Bisimulations Up-to for the Linear Time Branching Time Spectrum

Abstract

Abstract. Coinductive definitions of semantics based on bisimulations have rather pleasant properties and are simple to use. In order to get coinductive characterisations of those semantic equivalences that are weaker than strong bisimulation we use a variant of the bisimulation up-to technique in which we allow the use of a given preorder relation. We prove that under some technical conditions our bisimulations up-to characterise the kernel of the given preorder. It is remarkable that the adequate orientation of the ordering relation is crucial to get this result. As a corollary, we get nice coinductive characterisations of all the axiomatic semantic equivalences in Van Glabbeek's spectrum. Although we first prove our results for finite processes, reasoning by induction, then we see, by using continuity arguments, that they are also valid for infinite (finitary) processes.

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“…In that reference, independently of our work and building on their previous paper [10], de Frutos Escrig and Rodriguez show, amongst other things, how to generate an inequational axiomatization for preorders in the spectrum from equational axiomatizations for the corresponding equivalence. They generate this inequational axiomatization by simply adding the defining inequational axioms for the ready simulation preorder to the axiomatization for the equivalence-see Theorem 6 in [11].…”

Unifying the Linear Time-Branching Time Spectrum of Process Semantics

“…Likewise, the study of the concrete models has been usually undertaken paying little attention to the other semantics or to the relations among them, even though it is well-known that there exist "families" of semantics-such as the linear semantics-which are undoubtedly related. A unified study of semantics has both methodological and practical implications that have been explored along the last years by the authors of this work, for example in [22,24,25,20,21], and also in work by important researchers in the area [5,4,16,40]. This research shows that a unified view of process semantics is indeed possible.…”

“…Since Det(p) is deterministic for each a ∈ Act there is a unique transition Det(p) a =⇒ F Det( i p i a ). By applying the definition of a =⇒ F we have p a =⇒ F i p i a , and clearly we have (Det( i p i a ), i p i a ) ∈ R. Although this is a very simple example, it is interesting to compare the proof above with that in [22]. This proof is simpler and more natural, mainly because the proof obligations to check bisimulations forced us to remove the sub-terms that were not in the chosen transition when we had to simulate it.…”

“…Although the main results in this paper are also valid for infinite processes-as we showed in [22,25]-in order to simplify the presentation of the concepts, we will mainly consider finite processes generated by the basic process algebra BCCSP which contains only the basic process algebraic operators from CCS [42], and CSP [32], but is sufficiently powerful to express all finite synchronization trees [41]. This language has repeatedly been used in unification work, e.g.…”

“…Application: trace deterministic normal forms. As a simple application we present the example used by Klin in [36], that we already used in [22] to illustrate our coinductive characterization of the behavior preorders by means of our bisimulations up-to. Definition 9.12.…”