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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
A coalgebraic theory of reactive systems
, 1999
"... In this report we study the connection between two well known models for interactive systems. Reactive Systems à la Leifer and Milner allow to derive an interactive semantics from a reduction semantics guaranteeing, semantics (bisimilarity). Universal Coalgebra provides a categorical framework where ..."
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In this report we study the connection between two well known models for interactive systems. Reactive Systems à la Leifer and Milner allow to derive an interactive semantics from a reduction semantics guaranteeing, semantics (bisimilarity). Universal Coalgebra provides a categorical framework where bisimilarity can be characterized as final semantics, i.e., as the unique morphism to the final coalgebra. Moreover, if lifting a coalgebra to a structured setting is possible, then bisimilarity is compositional with respect to the lifted structure. Here we show that for every reactive system we can build a coalgebra. Furthermore, if bisimilarity is compositional in the reactive system, then we can lift this coalgebra to a structured coalgebra.
Coalgebraic Symbolic Semantics
"... The operational semantics of interactive systems is usually described by labeled transition systems. Abstract semantics (that is defined in terms of bisimilarity) is characterized by the final morphism in some category of coalgebras. Since the behaviour of interactive systems is for many reasons in ..."
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The operational semantics of interactive systems is usually described by labeled transition systems. Abstract semantics (that is defined in terms of bisimilarity) is characterized by the final morphism in some category of coalgebras. Since the behaviour of interactive systems is for many reasons infinite, symbolic semantics were introduced as a mean to define smaller, possibly finite, transition systems, by employing symbolic actions and avoiding some sources of infiniteness. Unfortunately, symbolic bisimilarity has a different “shape” with respect to ordinary bisimilarity, and thus the standard coalgebraic characterization does not work. In this paper, we introduce its coalgebraic models.
Tile Transition Systems as Structured Coalgebras
 FUNDAMENTALS OF COMPUTATION THEORY, VOLUME 1684 OF LNCS
, 1999
"... 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 ..."
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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...
On coalgebras over algebras
 In ”Proceedings of the Tenth Workshop on Coalgebraic Methods in Computer Science (CMCS 2010)”, Electr. Notes
"... We extend Barr’s wellknown characterization of the final coalgebra of a Setendofunctor H as the completion of its initial algebra to the EilenbergMoore category Alg(M) of algebras associated to a Setmonad M, if H can be lifted to Alg(M). As further analysis, we introduce the notion of commuting ..."
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We extend Barr’s wellknown characterization of the final coalgebra of a Setendofunctor H as the completion of its initial algebra to the EilenbergMoore category Alg(M) of algebras associated to a Setmonad M, if H can be lifted to Alg(M). As further analysis, we introduce the notion of commuting pair of endofunctors (T,H) with respect to a monad M and show that under reasonable assumptions, the final Hcoalgebra can be obtained as the completion of the free Malgebra on the initial Talgebra.
Coalgebra Morphisms Subsume Open Maps
, 1999
"... We relate two abstract notions of bisimulation, induced by open maps and by coalgebras morphisms, respectively. We show that open maps correspond to coalgebra morphisms for a suitable chosen endofunctor in a category of many sorted sets. This demonstrates that the notion of openmaps bisimilarity is ..."
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We relate two abstract notions of bisimulation, induced by open maps and by coalgebras morphisms, respectively. We show that open maps correspond to coalgebra morphisms for a suitable chosen endofunctor in a category of many sorted sets. This demonstrates that the notion of openmaps bisimilarity is of essentially coalgebraic nature. We nd moreover characterizations of subcategories of coalgebras corresponding to some relevant model categories: category of presheaves is isomorphic to the subcategory of consistent coalgebras with lax cohomomorphisms; a suitably chosen subcategory of many sorted algebras is equivalent to the full subcategory of presheaves preserving nite products.
Structured Coalbegras and Minimal HDAutomata for the πCalculus
, 2000
"... The coalgebraic framework developed for the classical process algebras, and in particular its advantages concerning minimal realizations, does not fully apply to the picalculus, due to the constraints on the freshly generated names that appear in the bisimulation. In this paper we propose to model ..."
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The coalgebraic framework developed for the classical process algebras, and in particular its advantages concerning minimal realizations, does not fully apply to the picalculus, due to the constraints on the freshly generated names that appear in the bisimulation. In this paper we propose to model the transition system of the πcalculus as a coalgebra on a category of name permutation algebras and to define its abstract semantics as the final coalgebra of such a category. We show that permutations are sufficient to represent in an explicit way fresh name generation, thus allowing for the definition of minimal realizations. We also link the coalgebraic semantics with a slightly improved version of history dependent (HD) automata, a model developed for verification purposes, where states have local names and transitions are decorated with names and name relations. HDautomata associated with agents with a bounded number of threads in their derivatives are finite and can be actually minimized. We show that the bisimulation relation in the coalgebraic context corresponds to the minimal HDautomaton.
www.elsevier.com/locate/jlap A compositional coalgebraic model of fusion calculus �
"... This paper is a further step in exploring the labelled transitions and bisimulations of fusion calculi. We follow a recent theory by the same authors and previously applied to the picalculus for lifting calculi with structural axioms to bialgebras and, thus, we provide a compositional model of the ..."
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This paper is a further step in exploring the labelled transitions and bisimulations of fusion calculi. We follow a recent theory by the same authors and previously applied to the picalculus for lifting calculi with structural axioms to bialgebras and, thus, we provide a compositional model of the fusion calculus with explicit fusions. In such a model, the bisimilarity relation induced by the unique morphism to the final coalgebra coincides with fusion hyperequivalence and it is a congruence with respect to the operations of the calculus. The key novelty in our work is that we give an account of explicit fusions through labelled transitions. Interestingly enough, this approach allows to exploit for the fusion calculus essentially the same algebraic structure used for the picalculus.