The Even More Irresistible SROIQ Ian Horrocks and Oliver Kutz and Ulrike Sattler School of Computer Science, The University of Manchester Kilburn Building, Oxford Road, Manchester M13 9PL, UK {Horrocks, Kutz, Sattler}@cs.man.ac.uk Abstract (Horrocks & Sattler, 2005). That is, we extend SHOIQ— which is SHOIN with qualified number restrictions—and We describe an extension of the description logic under- extend the work begun in Horrocks, Kutz, & Sattler (2005). lying OWL-DL, SHOIN , with a number of expressive means that we believe will make it more useful in prac- Since OWL-DL is becoming more widely used, it turns tice. Roughly speaking, we extend SHOIN with all out that it lacks a number of expressive means which— expressive means that were suggested to us by ontology when considered carefully—can be added without causing developers as useful additions to OWL-DL, and which, too much difficulties for automated reasoning. We extend additionally, do not affect its decidability and practica- SHOIQ with these expressive means and, although they bility. We consider complex role inclusion axioms of are not completely independent in that some of them can the form R ◦ S v˙ R or S ◦ R v˙ R to express prop- be expressed using others, first present them together with agation of one property along another one, which have some examples. Recall that, in SHOIQ, we can already proven useful in medical terminologies. Furthermore, state that a role is transitive or the subrole or the inverse of we extend SHOIN with reflexive, antisymmetric, and another one (and therefore also that it is symmetric). irreflexive roles, disjoint roles, a universal role, and con- structs ∃R.Self, allowing, for instance, the definition In addition, SROIQ allows for the following: of concepts such as a “narcist”. Finally, we consider 1. disjoint roles. Most DLs can be said to be “unbalanced” negated role assertions in Aboxes and qualified number since they allow to express disjointness on concepts but restrictions. The resulting logic is called SROIQ. not on roles, despite the fact that role disjointness is quite We present a rather elegant tableau-based reasoning al- gorithm: it combines the use of automata to keep track natural and can generate new subsumptions or inconsis- of universal value restrictions with the techniques de- tencies in the presence of role hierarchies and number re- veloped for SHOIQ. The logic SROIQ has been strictions. E.g., the roles sister and mother should be adopted as the logical basis for the next iteration of declared as being disjoint. OWL, OWL 1.1. 2. reflexive, irreflexive, and antisymmetric roles. These fea- tures are of minor interest when considering only TBoxes Introduction not using nominals, yet they add some useful constraints if we also refer to individuals, either by using nominals or SROIQ We describe an extension, called , of the descrip- ABoxes, especially in the presence of number restrictions. tion logics (DLs) SHOIN (Horrocks, Sattler, & Tobies, E.g., the roles knows, hasSibling, and properPartOf, 1999) underlying OWL-DL (Horrocks, Patel-Schneider, & should be declared as, respectively, reflexive, irreflexive, 1 RIQ van Harmelen, 2003) and (Horrocks & Sattler, 2004). and antisymmetric. SHOIN provides most expressive means that one could reasonably expect from the description-logical basis of an 3. negated role assertions. Most Abox formalisms only al- ontology language, and was designed to constitute a good low for positive role assertions (with few exceptions (Are- compromise between expressive power and computational ces et al., 2003; Baader et al., 2005)), whereas SROIQ complexity/practicability of reasoning. It lacks, however, also allows for statements like (John, Mary): ¬likes. e.g. qualified number restrictions which are present in the In the presence of complex role inclusions, negated role DL considered here since they are required in various appli- assertions can be quite useful and, like disjoint roles, they cations (Wolstencroft et al., 2005) and do not pose problems overcome a certain asymmetry in expressivity. 4. SROIQ provides complex role inclusion axioms of the Copyright c 2006, American Association for Artificial Intelli- form R ◦ S v˙ R and S ◦ R v˙ R that were first intro- gence (www.aaai.org). All rights reserved. 1 duced in RIQ. For example, w.r.t. the axiom owns ◦ OWL also includes datatypes, a simple form of concrete do- ˙ main (Baader & Hanschke, 1991). These can, however, be treated hasPart v owns, and the fact that each car contains an ˙ exactly as in SHOQ(D)/SHOQ(Dn) (Horrocks & Sattler, 2001; engine Car v ∃hasPart.Engine, an owner of a car is Pan & Horrocks, 2003), so we will not complicate our presentation also an owner of an engine, i.e., the following subsump- by considering them here. tion holds: ∃owns.Car v ∃owns.Engine. 5. SROIQ provides the universal role U. Together with To avoid considering roles such as R−−, we define a func- nominals (which are also provided by SHOIQ), this tion Inv on roles such that Inv(R) = R− if R ∈ R is a role role is a prominent feature of hybrid logics (Blackburn & name, and Inv(R) = S ∈ R if R = S−. Seligman, 1995). Nominals can be viewed as a powerful Since we will often work with finite strings of roles it is generalisation of ABox individuals (Schaerf, 1994; Hor- convenient to extend both ·I and Inv(·) to such strings: if rocks & Sattler, 2001), and they occur naturally in ontolo- w = R1 ...Rn is a string of roles Ri (1 ≤ i ≤ n), we set I I I gies, e.g., when describing a class such as EUCountries Inv(w) = Inv(Rn) ... Inv(R1) and w = R1 ◦ ... ◦ Rn, by enumerating its members. where ◦ denotes composition of binary relations. 6. Finally, SROIQ allows for concepts of the form ∃R.Self A role box R consists of two components. The first com- which can be used to express “local reflexivity” of a role ponent is a role hierarchy Rh which consists of (gener- R, e.g., to define the concept “narcist” as ∃likes.Self. alised) role inclusion axioms. The second component is a set Ra of role assertions stating, for instance, that a role R Besides a Tbox and an Abox, SROIQ provides a so-called must be interpreted as an irreflexive relation. Rbox to gather all statements concerning roles. We start with the definition of a (regular) role hierarchy SROIQ is designed to be of similar practicability as whose definition involves a certain ordering on roles, called SHOIQ. The tableau algorithm for SROIQ presented regular. A strict partial order ≺ on a set A is an irreflexive here is essentially a combination of the algorithms for RIQ and transitive relation on A. A strict partial order ≺ on the and SHOIQ. In particular, it employs the same technique set of roles R ∪ {R− | R ∈ R} is called a regular order using finite automata as in Horrocks & Sattler (2004) to han- if ≺ satisfies, additionally, S ≺ R ⇐⇒ S− ≺ R, for all dle role inclusions R ◦ S v˙ R and S ◦ R v˙ R. Even roles R and S. Note, in particular, that the irreflexivity of ≺ though the additional expressive means require certain ad- ensures that neither S− ≺ S nor S ≺ S− hold. justments, these adjustments do not add new sources of non- determinism and, subject to empirical verification, are be- Definition 2 ((Regular) Role Inclusion Axioms) Let ≺ be lieved to be “harmless” in the sense of not significantly de- a regular order on roles. A role inclusion axiom (RIA for grading typical performance as compared with the SHOIQ short) is an expression of the form w v˙ R, where w is a algorithm. Moreover, the algorithm for SROIQ has, simi- finite string of roles not including the universal role U, and lar to the one for SHOIQ, excellent “pay as you go” char- R 6= U is a role name. A role hierarchy Rh is a finite set acteristics. For instance, in case only expressive means of of RIAs. An interpretation I satisfies a role inclusion axiom SHIQ are used, the new algorithm will behave just like the w v˙ R, written I |= w v˙ R, if wI ⊆ RI . An interpretation algorithm for SHIQ. is a model of a role hierarchy Rh if it satisfies all RIAs in We believe that the combination of properties described Rh, written I |= Rh. above makes SROIQ a very useful basis for future exten- ˙ sions of OWL DL. A RIA w v R is ≺-regular if R is a role name, and 1. w = RR, or The Logic SROIQ 2. w = R−, or In this section, we introduce the DL SROIQ. This includes 3. w = S1 ...Sn and Si ≺ R, for all 1 ≤ i ≤ n, or the definition of syntax, semantics, and inference problems. 4. w = RS1 ...Sn and Si ≺ R, for all 1 ≤ i ≤ n, or 5. w = S ...S R and S ≺ R, for all 1 ≤ i ≤ n. Roles, Role Hierarchies, and Role Assertions 1 n i Finally, a role hierarchy Rh is regular if there exists a reg- Definition 1 Let C be a set of concept names including a ular order ≺ such that each RIA in Rh is ≺-regular. subset N of nominals, R a set of role names including the universal role U, and I = {a, b, c . .} a set of individual Regularity prevents a role hierarchy from containing names. The set of roles is R ∪ {R− | R ∈ R}, where a role cyclic dependencies. For instance, the role hierarchy R− is called the inverse role of R. ˙ ˙ ˙ ˙ As usual, an interpretation I = (∆I , ·I ) consists of a {RS v S, RT v R, V T v T,VS v V } I I set ∆ , called the domain of I, and a valuation · which is not regular because it would require ≺ to satisfy S ≺ V ≺ associates, with each role name R, a binary relation RI ⊆ I I T ≺ R ≺ S, which would imply S ≺ S, thus contradicting ∆ × ∆ , with the universal role U the universal relation the irreflexivity of ≺.