The Complexity of the Graded -Calculus Orna Kupferman , Ulrike Sattler , MosheY. Vardi School of Computer Science and Engineering, Hebrew University, Jerusalem, Israel orna@cs.huji.ac.il LuFG Theor. Informatik, RWTH Aachen, Germany sattler@cs.rwth-aachen.de Department of Computer Science, Rice University, Houston, TX 77251-1892, U.S.A. vardi@cs.rice.edu 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 for- mula. In many logics, these quantifiers have been generalized to express that, for a non-negative integer , at least individuals or all but individuals satisfy a for- mula. In modal logics, graded modalities generalize standard existential and univer- sal modalities in that they express, e.g., that there exist at least accessible worlds satisfying a certain formula. Graded modalities are useful expressive means in knowl- edge representation; they are present in a variety of other knowledge representation formalisms closely related to modal logic. A natural question that arises is how the generalization of the existential and uni- versal modalities affects the decidability problem for the logic and its computational complexity, especially when the numbers in the graded modalities are coded in bi- nary. In this paper we study the graded -calculus, which extens graded modal logic with fixed-point operators, or, equivalently, extends classical -calculus with graded modalities. We prove that the decidability problem for graded -calculus is EXPTIME-complete – not harder than the decidability problem for -calculus, even when the numbers in the graded modalities are coded in binary. 1 Introduction 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 , at least individuals or all but individuals satisfy a formula. For example, predicate logic has been extended with so-called counting quantifiers and [GOR97,GMV99,PST00]. In modal logics, graded modalities [Fin72,vdHD95,Tob01] generalize standard existential and universal modalities in that they express, e.g., that there exist at least accessible worlds satisfying a certain formula. In description logics, number restrictions have always played a central role; e.g., they are present in almost all knowledge-representation systems based on description logic [PSMB 91,BFH 94,Hor98,HM01]. Indeed, in a typical such system, one can describe cars as those vehicles having at least four wheels, and bicyles as those vehicles having exactly two wheels. A natural question that arises is how the generalization of the existential and universal quantifiers affects the decidability problem for the logic and its computational complexity. The complexity of a variety of description logics with different forms of number restrictions has been investigated; see, e.g. [DLNdN91,HB91,DL94b,BBH96,BS99,Tob00]. It turned