When deployed in practical applications, Ontologies and KBs often suffer various kinds of inconsistency, which limit the applications performances significantly. In this paper, we propose a framework to reason inconsistency between Ontology and KB and refine the inconsistency accordingly. To make our framework efficient, we only focus on reasoning a part responsible for the inconsistency, rather than the whole structures of Ontology and KB. Moreover, to improve the execution speed of algorithms employed in the framework, we also discuss an axiom-oriented strategy to reason on a reduced space of formula to be inferred in Ontology and KB.
The Semantic Web [
In this paper, we propose a framework to handle inconsistency between Ontology and KB. It is done by reasoning to find the part responsible for the inconsistency and then refining the detected inconsistencies accordingly. In addition, to reduce the complexity cost of algorithms employed in the framework, we also develop an axiomoriented strategy to isolate and detect the axioms responsible for the inconsistency. The rest of the paper is organized as follows. Section 2 presents formal definitions of Ontology and Knowledge Base. Section 3 discusses inconsistency between Ontology and KB. In Section 4, the general framework for inconsistency detecting and repairing is given. Section 5 gives discussion of the axiom-oriented strategy to deal with inconsistency. Finally, Section 6 concludes the paper.
Definition 1 (Ontology). An ontology is a structure O = (C; T; R; A; ≤C; ≤T; δR; δA; τT; SA). It consists of disjoint sets of concepts (or classes) C, types T, relations R, attributes A, and values V. The partial orders ≤C (on C) and ≤T (on T) define a concept hierarchy and a type hierarchy, respectively. The function δR: R → C2 provides relation signatures (i.e., for each Example 1. where C = {football-player, person, club, city } ≤C = {football-player ⊆ person} T = {integer} R = {live-in, locate-in, play-for, has-wife} A = {age, height, weight} δR = {live-in → football-player x city,livein → person x city,locate-in → club x city, play-for → football-player x club, has-wife → football-player x football-player} δA = {age→football-playerx integer,height → football-player x integer, weight → football-player x integer} SA = {(O1) football-player(x) Λ club(y) Λ city (z) Λ play-for(x, y) Λ locate-in(y, z) → relation, the function specifies which concepts may be linked by this relation); while the function δA: A → C x T provides attribute signatures (for each attribute, the function specifies to which concept the attribute belongs and what is its data type); and τT: T2V is the assignment of values to types. SA is a set of axioms, restrictions between concepts and attributes.
We define Football Ontology O = (C; T; R; A; ≤C; ≤T; δR; δA; τT; SA) live-in(x, z) // football player plays for club will live in the cty that the club locates. (O2) football-player(x) Λ city(y) Λ city (z) Λ live-in(x, y) Λ live-in(x, z) → y = z // football player is not living in more than one city. (O3) football-player(x) Λ has-wife(x, y)Λcity(z) Λ live-in(y, z) → livein(x,z) // football player who has wife will lives in the city will live in the same city as her wife’s. (O4) club(x) Λ locate-in(x, z) Λ club(y) Λ locate-in(y, z) → x = y // each city has not more than one club.} Definition 2 (Knowledge Base). A Knowledge Base (KB) is a structure K = (C; R; A; I; V; τC; τR; τA). It consists of disjoint sets of concepts (or classes) C, relations R, attributes A, individuals I and values V. The function τC: C2I is the assignment of instances to concepts), the function τR: R → 2I x I defines relations between instances, and τA: A → 2I x V defines attributes of instances.
Example 2. We define Football KB as K = (C; R; A; I; V; τC; τR; τA) where: I = {Beckham, MU, Manchester, Liverpool, Chelsea, Maria) τC = {(K5) football-player (Beckham), (K6) club (MU), (K7) city (Manchester),
(K8) city (Liverpool), (K9) club (Chelsea)} τR = {(K10) live-in (Beckham, Liverpool), (K11) play-for (Beckham, MU), (K12) locate-in (MU, Manchester), (K13) has-wife (Beckham, Maria), (K14) live-in (Maria, Manchester), (K15) locate-in (Chelsea, Manchester)} τA = { (K16) age (Beckham, 30), (K17) height (Beckham, 180),(K18) weight (Beckham,80)}
Although KB (containing concrete data) is always encoded with respect to an ontology (containing a general conceptual model of some domain knowledge), people may find it difficult to understand the logical meaning of the underlying ontology. Hence, people may fail to formulate precisely axioms, which are logically correct, or may specify contradictory statements.
Example 3. Between in Football Ontology and Football KB defined respectively in Example 1 and Example 2, from (K5), (K10), (K13), and (K14), we can infer that Beckham lives in Liverpool but has wife living in Manchester. However, from (O3) we can see that Beckham must live in the same city with his wife. Thus, Football Ontology and Football KB are inconsistent. 4 Framework for Diagnosing and Repairing Inconsistency
In this section, we present a framework to reason inconsistency between Ontology
and KB. The framework is conducted by incorporating the algorithm for debugging
inconsistency proposed in [
τA = {(K16) age (Beckham, 30), (K17) height (Beckham, 180), (K18) weight (Beckham, 80)}
In [
Rule 1 (Axiom-Related Selection). Only add to the final_set mentioned in Algorithm 1 formulae that are not only directly relevant to this set but also directly relevant to at least an axiom in Ontology.
Rule 2 (Onto-KB Selection). Only consider the subset S1 and subset T1 mentioned in Algorithm 1 if the formulae in them occur in both Ontology and KB.
Algorithm 1. Finding MUPS of an unsatisfied concept c.
In this paper, we first introduced inconsistency occurring between Ontology and KB. Then, we proposed some refinements and improvements for an effective framework to solve the inconsistency between Ontology and KB in the reasonable complexity and time. Generally, our proposed framework only focuses on axioms, rather than the whole structure of ontology. Hence, our approach is highly potential in terms of reducing computational cost, as compared to similar existing work.