BRIEF SURVEY OF APPROACHES TO REASONING WITH INCONSISTENCY in .NET

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BRIEF SURVEY OF APPROACHES TO REASONING WITH INCONSISTENCY
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5.2. BRIEF SURVEY OF APPROACHES TO REASONING WITH INCONSISTENCY 5.2.1. Paraconsistent Logics
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Reasoning with inconsistency is a well-known topic in logics and AI. Many approaches have been proposed to deal with inconsistency (Benferhat and Garcia, 2002; Beziau, 2000; Lang and Marquis, 2001). The development of paraconsistent logics was initiated to challenge the explosive problem of the standard logics. Paraconsistent logics (Beziau, 2000) allow theories that are inconsistent but nontrivial. There are many different paraconsistent logics, each of which weaken traditional logic in a different way. Nonadjunctive systems block the general inference a, b j a ^ b, so that in particular the combination of a and :a no longer entails a ^ :a. Relevace logics aim to block the explosive inference a ^ :a j b by requiring that the premises of an entailment must somehow be relevant to the conclusion. In the propositional calculus, this involves requiring that premises and conclusion share atomic sentences, which is not the case in the latter formula. Many relevant logics are multi-valued logics. They are de ned on a semantics which allows both a proposition and its negation to hold for an interpretation. Levesque s (1989) limited inference allows the interpretation of a language in which a truth assignment may map both a proposition l and its negation :l to true. Extending the idea of Levesque s limited inference, Schaerf and Cadoli (1995) propose S-3-entailment and S-1-entailment for approximate reasoning with tractable results. The main idea of Schaerf and Cadoli s approach is to introduce a subset S of the language, which can be used as a parameter in their framework and allows their reasoning procedure to focus on a part of the theory while the remaining part is ignored. However, how to construct and extend this subset S in speci c scenario s is still an open question (the problem of nding a general optimal strategy for S is known to be intractable). Based on Schaerf and Cadoli s S-3-entailment, Marquis and Porquet (2003) present a framework for reasoning with inconsistency by introducing a family of resource-bounded paraconsistent inference relations. In Marquis and Porquet s approach, consistency is restored by removing variables from the approximation set S instead of removing some explicit beliefs from the belief base, like the standard approaches do in belief revision. Their framework enables some forms of graded paraconsistency by explicit handling of preferences over the approximation set S. Marquis and Porquet (2003) propose several policies, for example, the linear order policy and the lexicographic policy, for the preference handling in paraconsistent reasoning.
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REASONING WITH INCONSISTENT ONTOLOGIES
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5.2.2. Ontology Diagnosis
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As mentioned in the introduction, an alternative approach to deal with inconsistencies is to repair them before reasoning, instead of reasoning in the presence of the inconsistencies. A long standing tradition in Arti cial Intelligence is that of belief revision, which we will discuss below. A more recent branch of work is explicitly tailored to diagnosis and repair of ontologies in particular. The rst in this line was done by Schlobach and Cornet (2003), who aimed at identifying a minimal subset of Description Logic axioms that is responsible for an inconsistency (i.e., such a minimal subset is inconsistent, but removal of any single axiom from the set makes the inconsistency go away). In later works by Friedrich and Shchekotykhin (2005) and Schlobach (2005b), this approach has been extended to deal with richer Description Logics, and has been rephrased in terms of Reiter s (1987) general theory of model-based diagnosis.
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5.2.3. Belief Revision
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Belief revision is the process of changing beliefs to take into account a new piece of information. What makes belief revision nontrivial is that several different ways for performing this operation may be possible. For example, if the current knowledge includes the three facts a, b, and a ^ b ! c, the introduction of the new information :c can be done preserving consistency only by removing at least one of the three facts. In this case, there are at least three different ways for performing revision. In general, there may be several different ways for changing knowledge. The main assumption of belief revision is that of minimal change: the knowledge before and after the change should be as similar as possible. The AGM postulates (Alchourron et al., 1985)1 are properties that an operator that performs revision should satisfy in order for being considered rational. Revision operators that satisfy the AGM postulates are computationally highly intractable. In an attempt to avoid this, Chopra et al. (2000) incorporate the local change of belief revision and relevance sensitivity by means of Schaerf and Cadoli s approximate reasoning method, and show how relevance can be introduced for approximate reasoning in belief revision. Incidently, recent work by Flouris et al. (2005) has shown that the AGM theory in its original form is not applicable to restricted logics such as the Description Logics that underly OWL, and that it is not trivial to nd alternative formulations of the AGM postulates that would work for OWL.
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