The Complexity of the Graded µ-Calculus.

Orna Kupferman, Ulrike Sattler, Moshe Y. Vardi*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

69 Scopus citations


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.

Original languageEnglish
Title of host publication Automated Deduction - CADE-18
Subtitle of host publication18th International Conference on Automated Deduction, Copenhagen, Denmark, July 27-30, 2002 Proceedings
EditorsAndrei Voronkov
PublisherSpringer Verlag
Number of pages15
ISBN (Print)3-540-43931-5
StatePublished - 2002
Event18th International Conference on Automated Deduction, CADE 2002 - Copenhagen, Denmark
Duration: 27 Jul 200230 Jul 2002

Publication series

NameLecture Notes in Computer Science
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference18th International Conference on Automated Deduction, CADE 2002

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2002.


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