Concurrent Dynamic Algebra

Abstract: We reconstruct Peleg's concurrent dynamic logic in the context of modal Kleene algebras. We explore the algebraic structure of its multirelational semantics and develop an abstract axiomatisation of concurrent dynamic algebras from that basis. In this axiomatisation, sequential composition is not associative. It interacts with concurrent composition through a weak distributivity law. The modal operators of concurrent dynamic algebra are obtained from abstract axioms for domain and antidomain operators; the Kleene star is modelled as a least fixpoint. Algebraic variants of Peleg's axioms are shown to be valid in these algebras and their soundness is proved relative to the multirelational model. Additional results include iteration principles for the Kleene star and a refutation of variants of Segerberg's axiom in the multirelational setting. The most important results have been verified formally with Isabelle/HOL.