A modal proof theory for final polynomial coalgebras

AbstractAn infinitary proof theory is developed for modal logics whose models are coalgebras of polynomial functors on the category of sets. The canonical model method from modal logic is adapted to construct a final coalgebra for any polynomial functor. The states of this final coalgebra are certain “maximal” sets of formulas that have natural syntactic closure properties.The syntax of these logics extends that of previously developed modal languages for polynomial coalgebras by adding formulas that express the “termination” of certain functions induced by transition paths. A completeness theorem is proven for the logic of functors which have the Lindenbaum property that every consistent set of formulas has a maximal extension. This property is shown to hold if the deducibility relation is generated by countably many inference rules.A counter-example to completeness is also given. This is a polynomial functor that is not Lindenbaum: it has an uncountable set of formulas that is deductively consistent but has no maximal extension and is unsatisfiable, even though all of its countable subsets are satisfiable

Coalgebras and Their Logics

Transition systems pervade much of computer science. This article outlines the beginnings of a general theory of specification languages for transition systems. More specifically, transition systems are generalised to coalgebras. Specification languages together with their proof systems, in the following called (logical or modal) calculi, are presented by the associated classes of algebras (e.g., classical propositional logic by Boolean algebras). Stone duality will be used to relate the logics and their coalgebraic semantics