Journal of Arti cial In telligence Researc h 6 (1997) 87-110 Submitted 7/96; published 3/97 A Uniform F ramew ork for Concept De nitions in Description Logics Giusepp e De Giacomo degia como@dis.unir oma1.it Universit a di R oma \L a Sapienza" Via Salaria 113, 00198 R oma, Italy Maurizio Lenzerini lenzerini@dis.unir oma1.it Universit a di R oma \L a Sapienza" Via Salaria 113, 00198 R oma, Italy Abstract Most mo dern formalisms used in Databases and Arti cial In telligence for describing an application domain are based on the notions of class (or concept) and relationship among classes. One in teresting feature of suc h formalisms is the p ossibilit y of de ning a class, i.e., pro viding a set of prop erties that precisely c haracterize the instances of the class. Man y recen t articles p oin t out that there are sev eral w a ys of assigning a meaning to a class de nition con taining some sort of recursion. In this pap er, w e argue that, instead of c ho osing a single st yle of seman tics, w e ac hiev e b etter results b y adopting a formalism that allo ws for di eren t seman tics to co exist. W e demonstrate the feasibilit y of our argumen t, b y presen ting a kno wledge represen tation formalism, the description logic  ALCQ , with the ab o v e c haracteristics. In addition to the constructs for conjunction, disjunction, negation, quan ti ers, and quali ed n um b er restrictions,  ALCQ includes sp ecial xp oin t constructs to express (suitably in terpreted) recursiv e de nitions. These constructs enable the usual frame-based descriptions to b e com bined with de nitions of recursiv e data structures suc h as directed acyclic graphs, lists, streams, etc. W e establish sev eral prop erties of  ALCQ , including the decidabilit y and the computational complexit y of reasoning, b y form ulating a corresp ondence with a particular mo dal logic of programs called the mo dal m u-calculus