Initial algebra and final coalgebra semantics for concurrency

Rütten, Jan; Turi, Daniele

doi:10.1007/3-540-58043-3_28

Cited by 58 publications

(53 citation statements)

References 27 publications

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“…A bialgebra is a combined algebra and coalgebra F (X ) ^ X ^ G (X ) on a common carrier (or state space) X , satisfying a certain compatibility re quirement wrt. a distributive law connecting the functors F, G. These bialgebras found application within the abstract, combined description of operational and denotational semantics started explicitly by Turi and Plotkin [34,33]-and more implicitly by Rutten and Turi [31]. This is now an active line of work [25,20,5,18].…”

Section: Introductionmentioning

confidence: 99%

A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages

Jacobs

2006

Algebra, Meaning, and Computation

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To Joseph Goguen on the occasion of his 65th birthday1.Abstract. This papers reviews the classical theory of deterministic automata and regular languages from a categorical perspective. The basis is formed by Rutten's description of the Brzozowski automaton structure in a coalgebraic framework. We enlarge the framework to a so-called bialgebraic one, by including algebras together with suitable distributive laws connecting the algebraic and coalgebraic structure of regular expressions and languages. This culminates in a reformulated proof via finality of Kozen's completeness result. It yields a complete axioma tisation of observational equivalence (bisimilarity) on regular expressions. We suggest that this situation is paradigmatic for (theoretical) computer science as the study of "generated behaviour".

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Section: Introductionmentioning

confidence: 99%

A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages

Jacobs

2006

Algebra, Meaning, and Computation

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“…Firstly, one can resort to the rich literature on standard measure theory for metric spaces (see, e.g., [KV84]). Secondly, we can use the recently developed coalgebraic theory on metric spaces [AR89,RT94], which seems to be better suited to describe (both ordinary and probabilistic) bisimulation than the corresponding theory for ordered spaces (cf. [RT94]).…”

Section: Introductionmentioning

confidence: 99%

Bisimulation for probabilistic transition systems: A coalgebraic approach

Vink

Rütten

1997

Automata, Languages and Programming

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The notion of bisimulation as proposed by Larsen and Skou for discrete probabilistic transition systems is shown to coincide with a coalgebraic definition in the sense of Aczel and Mendler in terms of a set functor, which associates to a set its collection of simple probability distributions. This coalgebraic formulation makes it possible to generalize the concepts of discrete probabilistic transition system and probabilistic bisimulation to a continuous setting involving Borel probability measures. A functor M 1 is introduced that yields for a metric space its collection of Borel probability measures. Under reasonable conditions, this functor exactly captures generalized probabilistic bisimilarity. Application of the final coalgebra paradigm to a functor based on M 1 then yields an internally fully abstract semantical domain with respect to probabilistic bisimulation, which is therefore well-suited for the interpretation of probabilistic specification and stochastic programming concepts.

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“…(Ideally, one should also be able to tell from the format of the equation whether its solution is rational and, hence, finitely representable by a nd-automaton.) The definition of such formats will be related to that of transition system specifications, for instance as described in [GV92] and used in [RT94]. See also [PE98].…”

Section: Discussionmentioning

confidence: 99%

“…Our way of solving differential equations is essentially based on the processes as terms methodology, used in [RT94]. The notion of input derivative of formal power series, generalizes Brzozowski's original definition for regular expressions [Brz64,Con71].…”

Section: Introductionmentioning

confidence: 99%

Automata, Power Series, and Coinduction: Taking Input Derivatives Seriously (Extended Abstract)

Rütten

1999

Automata, Languages and Programming

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REPORTRAPPORT AbstractFormal power series, which are functions from the set of words over an alphabet A to a semiring k, are viewed coalgebraically. In summary, this amounts to supplying the set of all power series with a deterministic automaton structure, which has the universal property of being final. Finality then forms the basis for both definitions and proofs by coinduction, the coalgebraic counterpart of induction. Coinductive definitions of operators on power series take the shape of what we have called behavioural differential equations, after Brzozowski's notion of input derivative, and include many classical differential equations for analytic functions. The use of behavioural differential equations leads, amongst others, to easy definitions of and proofs about both existing and new operators on power series, as well as to the construction of finite (syntactic) nondeterministic automata, implementing them.

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Initial algebra and final coalgebra semantics for concurrency

Cited by 58 publications

References 27 publications

A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages

A Bialgebraic Review of Deterministic Automata, Regular Expressions and Languages

Bisimulation for probabilistic transition systems: A coalgebraic approach

Automata, Power Series, and Coinduction: Taking Input Derivatives Seriously (Extended Abstract)

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