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5.7 SYNTACTIC RELEVANCE-BASED SELECTION FUNCTIONS As we have pointed out in Section 5.4, the definition of the selection function should be independent of the general procedure of the incon- sistency 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 inconsis- tency 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. Definition 9 (Direct Relevance and k-Relevance (Chopra et al. 2000)). Given a formula set Æ, two atoms p, q are directly relevant, denoted by R(p, q, Æ) if there is a formula a 2 Æ such that p, q appear in a. A pair of atoms p and q are k-relevant with respect to Æ if there exist p 1 ,p 2 ,ÁÁÁ,pk2Lsuch that:  p, p 1 are directly relevant;  p i ,p i+1 are directly relevant, i ¼ 1,ÁÁÁ, k À 1;  p k , q are directly relevant. The notions of relevance are based on propositional logics. However, ontology languages are usually written in some subset of first order logic. It would not be too difficult to extend the ideas of relevance to those first- order logic-based languages by considering an atomic formula in first- order logic as a primitive proposi tion 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. Definition 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) 6¼ Ø _ C (f) \ C(c) 6¼ Ø _ R(f) \ R(c) 6¼ Ø. Definition 11 (Direct Relevance to a Set). A formula f is relevant to a set of formula Æ if there exists a formula c 2 Æ such that f and c are directly relevant. We can similarly specialize the notion of k-relevance. Definition 12 (k-Relevance). Two formulas f, f’ are k-relevant with respect to a formula set Æ if there exist formulas c 0 ,ÁÁÁc k 2 Æ such that f and c 0 , c 0 and c 1 , ÁÁÁ, and ck and f’ are directly relevant. Definition 13 (k-Relevance to a set). A formula f is k-relevant to a formula set Æ if there exists formula c 2 Æ such that f and c are k-relevant with respect to Æ. In inconsistency reasoning we can use syntactic relevance to define a selection function s to extend the query ‘Æ j% f?’ as follows: We start with SYNTACTIC RELEVANCE-BASED SELECTION FUNCTIONS 85 the query formula f as a starting point for the selection based on syntactic relevance. Namely, we define: s(Æ, f,0)¼ Ø. Then the selection function selects the formulas c 2 Æ which are directly relevant to f as a working set (i.e., k ¼ 1) to see whether or not they are sufficient to give an answer to the query . Namely, we define: s(Æ, f,1)¼ {c 2 Æ 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(Æ, f, k) ¼ {c 2 Æ j c is directly relevant to s(Æ, f, k – 1)}, for k > 1. This leads to a ‘fan out’ behavior of the selection function: the first 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 mono- tonically increasing. Proposition 4. If k ! 1, then s(Æ, f, k) ¼ {fjf is (k-1)-relevant to Æ} The syntactic relevance-based selection functions defined 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 Æ 00 at the step k when s(Æ, f, k) is inconsistent such that s (Æ , f, k À 1) & Æ 00 & s(Æ, 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(Æ, f, k) À s(Æ, f, k À 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 jÆj and k be n – jSj, that is the cardinality difference between the ontology Æ 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 proces- sing is polynomial to the complexity of the consistency checking (Huang et al., 2005). 86 REASONING WITH INCONSISTENT ONTOLOGIES Note that ODP introduces a degree of non-determinism: selecting different maximal consistent subsets of s(Æ, f, k) may yield different answers to the query Æ j% f. The simplest example of this is Æ ={f, :f}. 5.8. PROTOTYPE OF PION 5.8.1. Implementation We are implementing the prototype of PION by using SWI-Prolog. 5 PION implements an inconsistency reasoner based on a linear extension strategy and the syntactic relevance-based selection function as discussed in Sections 5.6 and 5.7. PION is powered by XDIG, an extended DIG Description Logic interface for Prolog (Huang and Visse r, 2004). PION supports the TELL requests in DIG data format and in OWL, and the ASK requests in DIG data format. A prototype of PION is available for download at the website: http://wasp.cs.vu.nl/sekt/pion. The architecture of a PION is designed as an extension of the XDIG framework, and is shown in Figure 5.3. A PION consists of the following components:  DIG Server: The standard XDIG server acts as PION’s XDIG server, which deals with requests from other ontology applications. It not only supports standard DIG requests, like ‘tell’ and ‘ask,’ but also provides additional reasoning facilities, like the identification of the reasoner or change of the selected selection functions.  Main Control Component: The mai n control component performs the main processing, like query analysis, query pre-processing, and the extension strategy, by calling the selection function and interacting with the ontology repositories. 5 http://www.swi-prolog.org Figure 5.3 Architecture of PION. PROTOTYPE OF PION 87  Selection Functions: The selection function component is an enhanced component to XDIG, it defines the selection functions that may be used in the reasoning process.  DIG Client: PION’s DIG client is the standard DIG client, which calls external description Logic reasoners that support the DIG interface to obtain the standard Description Logic reasoning capabilities.  Ontology Repositories: The ontology repositories are used to store ontology statements, provided by external ontology applications. The current version of the PION implements a reasoner based on a linear extension strategy and a k-relevance selection function as discussed in Sections 5.2 and 5.5. A screenshot of the PION testbed, is shown in Figure 5.4. 5.8.2. Experiments and Evaluation We have tested the prototype of PION by applying it on several example ontologies. These example ontologies are the bird example, the brain example, the Married-Woman example, and the MadCow Ontology, which are discussed in Section 5.3. We compare PION’s answers with their intuitive answers which is supposed by a human to see to what extend PION can provide intended answers. For a query, there might exist the following difference between an answer by PION and its intuitive answer.  Intended Answer: PION’s answer is the same as the intuitive answer.  Counter-Intuitive Answer: PION’s answer is opposite to the intuitive answer. Namely, the intuitive answer is ‘accepted’ whereas PION’s answer is ‘rejected,’ or vice versa.  Cautious Answer: The intuitive answer is ‘accepted’ or ‘rejected,’ but PION’s answer is ‘undetermined.’  Reckless Answer: PION’s answer is ‘accepted’ or ‘rejected’ whereas the intuitive answer is ‘undetermined.’ We call it a reckless answer because under this situation PION returns just one of the possible answers without seeking other possibly opposite answers, which may lead to ‘undetermined.’ For each concept C in those ontologies, we create an instance ‘the_C’on them. We make both a positive instance query and a negative instance query of the instance ‘the_C’ for some concepts D in the ontologies, like a query is ‘the_C a D?’ PION test results are shown in Figure 5.5. Of the four test examples, PION can return at least 85.7 % intended answers. Of the 396 queries, PION returns 24 cautious answers or reckless answers, and 2 counter-intuitive answers. However, we would like to point out that the high rate of the intended answers includes many ‘undetermined’ answers. One interesting (and we believe realistic) property of the Mad 88 REASONING WITH INCONSISTENT ONTOLOGIES Figure 5.4 PION testbed. Cows ontology is that many concepts which are intuitively disjoint (such as cows and sheep) are not actually declared as being disjoint (keep in mind that OWL has an open world semantics, and does not make the unique name assumption). As a result, many queries such as ‘is the_cow a sheep’ are indeed undetermined on the basis of the ontology, and PION correctly reports them as undetermined. The average time cost of the tested queries is about 5 seconds even on a low-end PC (with 550 mHz CPU, 256 MB memory under Windows 2000). The counter-intuitive results occur in the MadCows Example. PION returns the ‘accepted’ answer to the query ‘is the_mad_cow a vege- tarian?’ This counter-intuitive answer results from the weakness of the syntactic relevance-based selection function because it always pre- fers a shorter relevance path when a conflict occurs. In the mad cow example, the path ‘mad cow – cow – vegetarian’ is shorter than the path ‘mad cow –.eat brain – eat bodypart – sheep are animals – eat animal – NOT vegetarian.’ Therefore, the syntactic relevance-based selection function finds a consistent subtheory by simply ignoring the fact ‘sheep are animals.’ The problem results from the unbalanced specifica- tion between Cow and MadCow, in which Cow is directly specified as a vegetarian whereas there is no direct statement ‘a MadCow is not a vegetarian.’ There are several alternative approaches to solve this kind of problems. One is to introduce the locality requirement. Namely, the selection function starts with a certain subtheory which must always be selected. For example, the statement ‘sheep are animals’ can be considered to be a knowledge statement which cannot be ignored. Another approach is to add a shortcut path, like the path ‘mad cow – eat animal – NOT vegetarian’ to achieve the relevance balance between the concepts ‘vegetarian’ and NOT vegetarian,’ as shown in the second mad cow example of PION testbe d. The latter approach can be achieved auto- matically by accommodation of the semantic relevance from the user queries. The hypothesis is that both concepts appear in a query more frequently, when they are semantically more relevant. Therefore, from a semantic point of view, we can add a relevance shortcut path between strongly relevant concepts. Example Queries IA CA RA CIA IA Rate(%) ICR Rate(%) Bird 50 50 0 0 0 100 100 Brain 42 36 4 2 0 85.7 100 MarriedWoman 50 48 0 2 0 96 100 MadCow 254 236 16 0 2 92.9 99 IA, intended answers; CA, cautious answers; RA, reckless answers; CIA, counter-intuitive answers; IA Rate, intended answers (%); ICR rate, IA+CA+RA (%). Figure 5.5 PION test results. 90 REASONING WITH INCONSISTENT ONTOLOGIES 5.8.3. Future Experiments As noted in many surveys of current Semantic Web work, most Semantic Web applications to date (including those included in this volume) use rather lightweight ontologies. These lightweight ontologies are often expressed in RDF Schema, which means that by definition they will not contain any inconsistencies. However, closer inspection by Schlobach (2005a) revealed that such lightweight ontologies contain many implicit assumptions (such as disjointness of siblings in the class hierarchy) that have not been modeled explicitly because of the limitations of the lightweight representation language Schlobach’s (2005a) study reveals that after making such implicit disjointness assumptions explicit (a process called semantic clarification ), many of the ontologies do reveal internal inconsistencies. In future experiments, we intend to determine to which extent it is still possible to locally reason in such semantically clarified inco nsistent ontologies using the heuri stics described in this chapter. 5.9. DISCUSSION AND CONCLUSIONS In this chapter, we have presented a framework for reasoning with inconsistent ontologies. We have introduced the formal definitions of the selection functions, and investigated the strategies of inconsistency reasoning processing based on a linear extension strategy. One of the novelties of our approach is that the selection functions depend on individual queries. Our approach differs from the traditional one in paraconsistent reasoning, nonmonotonic reasoning, and belief revision, in which a pre-defined preference ordering for all of the queries is required. This makes our approach more flexible, and less inefficient to obtain intended results. The selection functions can be viewed as ones creating query-specific preference orderings. We have implemented and presented a prototype of PION. In this chapter, we have provided the evaluation report of the prototype by applying it to the several inconsistent ontology examples. The tests show that our approach can obtain intuitive results in most cases for reasoning with inconsistent ontologies. Considering the fact that standard reason- ers always result in either meaningless answers or incoherence errors for queries on inconsistent ontologies, we can claim that PION can do much better because it can provide a lot of intuitive, thus meaningful answers. This is a surprising result given the simplicity of our selectio n function. We are also working on a framework for inconsistent ontology diagnosis and repair by defining a number of new nonstandard reason- ing services to explain inconsistencies through pinpointing (Schlobach and Huang, 2005). An informed bottom-up approach to calculate DISCUSSION AND CONCLUSIONS 91 minimally inconsistent sets by the support of an external Description Logic reasoner has been proposed in Schlobach and Huang (2005). That approach has been prototypically implemented as the DION (Debugger of Inconsistent Ontologies). DION uses the relevance relation which has been used in PION as its heuristic information to guide the selecting procedure for finding minimally inconsistent sets. That justifies to some extent that the notion of ‘concept relevance’ is useful for inconsistent ontology processing. In future work, we are going to test PION with more large-scale ontology examples. We are also going to inve stigate different approaches for selection functions (e.g., semantic-relevance based) and different extension strategies as alternatives to the linear extensio n strategy in combination with different selection functions, and test their perfor- mance. ACKNOWLEDGMENT We are indebted to Peter Haase for so carefully proofreading this chapter. REFERENCES Alchourron C, Gaerdenfors P, Makinson D. 1985. On the logic of theory change: partial meet contraction and revision functions. The Journal of Symbolic Logic 50: 510–530. Belnap N. 1977. A useful four-valued logic. 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