Concurrent Kripke Structures

We consider a class of Kripke Structures in which the atomic propositions are events. This enables us to represent worlds as sets of events and the transition and satisfaction relations of Kripke structures as the subset and membership relations on sets. We use this class, called event Kripke structures, to model concurrency. The obvious semantics for these structures is a true concurrency semantics. We show how several aspects of concurrency can be easily defined, and in addition get distinctions between causality and enabling, and choice and nondeterminism. We define a duality for event Kripke structures, and show how this duality enables us to convert between imperative and declarative views of programs, by treating states and events on the same footing. We provide pictorial representations of both these views, each encoding all the information to convert to the other. We define a process algebra of event Kripke structures, showing how to combine them in the usual ways—parallel composition, sequential composition, choice, interaction and iteration. Various properties of these connectives like associativity and distributivity are proved. We then show that Winskel’s event structures can be embedded in the class of event Kripke structures, and define partial synchronous composition, the primary connective for event structures, for event Kripke structures, and show its equivalence to Winskel’s definition. Note: This work was done jointly with Vaughan Pratt, Stanford University. ∗This work was supported by ONR under grant number N00014-92-J-1974, a gift from Mitsubishi and an Accel Partners Fellowship.