Bounded Game-Theoretic Semantics for Modal Mu-Calculus and Some Variants

Abstract

We introduce a new game-theoretic semantics (GTS) for the modal mu-calculus. Our so-called bounded GTS replaces parity games with alternative evaluation games where only finite paths arise; infinite paths are not needed even when the considered transition system is infinite. The novel games offer alternative approaches to various constructions in the framework of the mu-calculus. For example, they have already been successfully used as a basis for an approach leading to a natural formula size game for the logic. While our main focus is introducing the new GTS, we also consider some applications to demonstrate its uses. For example, we consider a natural model transformation procedure that reduces model checking games to checking a single, fixed formula in the constructed models, and we also use the GTS to identify new alternative variants of the mu-calculus with PTime model checking.

The power of clockings

“…That is, label symbols C must be written in the form C(t) where t is a kExp-term. 5 We denote this logic by T [kExp]. Note that in this logic, the use of constructs Ix is trivially limited to the kExp case because in fact all looping is Let k ≥ 1.…”

“…The paper [3] has similar motivations, but investigates the more involved case of ATL + . The article [5] concentrates on the µ-calculus. The clockings are part of the semantics and help control the semantic game.…”

“…There exist several works that make use of clockings in the literature. Much of the work in [2] and [3] and [5], including the background motivations for those studies, relates to investigating the properities of T with clockings. The paper [2] provides a game-theoretic semantics for ATL with clock values declared on the fly.…”