Many-Sorted Coalgebraic Modal Logic: a Model-theoretic Study

Abstract

This paper gives a semantical underpinning for a many- sorted modal logic associated with certain dynamical systems, like tran- sition systems, automata or classes in object-oriented languages. These systems will be described as coalgebras of so-called polynomial func- tors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Many- sorted Boolean Algebras with Operators, combining standard (categor- ical) models of modal logic and of many-sorted predicate logic. In this setting we will see Lindenbaum MBAO models as initial objects, and canonical coalgebraic models of maximally consistent sets of formulas as nal objects. They will be used to (re)prove completeness results, and Hennessey{Milner style characterisation results for the modal logic, rst established by Roiger. Mathematics Subject Classication. 03G05, 03G30, 06E25.