The meaning of negative premises in transition system specifications

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 [Bloom et al. 1995; Groote 1993]. Finally, we provide a sound and complete axiomatization for this example.