Decidability and Undecidability Results of Modal μ-calculi with N1 Semantics

In our previous study, we defined the semantics of modal μ-calculus on minplus 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 the semantics, disjunctions are interpreted by min and conjunctions by plus. This semantics allows interesting properties of a Kripke structure, such as the shortest path to some state or the number of states that satisfy a specified condition, to be expressed using simple formulae. In this study, we investigate the satisfiability problem in N∞ semantics and prove decidability and undecidability results. First, the problem is decidable if the logic does not contain the implication operator. We prove this result by defining a translation tr(φ) of formula φ such that the satisfiability of φ in N∞ semantics is equivalent to that of tr(φ) in ordinary semantics. Second, the satisfiability problem becomes undecidable if the logic contains the implication operator.