The meaning of negative premises in transition system specifications

Abstract

Abstract. We present a general theory for the use of negative premises in the rules of Transition System Specifications (TSSs). We formulate a criterion that should be satisfied by a TSS in order to be meaningful, that is, to unequivocally define a transition relation. We also provide powerful techniques for proving that a TSS satisfies this criterion, meanwhile constructing this transition relation. Both the criterion and the techniques originate from logic programming [van Gelder et al. 1988;Gelfond and Lifschitz 1988] to which TSSs are close. In an appendix we provide an extensive comparison between them.As in Groote [1993], we show that the bisimulation relation induced by a TSS is a congruence, provided that it is in ntyft/ntyxt-format and can be proved meaningful using our techniques. We also considerably extend the conservativity theorems of Groote [1993] and Groote and Vaandrager [1992]. As a running example, we study the combined addition of priorities and abstraction to Basic Process Algebra (BPA). Under some reasonable conditions we show that this TSS is indeed meaningful, which could not be shown by other methods Groote 1993]. Finally, we provide a sound and complete axiomatization for this example.

Rooted branching bisimulation as a congruence for probabilistic transition systems

“…As has been argued elsewhere (e.g. [18,8,16]), transition system specifications with negative premises do not uniquely define a transition relation and different reasonable techniques may lead to incomparable models. For instance, the PTSS with the single constant f , set of labels {a, b} and the two rules…”

“…− → f f f , has two models that are justifiably compatible with the rules (so called supported models [7,8,16]…”

“…the second item follows from the Knaster-Tarski theorem. PTSS with a least 3-valued stable model that are also a 2-valued model are of particular interest, since such a model is actually the only 3-valued stable model [8,16].…”

SOS rule formats for convex and abstract probabilistic bisimulations

“…PTSSs with least 3-valued stable model that are also a 2-valued model are particularly interesting, since this model is actually the only 3-valued stable model [7,13]. A PTSS P is said to be complete if its least 3-valued stable model CT, PT satisfies that CT = PT (i.e., the model is also 2-valued).…”

Priorities Without Priorities: Representing Preemption in Psi-Calculi

“…The second is that the rules are augmented with side conditions that prevent a process from taking low priority actions. This has a natural formulation in terms of negative premises [5], but in order to make implementation in Isabelle easier we instead define the semantics in two layers, following [9,10,22].…”