🤖 AI Summary
This work establishes a duality between relations among computational systems—such as bisimulation—and relations among logical predicates, thereby enabling cross-system logical reasoning. By extending Tarski duality and Thomason duality to the relational level for the first time, and integrating tools from category theory, Kripke semantics, and infinitary modal logic, the authors construct a dual framework that systematically links system relations with predicate relations. Building on this foundation, they develop a novel proof system capable of formally relating formulas across distinct systems. The resulting framework provides a robust theoretical basis for program logics and verification of concurrent systems, while significantly broadening the scope of classical duality theory within relational semantics.
📝 Abstract
The Kripke semantics of various logics arises via categorical dualities between a category of relational frames and their maps, and a category of algebras and logical homomorphisms. When the relational frames are considered as computational systems (e.g. the states of a machine), the corresponding algebra is one of logical predicates on these systems (e.g. predicates on these states, i.e. program logics). Our aim is to extend this phenomenon to relations, putting well-behaved relations between systems (e.g. bisimulations) in correspondence with relations between predicates. This is achieved by constructing particular relational extensions of Tarski duality (for infinitary classical propositional logic) and Thomason duality (for infinitary classical modal logic). We sketch how these dualities give rise to a proof system that relates formulae between different systems.