Modeling Ontologies using OWL, Description Graphs, and Rules

structure common to all fingers as shown in Figure 1c; then, we should be able to specialize this structure for the index finger and introduce the middle phalanx, as in Figure 1e. The graph specialization Gfinger ⊳ Gthumb states that the graph for the thumb specializes the graph for a finger. Definition 3 (Graph Specialization). A graph specialization is an axiom of the form G1 ⊳ G2, where G1 = (V1, E1, λ1,M1) and G2 = (V2, E2, λ2,M2) are description graphs with V1 ⊆ V2. Next, we introduce axioms that allow us to properly connect graph instances. For example, Ghand contains the vertices 3 and 4 for the thumb and its proximal phalanx, which correspond to the vertices 1 and 3 of Gthumb . We can specify this correspondence using a graph alignment of the form Ghand [3, 4] ↔ Gthumb [1, 3]. Intuitively, this ensures that it is not possible for Ghand and Gthumb to share the thumb without sharing the proximal phalanx as well. Definition 4 (Graph Alignment). A graph alignment is an expression of the form G1[v1, . . . , vn] ↔ G2[w1, . . . wn], where G1 and G2 are description graphs with sets of vertices V1 and V2, respectively, vi ∈ V1 and wi ∈ V2 for 1 ≤ i ≤ n. Finally, we define GBoxes and graph-extended KBs. Definition 5 (Formalism). A graph box (GBox) is a tuple G = (GG,GS ,GA) where GG, GS , and GA are finite sets of description graphs, graph specializations over GG, and graph alignments over GG. ABoxes are extended to allow for graph assertions of the form G(a1, . . . , al) for G an l-ary graph. A graph-extended knowledge base is a 4-tuple K = (T ,P ,G,A) where T is a TBox, P is a program with a finite number of connected rules, G is a GBox, and A is an ABox. Next, we define the semantics of the formalism. Definition 6 (Semantics). An interpretation I = (△ , ·) is defined as usual, and it interprets each l-ary description graph G as an l-ary relation over △ ; that is, G ⊆ (△). A graph assertion is satisfied in I, written I |= G(a1, . . . , al), iff 〈a1, . . . , a I l 〉 ∈ G I . Satisfaction of a description graph, graph specialization, and graph alignment is defined in Table 1. Satisfaction of T , P and A is standard. A knowledge base K = (T ,P ,G,A) is satisfied in I, written I |= K, if all its components are satisfied in I. Thus, each l-ary graph G is interpreted as an l-ary relation G in which each tuple corresponds to an instance of G in the interpretation. The key and disjointness properties in Table 1 ensure that no two distinct instances of G can share a vertex; for example, no two distinct instances of Ghand can share the vertex for the thumb. These properties prevent, for example, the occurrence of infinite ‘chains’ of Ghand and therefore are needed to ensure that the representation of the structured objects is bounded. The start property in Table 1 ensures that each instance of a main concept A of G occurs in an instance of G. For example, since Hand is a main concept for Ghand , each instance of Hand must occur as vertex 1 in an instance of Ghand . Table 1: Satisfaction of GBox Elements in an Interpretation I |= G for G = (V,E, λ,M) iff Key property : ∀x1, . . . , xl, y1, . . . , yl ∈ △ I : 〈x1, . . . , xl〉 ∈ G I ∧ 〈y1, . . . , yl〉 ∈ G I ∧ W