Knowledge Representation in Description Logic with Rules

Description logic SHOIQ is widely used for defining ontologies. Recently, rules are added to DL to increase its expressive power. We present a reasoner for DL with rules that translates the DL reasoning task into first order logic, performs reasoning in FOL, and interprets the result in terms of DL with rules. We compare the performance of our reasoner with tableax-based reasoners for DL (which do not support rules). We discuss the undecidability of DL with rules. One of practical disadvantages of first order logic (FOL) is its undecidability. Various description logics (DL) were presented as decidable fragments of FOL. The Web Ontology Language (OWL) is widely used for describing ontologies. OWL provides constructs equivalent to presented SHOIQ DL. Due to limitations of OWL, World Wide Web Consortium proposed Semantic Web Rule Language (SWRL) – a Horn clause extension to OWL that overcomes many of the limitations. In this paper we present a reasoner for OWL and SWRL that translates a given ontology into FOL, performs reasoning using a FOL reasoner and interprets the result in terms of the given ontology. Description Logics In this section we present syntax and semantics of DL. We also define the related reasoning problems. Presented logic is based on well known DL ALC [5]. This logic is also referred as S . Adding more expressive properties of roles and constructors for concepts we finally obtain SHOIQ [4] DL. Table 1 lists the syntax and semantics of role expressions. Definition 1 (Roles) Let R be a set of role names which includes functional role names Rf ⊆ R, inverse functional role names Rif ⊆ R, symmetric role names Rs ⊆ R and transitive role names R+ ⊆ R. The set of SI roles is R ∪ {R | R ∈ R}. To avoid considering roles such as R we define a function inv on roles such that inv(R) = R if R is role name, and inv(R) = S if R = S. SHI is obtained from SI by adding a set of role inclusion axioms R = {Ri ⊑ Si | 1 ≤ i ≤ n}, where Ri and Si are (possibly inverse) role names (RBox). For a set of role inclusion axioms R we define role hierarchy R = (R∪ {inv(R) ⊑ inv(S) | (R ⊑ S) ∈ R},⊑) where ⊑ is transitive reflexive closure of ⊑ over R∪ {inv(R) ⊑ inv(S) | (R ⊑ S) ∈ R}. SHIQ is obtained from SHI by adding qualifying number restrictions, i.e. concepts of the form (≤ nR.C) and (≥ nR.C), where n is non-negative integer, R is simple (possibly inverse) role and C is concept. A role is called simple iff it is neither transitive nor has transitive subroles. SHOIQ is obtained from SHIQ by allowing also nominals in concepts. Table 1. Syntax and semantics of role expressions in DLs. Construct Name Syntax Semantics (in FOL) Atomic role R R ⊆ ∆ ×∆ S Functional role R ∈ Rf 〈x, y〉 ∈ R I ∧ 〈x, z〉 ∈ R ⇒ y = z Inverse functional role R ∈ Rif 〈y, x〉 ∈ R I ∧ 〈z, x〉 ∈ R ⇒ y = z Symmetric role R ∈ Rs 〈x, y〉 ∈ R I ⇒ 〈y, x〉 ∈ R Transitive role R ∈ R+ R I ⊆ (R) Inverse role R {〈x, y〉 | 〈y, x〉 ∈ R} I Role hierarchy R ⊑ S R ⊆ S H WDS'06 Proceedings of Contributed Papers, Part I, 172–178, 2006. ISBN 80-86732-84-3 © MATFYZPRESS