The complexity of the graded μ-calculus

Abstract

In classical logic, existential and universal quantifiers express that there exists at least one individual satisfying a formula, or that all individuals satisfy a formula. In many logics, these quantifiers have been generalized to express that, for a non-negative integer n, at least n individuals or all but n individuals satisfy a formula. In modal logics, graded modalities generalize standard existential and universal modalities in that they express, e.g., that there exist at least n accessible worlds satisfying a certain formula. Graded modalities are useful expressive means in knowledge representation; they are present in a variety of other knowledge representation formalisms closely related to modal logic. Anatural question that arises is howthe generalization of the existential and universal modalities affects the satisfiability problem for the logic and its computational complexity, especially when the numbers in the graded modalities are coded in binary. In this paper we study the graded μ-calculus, which extends graded modal logic with fixed-point operators, or, equivalently, extends classical μ-calculus with graded modalities.We prove that the satisfiability problem for graded μ-calculus is EXPTIME-complete -not harder than the satisfiability problem for μ-calculus, even when the numbers in the graded modalities are coded in binary.