SYNTACTIC RELEVANCE-BASED SELECTION FUNCTIONS

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5.7 SYNTACTIC RELEVANCE-BASED SELECTION FUNCTIONS

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As we have pointed out in Section 5.4, the de nition of the selection function should be independent of the general procedure of the inconsistency processing (i.e., strategy). Further research will focus on a formal development of selection functions. However, we would like to point out that there exist several alternatives which can be used for an inconsistency reasoner. Chopra et al. (2000) propose syntactic relevance to measure the relationship between two formulas in belief sets, so that the relevance can be used to guide the belief revision based on Schaerf and Cadoli s method of approximate reasoning. We will exploit their relevance measure as selection function and illustrate them on two examples. De nition 9 (Direct Relevance and k-Relevance (Chopra et al. 2000)). Given a formula set , two atoms p, q are directly relevant, denoted by R(p, q, ) if there is a formula a 2 such that p, q appear in a. A pair of atoms p and q are k-relevant with respect to if there exist p1,p2, , pk 2 L such that: p, p1 are directly relevant; pi, pi+1 are directly relevant, i 1, , k 1; pk, q are directly relevant. The notions of relevance are based on propositional logics. However, ontology languages are usually written in some subset of rst order logic. It would not be too dif cult to extend the ideas of relevance to those rstorder logic-based languages by considering an atomic formula in rstorder logic as a primitive proposition in propositional logic. Given a formula f, we use I(f), C(f), R(f) to denote the sets of individual names, concept names, and relation names that appear in the formula f, respectively. De nition 10 (Direct Relevance). Two formula f and c are directly relevant if there is a common name which appears both in formula f and formula c, that is I(f) \ I(c) 6 _ C (f) \ C(c) 6 _ R(f) \ R(c) 6 . De nition 11 (Direct Relevance to a Set). A formula f is relevant to a set of formula if there exists a formula c 2 such that f and c are directly relevant. We can similarly specialize the notion of k-relevance. De nition 12 (k-Relevance). Two formulas f, f are k-relevant with respect to a formula set if there exist formulas c0, ck 2 such that f and c0, c0 and c1, , and ck and f are directly relevant. De nition 13 (k-Relevance to a set). A formula f is k-relevant to a formula set if there exists formula c 2 such that f and c are k-relevant with respect to . In inconsistency reasoning we can use syntactic relevance to de ne a selection function s to extend the query j% f as follows: We start with

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REASONING WITH INCONSISTENT ONTOLOGIES

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the query formula f as a starting point for the selection based on syntactic relevance. Namely, we de ne: s( , f, 0) . Then the selection function selects the formulas c 2 which are directly relevant to f as a working set (i.e., k 1) to see whether or not they are suf cient to give an answer to the query. Namely, we de ne: s( , f, 1) {c 2 j f and c are directly relevant}. If the reasoning process can obtain an answer to the query, it stops. otherwise the selection function increases the relevance degree by 1, thereby adding more formulas that are relevant to the current working set. Namely, we have: s( , f, k) {c 2 j c is directly relevant to s( , f, k 1)}, for k > 1. This leads to a fan out behavior of the selection function: the rst selection is the set of all formulae that are directly relevant to the query; then all formulae are selected that are directly relevant to that set, etc. This intuition is formalized in the following: Proposition 3. The syntactic relevance-based selection function s is monotonically increasing. Proposition 4. If k ! 1, then s( , f, k) {fjf is (k-1)-relevant to } The syntactic relevance-based selection functions de ned above usually grows up to an inconsistent set rapidly. That may lead to too many undetermined answers. In order to improve it, we require that the selection function returns a consistent subset 00 at the step k when s( , f, k) is inconsistent such that s( , f, k 1) & 00 & s( , f, k). It is actually a kind of backtracking strategy which is used to reduce the number of undetermined answers to improve the linear extension strategy. We call the procedure an over-determined processing (ODP) of the selection function. Note that the over-determined processing does not need to exhaust the powerset of the set s( , f, k) s( , f, k 1) because of the fact that if a consistent set S cannot prove or disprove a query, then nor can any subset of S. Therefore, one approach of ODP is to return just a maximally consistent subset. Let n be j j and k be n jSj, that is the cardinality difference between the ontology and its maximal consistent subset S (note that k is usually very small), and let C be the complexity of the consistency checking. The complexity of the over-determined processing is polynomial to the complexity of the consistency checking (Huang et al., 2005).

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