The Interval Domain: A Matchmaker for aCTL and aPCTL

Abstract We present aPCTL, a version of PCTL with an action-based semantics which coincides with the ordinary PCTL in case of a sole action type. We point out what aspects of aPCTL may be improved for its application as a probabilistic logic in a tool modeling large probabilistic system. We give a non-standard semantics to the action-based temporal logical aCTL, where the propositional clauses are interpreted in a fuzzy and the modalities in a probabilistic way; the until-construct is evaluated as a least fixed-point over these meanings. We view aCTL formulas ⊘ as templates for aPCTL formulas (which still need vectors of thresholds as annotations for all subformulas which are path formulas). Since [⊘]s, our non-standard meaning of o at state s, is an interval [a, b], we may craft aPCTL formulas o from using the information a and b respectively. This results in two aPCTL formulas o and o1. This translation defines a critical region of such thresholds for ⊘ in the following sense: if a > 0 then a satisfies the aPCTL formula o1 dually, if b we would like to thank Martin Hotzel Escardo for suggesting to look at the interval domain at the LICS'97 meeting in Warsaw. He also pointed to work in his PhD thesis about the universality of I. we also acknowledge Marta Kwaitkowska, Christel Baier, Rance Cleaveland, and Scott Smolka for fruitful discussion on this subject matter.