able to express formalisms different from the standard, treelike presentation of terms. Many of these approaches reveal a specific interest towards their application in the "distributed and concurrent systems" ..."
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Cited by 14 (11 self)
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Recently there has been a growing interest towards algebraic structures that are able to express formalisms different from the standard, treelike 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.
From SOS specifications to structured coalgebras: How to make bisimulation a congruence
 ENTCS, 19(0):118 – 141
, 1999
"... In this paper we address the issue of providing a structured coalgebra presentation of transition systems with algebraic structure on states determined by an equational specification Γ. More precisely, we aim at representing such systems as coalgebras for an endofunctor on the category of Γalgebras ..."
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In this paper we address the issue of providing a structured coalgebra presentation of transition systems with algebraic structure on states determined by an equational specification Γ. More precisely, we aim at representing such systems as coalgebras for an endofunctor on the category of Γalgebras. The systems we consider are specified by using a quite general format of SOS rules, the algebraic format, which in general does not guarantee that bisimilarity is a congruence. We first show that the structured coalgebra representation works only for systems where transitions out of complex states can be derived from transitions out of corresponding component states. This decomposition property of transitions indeed ensures that bisimilarity is a congruence. For a system not satisfying this requirement, next we propose a closure construction which adds context transitions, i.e., transitions that spontaneously embed a state into a bigger context or viceversa. The notion of bisimulation for the enriched system coincides with the notion of dynamic bisimilarity for the original one, that is, with the coarsest bisimulation which is a congruence. This is sufficient to ensure that the structured coalgebra representation works for the systems obtained as result of the closure construction. 1
Coalgebraic Monads
, 2002
"... 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 rat ..."
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Cited by 7 (5 self)
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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.
Symmetric and Cartesian Double Categories as a Semantic Framework for Tile Logic
, 1995
"... 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 eff ..."
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Cited by 6 (5 self)
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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
Normal Forms for Algebras of Connections
 Theoretical Computer Science
, 2000
"... Recent years have seen a growing interest towards algebraic structures that are able to express formalisms different from the standard, treelike presentation of terms. Many of these approaches reveal a specific interest towards the application to the `distributed and concurrent systems' field, ..."
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Cited by 5 (4 self)
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Recent years have seen a growing interest towards algebraic structures that are able to express formalisms different from the standard, treelike 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.