An algebraic characterization of transition system equivalences

Arnold, André; Dicky, Anne

doi:10.1016/0890-5401(89)90054-0

Cited by 19 publications

(8 citation statements)

References 9 publications

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“…The reverse inclusion amounts to the condition that if f(x) a −→ y for some y in X , then there exists a state x in X such that x a −→ x and f(x ) = y. Morphisms of transition systems that verify this extra condition are exactly those whose graph is a strong bisimulation. They are also known in the literature as the 'zig-zag morphisms' (van Benthem 1984) (or as the 'saturating morphisms' (Arnold and Dicky 1989)), which, as shown in Joyal et al (1993), are the 'P-open morphisms', for a suitable subcategory P of the category of transition systemscf. the next section.…”

Section: Operational Models As Coalgebras Of Behavioursmentioning

confidence: 99%

On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces

Turi

Rütten

1998

Math. Struct. Comp. Sci.

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This paper, a revised version of Rutten and Turi (1993), is part of a programme aiming at formulating a mathematical theory of structural operational semantics to complement the established theory of domains and denotational semantics to form a coherent whole (Turi 1996; Turi and Plotkin 1997). The programme is based on a suitable interplay between the induction principle, which pervades modern mathematics, and a dual, non-standard ‘coinduction principle’, which underlies many of the recursive phenomena occurring in computer science.The aim of the present survey is to show that the elementary categorical notion of a final coalgebra is a suitable foundation for such a coinduction principle. The properties of coalgebraic coinduction are studied both at an abstract categorical level and in some specific categories used in semantics, namely categories of non-well-founded sets, partial orders and metric spaces.

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Section: Operational Models As Coalgebras Of Behavioursmentioning

confidence: 99%

On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces

Turi

Rütten

1998

Math. Struct. Comp. Sci.

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“…Both important properties have been extended in [16] to partial ordering labelings, and they are proved to hold under significantly milder conditions. Similar results (but apparently applicable only to transition systems without T) have been generalized to saturating quasi-homomorphisms of ~ algebras in [2] (following [19]). In the present paper, the main result about abstraction homomorphisms is the evidence of their flexibility in expressing several notions of equivalence: we show that observation equivalence can be characterized in terms of abstraction homomorphims that preserve successors, whereas branching bisimulation corresponds with abstraction homomorphisms which preserve both successors and predecessors.…”

Section: Introductionmentioning

confidence: 58%

Back and forth bisimulations

Nicola

Montanari

Vaandrager³

1990

Lecture Notes in Computer Science

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This paper is concerned with bisimulation relations which do not only require related agents to simulate each others behavior in the direction of the arrows, but also to simulate each other when going back in history. First it is demonstrated that the back and forth variant of strong bisimulatlon leads to the same equivalence as the ordinary notion of strong bisimulation. Then it is shown that the back and forth variant of Milner's observation equivalence is different from (and finer than) observation equivalence. In fact we prove that it coincides with the branching bisimulation equivalence of Van Glabbeek & Weljland. Also the back and forth variants of branching, 1J and delay bisimulation lead to branching bisimuiation equivalence. The notion of back and forth bisimulation moreover leads to characterizations of branching bisimulation in terms of abstraction homomorphisms and in terms of Hennessy-Milner logic with backward modalities. In our view these results support the claim that branching blsimulation Is a natural and important notion.

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“…They proved that bisimulation is expressed through a span of open maps (in the suitable category). Moreover, the pure-morphisms of Benson and Ben-Shachar (1988), the abstraction homomorphisms of Castellani (1987) and Montanari and Sgamma (1989), and the saturating quasi-homomorphism of Arnold and Dicky (1989) are special instances of open maps. Moreover, the pure-morphisms of Benson and Ben-Shachar (1988), the abstraction homomorphisms of Castellani (1987) and Montanari and Sgamma (1989), and the saturating quasi-homomorphism of Arnold and Dicky (1989) are special instances of open maps.…”

Section: Discussionmentioning

confidence: 99%

Structured transition systems with parametric observations: observational congruences and minimal realizations

Ferrari

Montanari

Mowbray

1997

Math. Struct. Comp. Sci.

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Link to this article: http://journals.cambridge.org/abstract_S0960129597002284How to cite this article: GIANLUIGI FERRARI, UGO MONTANARI and MIRANDA MOWBRAY (1997). Structured transition systems with parametric observations: observational congruences and minimal realizations .A large number of observational semantics for process description languages have been developed, many of which are based on the notion of bisimulation. In this paper, we consider in detail the problem of defining a semantic framework to unify these. The discussion takes place in a purely algebraic setting. We introduce a special class of algebras called Structured Transition Systems. A structured transition system can be viewed as a transition system with an algebraic structure both on states and transitions. In this framework, observations of behaviours are dealt with by means of maps from the transitions to some algebra of observations.Using several examples, we show that this framework allows us to describe a range of observational semantics within a single underlying presentation: it is enough to consider different mappings and algebras of observations. Furthermore, we introduce a notion of bisimulation that is parameterized with respect to the choice of the algebra of observations, and we find circumstances under which a Structured Transition System has good properties with respect to this parameterized bisimulation.First, some general syntactic constraints, independent from the choice of the algebra of the observations, are given for Structured Transition System presentations. We show that these constraints ensure that parameterized bisimulation is always a congruence. Next, we address the problem of Minimal Realizations. We show that when the presentation satisfies the syntactic constraints there exists a minimal realization, i.e., there is a model of the presentation whose elements fully characterize congruence classes under bisimulation. Structured transition systems 275Because of the previous theorems, in order to study the properties of the preorder of tp-morphisms it is enough to study the partially ordered set (C tp , ⊆) of congruences induced on the initial object by tp-morphisms and ordered by inclusion.Definition 48. (The Maximal Congruence) Let Θ be a relation over the elements of the initial object defined as ( i∈J R(f i )) * where {f i : (I, I ) → (T i , i ) | i ∈ J} is the set of all tp-morphisms starting at (I, I ).Clearly, Θ contains each relation in C tp . It is easy to prove that it is a congruence since every R(f i ) is a congruence relation. Furthermore, e 1 Θe 2 implies that I (e 1 ) = I (e 2 ).Definition 49. (The Quotient Construction) Let Q be the quotient of the initial object I with respect to the congruence Θ. Q is an equational type algebra whose elements are the congruences classes [e] Θ . The typing structure on Q is given by: [e] Θ : a provided that there exists some e ∈ [e] Θ such that e : a.Notice that because O has no confusion on types, if a is a type in I and I (a) = I (e), then e = a. Therefore, [a] ...

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An algebraic characterization of transition system equivalences

Cited by 19 publications

References 9 publications

On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces

On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces

Back and forth bisimulations

Structured transition systems with parametric observations: observational congruences and minimal realizations

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