Terminological Cycles in a Description Logic with Existential Restrictions* Franz Baader Theoretical Computer Science TU Dresden D-01062 Dresden, Germany [email protected] Abstract of technology rather than need. In fact, the Galen medical knowledge base contains many cyclic dependencies [Rector Cyclic definitions in description logics have un• and Horrocks, 1997]. Also, even in the case of acyclic ter• til now been investigated only for description log• minologies, our polynomial subsumption algorithm improves ics allowing for value restrictions. Even for the on the usual approach that first unfolds the TBox (a poten• most basic language which allows for con• tially exponential step) and then applies the polynomial sub• junction and value restrictions only, deciding sub- sumption algorithm for -concept descriptions [Baader et sumption in the presence of terminological cycles al., 1999]. is a PSPACE-complete problem. This paper inves• The first thorough investigation of cyclic terminologies in tigates subsumption in the presence of terminolog• description logics (DL) is due to Nebel [1991], who intro• ical cycles for the language . , which allows for duced three different semantics for such terminologies: least conjunction, existential restrictions, and the top- fixpoint (lfp) semantics, which considers only the models that concept. In contrast to the results for , sub- interpret the defined concepts as small as possible; greatest sumption in remains polynomial, independent fixpoint (gfp) semantics, which considers only the models of whether we use least fixpoint semantics, greatest that interpret the defined concepts as large as possible; and fixpoint semantics, or descriptive semantics. descriptive semantics, which considers all models. In [Baader, 1990; 1996], subsumption w.r.t. cyclic termi• 1 Introduction nologies in the small DL which allows for conjunction and value restrictions only, was characterized with the help of Early description logic (DL) systems allowed the use of value finite automata. This characterization provided PSPACE de• restrictions but not of existential restrictions cision procedures for subsumption in with cyclic termi• Thus, one could express that all children are male using the nologies for the three types of semantics introduced by Nebel. value restriction Male, but not that someone has a son In addition, it was shown that subsumption is PSPACE-hard. using the existential restriction The main rea• The results for cyclic -terminologies were extended by son was that, when clarifying the logical status of property Kiisters [ 1998] to ALN, which extends by atomic nega• arcs in semantic networks and slots in frames, the decision tion and number restrictions. was taken that arcs/slots should be read as value restrictions The fact that the DL ACC (which extends FL by full (see, e.g., [Nebel, 1990]). Once one considers more expres• 0 negation) is a syntactic variant of the multi-modal logic K sive DLs allowing for full negation, existential restrictions opens a way for treating cyclic terminologies and more gen• come in as the dual of value restrictions [Schmidt-SchauB eral recursive definitions in more expressive languages like and Smolka, 1991]. Thus, for historical reasons, DLs that ACC and extensions thereof by a reduction to the modal mu- allow for existential, but not for value restrictions, are largely calculus [Schild, 1994; De Giacomo and Lenzerini, 1994]. In unexplored. In the present paper, we investigate termino• this setting, one can use a mix of the three types of semantics logical cycles in the DL which allows for conjunction, introduced by Nebel. However, the complexity of the sub• existential restrictions, and the top-concept. In contrast to sumption problem is EXPTIME-complete. (even very inexpressive) DLs with value restrictions, sub- sumption in remains polynomial in the presence of ter• In spite of these very general results for cyclic definitions minological cycles. It should be noted that there are in• in expressive languages, there are still good reasons to look deed applications where the small appears to be suffi• at cyclic terminologies in less expressive (in particular sub- cient. In fact, SNOMED, the Systematized Nomenclature of Boolean) description logics. One reason is, of course, that Medicine [Cote et a/., 1993] employs [Spackman, 2001; one can hope for a lower complexity of the subsumption prob• Spackman et al., 1997]. Even though SNOMED does not lem. For DLs with value restrictions, this hope is not fulfilled, appear to use cyclic definitions, this may be due to a lack though. Even in the inexpressive DL subsumption be• comes PSPACE-complete if one allows for cyclic definitions. * Partially supported by the DFG under grant BA 1122/4-3. This is still better than the EXPTIME-completeness that one DESCRIPTION LOGICS 325 For some applications, it is more appropriate to interpret cyclic concept definitions with the help of an appropriate fix- point semantics. Before defining least and greatest fixpoint semantics formally, let us illustrate their effect on an exam• ple. Example 1 Assume that our interpretations are graphs where we have nodes (elements of the concept name Node) and edges (represented by the role edge), and we want to define the concept I node of all nodes lying on an infinite (possibly has in ACC with cyclic definitions, but from the practical cyclic) path of the graph. The following is a possible defini• point of view it still means that the subsumption algorithm tion of Inode: may need exponential time. Now consider the following interpretation of the primitive In contrast, the subsumption problem in can be decided concepts and roles: in polynomial time w.r.t. the three types of semantics intro• duced by Nebel. The main tool used to show these results is a characterization of subsumption through the existence of so-called simulation relations. In the next section we will introduce the DL as well as cyclic terminologies and the three types of semantics for Where there these terminologies. Then we will show in Section 3 how are four possible ways of extending this interpretation of the such terminologies can be translated into description graphs. primitive concepts and roles to a model of the TBox consist• In this section, we will also define the notion of a simulation ing of the above concept definition: Inode can be interpreted between nodes of a description graph, and mention some use• by M U N, M, N, or All these models are admissible ful properties of simulations. The next three sections are then w.r.t. descriptive semantics, whereas the first is the gfp-model devoted to the characterization of subsumption in w.r.t. and the last is the lfp-model of the TBox. Obviously, only gfp, Ifp, and descriptive semantics, respectively. the gfp-model captures the intuition underlying the definition (namely, nodes lying on an infinite path) correctly. It should be noted, however, that in other cases descrip• 2 Cyclic terminologies in the DL tive semantics appears to be more appropriate. For example, Concept descriptions arc inductively defined with the help of consider the definitions a set of constructors, starting with a set Nc ofconcept names and a set NR of role names. The constructors determine the expressive power of the DL. We restrict the attention to the whose concept descriptions are formed using the con• With respect to gfp-semantics, the defined concepts Tiger and structors top-concept (T), conjunction , and existen• Lion must always be interpreted as the same set whereas this tial restriction . The semantics of £L-concept descrip• is not the case for descriptive semantics.l tions is defined in terms of an interpretation Before we can define lfp- and gfp-semantics formally, we The domain is a non-empty set of individuals and the must introduce some notation. Let T be an con• interpretation function I maps each concept name A £ N c taining the roles the primitive concepts N and the to a subset of _ and each role to a binary re• prim defined concepts . A primitive inter• lation The extension of to arbitrary concept descriptions is inductively defined, as shown in the third col• pretations J for T is given by a domain , an interpreta• umn of Table 1. tion of the roles . by binary relations A terminology (or TBox for short) is a finite set of con• and an interpretation of the primitive concepts cept definitions of the form where A is a concept by subsets . Obviously, a primitive interpretation name and D a concept description. In addition, we require differs from an interpretation in that it does not interpret the that TBoxes do not contain multiple definitions, i.e., there defined concepts in We say that the interpretation X is based on the primitive interpretation J iff it has the same cannot be two distinct concept descriptions D\ and D2 such that both belongs to the TBox. Con• domain as J and coincides with J on Nrole and Nprim. For cept names occurring on the left-hand side of a definition are a fixed primitive interpretation J, the interpretations X based called defined concepts. All other concept names occurring on it are uniquely determined by the tupleof in the TBox are called primitive concepts. Note that we al• the interpretations of the defined concepts in , We define low for cyclic dependencies between the defined concepts, lnt{J) := {X I X is an interpretation based on J}. i.e., the definition of A may refer (directly or indirectly) to A itself. An interpretation X is a model of the TBox T iff Interpretations based on J can be compared by the following it satisfies all its concept definitions, i.e., for all ordering, which realizes a pairwise inclusion test between the definitions 'This example is similar to the "humans and horses" example The semantics of (possibly cyclic) we have used by Nebel [ 1991 ] to illustrate the difference between descriptive just defined is called descriptive semantic by Nebel [1991].