Modal μ-calculus on Min-plus Algebra N_∞

Abstract

We have developed an interpretation of modal μ-calculus using min-plus algebra N_∞, the set of all natural numbers and infinity ∞. Disjunctions are interpreted by min, and conjunctions by plus. This interpretation allows complex properties, such as the shortest path on a Kripke structure or the number of states that satisfy a specified condition, to be expressed with simple formulas. We defined the semantics of modal μ-calculus on min-plus algebra, and then described a model-checking algorithm for the semantics and its implementation. Although simple iterative computation of the least fixed-point generally does not terminate in N_∞, due to abstraction, we made model-checking possible by reducing the least fixed-point computation to the greatest fixed-point computation. Finally, we discuss the relationship between our semantics and the theory of Kripke structures on complete Heyting algebra.