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12
Generalizing the powerset construction, coalgebraically
, 2010
"... Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the type of systems (Fcoalgebras) and a notion of behavioral equivalence (∼F) amongst them. Many types of transition systems and their equivalences can be captured by a ..."
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Cited by 21 (7 self)
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Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the type of systems (Fcoalgebras) and a notion of behavioral equivalence (∼F) amongst them. Many types of transition systems and their equivalences can be captured by a functor F. For example, for deterministic automata the derived equivalence is language equivalence, while for nondeterministic automata it is ordinary bisimilarity. The powerset construction is a standard method for converting a nondeterministic automaton into an equivalent deterministic one as far as language is concerned. In this paper, we lift the powerset construction on automata to the more general framework of coalgebras with structured state spaces. Examples of applications include partial Mealy machines, (structured) Moore automata, and Rabin probabilistic automata.
Coalgebraic automata theory: Basic results
 Logical Methods in Computer Science
"... Vol. 4 (4:10) 2008, pp. 1–43 www.lmcsonline.org ..."
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A Coalgebraic Perspective on Linear Weighted Automata
, 2011
"... Weighted automata are a generalization of nondeterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for nondeterministic automata, their behaviours can be expressed in terms of either ( ..."
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Cited by 11 (6 self)
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Weighted automata are a generalization of nondeterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for nondeterministic automata, their behaviours can be expressed in terms of either (weighted) bisimilarity or (weighted) language equivalence. Coalgebras provide a categorical framework for the uniform study of statebased systems and their behaviours. In this work, we show that coalgebras can suitably model weighted automata in two different ways: coalgebras on
Deriving syntax and axioms for quantitative regular behaviours
, 2009
"... We present a systematic way to generate (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of quantitative systems. Our quantitative systems include weighted versions of automata and transition systems, in which transitions ar ..."
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Cited by 10 (4 self)
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We present a systematic way to generate (1) languages of (generalised) regular expressions, and (2) sound and complete axiomatizations thereof, for a wide variety of quantitative systems. Our quantitative systems include weighted versions of automata and transition systems, in which transitions are assigned a value in a monoid that represents cost, duration, probability, etc. Such systems are represented as coalgebras and (1) and (2) above are derived in a modular fashion from the underlying (functor) type of these coalgebras. In previous work, we applied a similar approach to a class of systems (without weights) that generalizes both the results of Kleene (on rational languages and DFA’s) and Milner (on regular behaviours and finite LTS’s), and includes many other systems such as Mealy and Moore machines. In the present paper, we extend this framework to deal with quantitative systems. As a consequence, our results now include languages and axiomatizations, both existing and new ones, for many different kinds of probabilistic systems.
GENERALIZING DETERMINIZATION FROM AUTOMATA TO COALGEBRAS
"... The powerset construction is a standard method for converting a nondeterministic automaton into a deterministic one recognizing the same language. In this paper, we lift the powerset construction from automata to the more general framework of coalgebras with structured state spaces. Coalgebra is an ..."
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The powerset construction is a standard method for converting a nondeterministic automaton into a deterministic one recognizing the same language. In this paper, we lift the powerset construction from automata to the more general framework of coalgebras with structured state spaces. Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the type of systems (Fcoalgebras) and a notion of behavioural equivalence (∼F) amongst them. Many types of transition systems and their equivalences can be captured by a functor F. For example, for deterministic automata the derived equivalence is language equivalence, while for nondeterministic automata it is ordinary bisimilarity. We give several examples of applications of our generalized determinization construction, including partial Mealy machines, (structured) Moore automata, Rabin probabilistic automata, and, somewhat surprisingly, even pushdown automata. To further witness the generality of the approach we show how to characterize coalgebraically several equivalences which have been object of interest in the concurrency community, such as failure or ready
TYPES AND COALGEBRAIC STRUCTURE
"... We relate weak limit preservation properties of coalgebraic type functors F to structure theoretic properties of the class of all Fcoalgebras. ..."
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We relate weak limit preservation properties of coalgebraic type functors F to structure theoretic properties of the class of all Fcoalgebras.
Resource bisimilarity and graded bisimilarity coincide ✩
"... Resource bisimilarity has been proposed in the literature on concurrency theory as a notion of bisimilarity over labeled transition systems that takes into account the number of choices that a system has. Independently, gbisimilarity has been defined over Kripke models as a suitable notion of bisim ..."
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Resource bisimilarity has been proposed in the literature on concurrency theory as a notion of bisimilarity over labeled transition systems that takes into account the number of choices that a system has. Independently, gbisimilarity has been defined over Kripke models as a suitable notion of bisimilarity for graded modal logic. This note shows that these two notions of bisimilarity coincide over imagefinite Kripke frames. Keywords: coalgebras Concurrency, resource bisimilarity, graded bisimilarity, graded modal logic, Kripke frames, 1.
Latest version of Annex I: 7.6.2010
, 2013
"... Labeled statetofunction transition systems, FuTSs for short, capture transition schemes incorporating multiplicities from states to functions of finite support over general semirings. As such FuTSs constitute a convenient modeling instrument to deal with process languages and their stochastic ex ..."
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Labeled statetofunction transition systems, FuTSs for short, capture transition schemes incorporating multiplicities from states to functions of finite support over general semirings. As such FuTSs constitute a convenient modeling instrument to deal with process languages and their stochastic extensions in particular. In this paper, the notion of bisimulation induced by a FuTS is addressed from a coalgebraic point of view. A correspondence result is established stating that FuTSbisimilarity coincides with behavioural equivalence of the associated functor. Moreover, it is shown that for FuTSs involving a specific type of semiring only, weak pullbacks are preserved. As a consequence, for these FuTSs, behavioural equivalence coincides with coalgebraic bisimilarity. As generic examples, the equivalences underlying the stochastic process algebras PEPA and IML are related to the bisimilarity of specific FuTSs. By the correspondence result coalgebraic justification of the equivalences of these calculi is obtained. Further illustrations of FuTS semantics are discussed for deterministically (discrete) timed process algebras and Markov Automata.
Algebraic Enriched Coalgebras?
"... Abstract. Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the types of systems (Fcoalgebras) and a notion of behavioral equivalence (∼F) amongst them. Many types of transition systems and their equivalences can be ca ..."
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Abstract. Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the types of systems (Fcoalgebras) and a notion of behavioral equivalence (∼F) amongst them. Many types of transition systems and their equivalences can be captured by a functor F. For example, for deterministic automata the derived equivalence is language equivalence, while for nondeterministic automata is ordinary bisimilarity. The powerset construction is a standard method for converting a nondeterministic automaton into an equivalent deterministic one as far as language is concerned. In this paper, we lift the powerset construction on automata to the more general framework of coalgebras with enriched state spaces. Examples of application include partial Mealy machines, (enriched) Moore automata, and Rabin probabilistic automata.
Abstract Copower
"... We give a common generalization of two earlier constructions in [2], that yielded coalgebraic type functors for weighted, resp. fuzzy transition systems. Transition labels for these systems were drawn from a commutative monoid M or a complete semilattice L, with the transition structure interacting ..."
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We give a common generalization of two earlier constructions in [2], that yielded coalgebraic type functors for weighted, resp. fuzzy transition systems. Transition labels for these systems were drawn from a commutative monoid M or a complete semilattice L, with the transition structure interacting with the algebraic structure on the labels. Here, we show that those earlier signature functors are in fact instances of a more general construction, provided by the socalled copower functor. Exemplarily, we instantiate this functor in categories given by varieties V of algebras. In particular, for the variety S of all semigroups, or the variety M of all (not necessarily commutative) monoids, and with M any monoid, we find that the resulting copower functors MS[−] (resp MM[−]) weakly preserve pullbacks if and only if M is equidivisible (resp. conical and equidivisible). Finally, we show that copower functors are universal in the sense that every Setfunctor can be seen as an instance of an appropreiate copower functor. 1