Complete Abstractions and Subclassical Modal Logics

Abstract

Forwards-completeness is a concept in abstract interpretation expressing that an abstract and a concrete transformer have the same semantics with respect to an abstraction. When the set of transformers is generated by the signature of a logic, a forwards-complete abstraction of a structure is one that satisfies the same formulae in a given logic. We highlight a connection between models of positive modal logic, which are logics that lack negation and implication, and forwards-completeness. These models, which were discovered independently by researchers in modal logic, model checking, and static analysis of logic programs, correspond to Kripke structures with an order on their states. We show that forwards-completeness provides a new way to synthesize both models for positive modal logics and a notion of simulation for these models. The Kripke structures that can be synthesized using forwards-completeness satisfy a saturation condition which ensures that transition relations behave like best abstract transformers.