Decidability and undecidability results on the modal μ-calculus with a natural number-valued semantics

Abstract

In our previous study, we defined a semantics of modal μ-calculus based on min-plus algebra N∞ and developed a model-checking algorithm. N∞ is the set of all natural numbers and infinity (∞), and has two operations min and plus. In our semantics, disjunctions are interpreted by min and conjunctions by plus. This semantics allows interesting properties of a Kripke structure to be expressed using simple formulae. In this study, we investigate the satisfiability problem in the N ∞ semantics and show decidability and undecidability results: the problem is decidable if the logic does not contain the implication operator, while it becomes undecidable if we allow the implication operator. © 2010 Springer-Verlag.