Parallel action: Concurrent dynamic logic with independent modalities

Abstract

Regular dynamic logic is extended by the program constructα∩β, meaning “α andβ executed in parallel”. In a semantics due to Peleg, each commandα is interpreted as a set of pairs (s,T), withT being the set of states “reachable” froms by a single execution ofα, possibly involving several processes acting in parallel. The modalities <α< and [α] are given the interpretations

<α>A is true ats iff there existsT withsRβT andA true throughoutT, and

[α]A is true ats iff for allT, ifsRβT thenA is true throughoutT, which make <α> and [α] no longer interdefinable via negation, as they are in the regular case.

We prove that the logic defined by this modelling is finitely axiomatisable and has the finite model property, hence is decidable. This requires the development a new theory of canonical models and filtrations for “reachability” relations.