Modal Knowledge and Game Semirings1

The aim of algebraic logic is to compact series of small steps of general logical inference into larger (in)equational steps. Algebraic structures that have proved very useful in this context are modal semirings and modal Kleene algebras. We show that they can also model knowledge and belief logics as well as games without additional effort; many of the standard logical properties are theorems rather than axioms in this setting. As examples of the first area, we treat the classical puzzles of the Wise Men and the Muddy Children. Moreover, we show possibilities of handling knowledge update and revision algebraically. For the area of games, we generalize the well-known connection between game logic and dynamic logic to the setting of modal semirings and link it to predicate transformer semantics, in particular to demonic refinement algebra. We think that our study provides evidence that modal semirings are capable of handling a wide variety of (multi-)modal logics in a uniform algebraic fashion.