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56
Default Reasoning from Conditional Knowledge Bases: Complexity and Tractable Cases
 Artif. Intell
, 2000
"... Conditional knowledge bases have been proposed as belief bases that include defeasible rules (also called defaults) of the form " ! ", which informally read as "generally, if then ." Such rules may have exceptions, which can be handled in different ways. A number of entailment semantics for condi ..."
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Cited by 21 (13 self)
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Conditional knowledge bases have been proposed as belief bases that include defeasible rules (also called defaults) of the form " ! ", which informally read as "generally, if then ." Such rules may have exceptions, which can be handled in different ways. A number of entailment semantics for conditional knowledge bases have been proposed in the literature. However, while the semantic properties and interrelationships of these formalisms are quite well understood, about their computational properties only partial results are known so far. In this paper, we fill these gaps and first draw a precise picture of the complexity of default reasoning from conditional knowledge bases: Given a conditional knowledge base KB and a default ! , does KB entail ! ? We classify the complexity of this problem for a number of wellknown approaches (including Goldszmidt et al.'s maximum entropy approach and Geffner's conditional entailment), where we consider the general propositional case as well as natural syntactic restrictions (in particular, to Horn and literalHorn conditional knowledge bases). As we show, the more sophisticated semantics for conditional knowledge bases are plagued with intractability in all these fragments. We thus explore cases in which these semantics are tractable, and find that most of them enjoy this property on feedbackfree Horn conditional knowledge bases, which constitute a new, meaningful class of conditional knowledge bases. Furthermore, we generalize previous tractability results from Horn to qHorn conditional knowledge bases, which allow for a limited use of disjunction. Our results complement and extend previous results, and contribute in refining the tractability/intractability frontier of default reasoning from conditional know...
Weak nonmonotonic probabilistic logics
"... Towards probabilistic formalisms for resolving local inconsistencies under modeltheoretic probabilistic entailment, we present probabilistic generalizations of Pearl’s entailment in System Z and Lehmann’s lexicographic entailment. We then analyze the nonmonotonic and semantic properties of the new ..."
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Cited by 21 (6 self)
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Towards probabilistic formalisms for resolving local inconsistencies under modeltheoretic probabilistic entailment, we present probabilistic generalizations of Pearl’s entailment in System Z and Lehmann’s lexicographic entailment. We then analyze the nonmonotonic and semantic properties of the new notions of entailment. In particular, we show that they satisfy the rationality postulates of System P and the property of Rational Monotonicity. Moreover, we show that modeltheoretic probabilistic entailment is stronger than the new notion of lexicographic entailment, which in turn is stronger than the new notion of entailment in System Z. As an important feature of the new notions of entailment in System Z and lexicographic entailment, we show that they coincide with modeltheoretic probabilistic entailment whenever there are no local inconsistencies. We also show that the new notions of entailment in System Z and lexicographic entailment are proper generalizations of their classical counterparts. Finally, we present algorithms for reasoning under the new formalisms, and we give a precise picture of its computational complexity.
Probabilistic Logic Programming under Inheritance with Overriding
 In Proceedings UAI01
, 2001
"... We present probabilistic logic programming under inheritance with overriding. This approach is based on new notions of entailment for reasoning with conditional constraints, which are obtained from the classical notion of logical entailment by adding inheritance with overriding. This is done by usin ..."
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Cited by 20 (13 self)
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We present probabilistic logic programming under inheritance with overriding. This approach is based on new notions of entailment for reasoning with conditional constraints, which are obtained from the classical notion of logical entailment by adding inheritance with overriding. This is done by using recent approaches to probabilistic default reasoning with conditional constraints. We analyze the semantic properties of the new entailment relations. We also present algorithms for probabilistic logic programming under inheritance with overriding, and we analyze its complexity in the propositional case. 1
Iterated revision as prioritized merging
 In Proc. of KR’06
, 2006
"... Standard accounts of iterated belief revision assume a static world, about which an agent receives a sequence of observations. More recent items are assumed to have priority over less recent items. We argue that there is no reason, given a static world, for giving priority to more recent items. Inst ..."
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Cited by 18 (2 self)
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Standard accounts of iterated belief revision assume a static world, about which an agent receives a sequence of observations. More recent items are assumed to have priority over less recent items. We argue that there is no reason, given a static world, for giving priority to more recent items. Instead we suggest that a sequence of observations should be merged with the agent’s beliefs. Since observations may have differing reliability, arguably the appropriate belief change operator is prioritized merging. We develop this view here, suggesting postulates for prioritized merging, and examining existing merging operators with respect to these postulates. As well, we examine other suggested postulates for iterated revision, to determine how well they fit with the prioritized merging interpretation. All postulates for iterated revision that we examine, except for Darwiche and Pearl’s controversial C2, are consequences of our suggested postulates for prioritized merging.
Managing uncertainty and vagueness in description logics, logic programs and description logic programs
, 2008
"... Managing uncertainty and/or vagueness is starting to play an important role in Semantic Web representation languages. Our aim is to overview basic concepts on representing uncertain and vague knowledge in current Semantic Web ontology and rule languages (and their combination). ..."
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Cited by 16 (5 self)
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Managing uncertainty and/or vagueness is starting to play an important role in Semantic Web representation languages. Our aim is to overview basic concepts on representing uncertain and vague knowledge in current Semantic Web ontology and rule languages (and their combination).
NonMonotonic Reasoning: From Complexity to Algorithms
 ANNALS OF MATHEMATICS AND ARTIFICIAL INTELLIGENCE
, 1996
"... We focus on coherencebased approaches to nonmonotonic reasoning. These approaches handle syntactical belief bases equipped with symbolic ordering structures and rely upon classical logic. The belief base, a finite set of propositional formulae, is considered syntactically, as in (Nebel 1991): eac ..."
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Cited by 15 (0 self)
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We focus on coherencebased approaches to nonmonotonic reasoning. These approaches handle syntactical belief bases equipped with symbolic ordering structures and rely upon classical logic. The belief base, a finite set of propositional formulae, is considered syntactically, as in (Nebel 1991): each belief is a distinct piece of information and only beliefs which are explicitly present in the base are taken into account. It departs from the logical point of view where a base is identified with the set of its models. The belief base is not assumed consistent. Moreover, we assume that the belief base is equipped with a complete preordering structure (a priority relation) which, contrarily to (Gardenfors & Makinson 1994), is not related to the semantical entailment ordering. So, the base is stratified in a collection of subbases of different priority levels. Coherencebased nonmonotonic entailment from a stratified belief base can be described as a twostep procedur
Nonmonotonic SyntaxBased Entailment: A Classification of Consequence Relations
 in ECSQARU’95
, 1995
"... . The purpose of this paper is to provide a comparative study of nonmonotonic syntaxbasedconsequencerelations, from different points of view. Starting from a (not necessarily consistent) belief base E and a preordering on E, we first remind different mechanisms for selecting preferred consistent s ..."
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Cited by 12 (1 self)
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. The purpose of this paper is to provide a comparative study of nonmonotonic syntaxbasedconsequencerelations, from different points of view. Starting from a (not necessarily consistent) belief base E and a preordering on E, we first remind different mechanisms for selecting preferred consistent subbases in syntaxbased approaches. Then, we present three entailment principles in order to cope with these multiple subbases. The crossing point of each generation mechanism and each principle defines a syntaxbased consequence relation. Pursuing previous work of the authors concerning the computational complexity point of view, we first provide a comparison from the cautiousness point of view. Our proposal restates previous results [18] in a single framework and provides new results. In the last part of this paper, we study the validity of deduction rules (such as those introduced by [15, 12]). Results are discussed in the conclusion. 1 Introduction In this paper, we focus on syntactic a...
Penalty logic and its link with DempsterShafer theory
 In Proc. of the 10 th Conf. on Uncertainty in Artificial Intelligence
, 1994
"... Penalty logic, introduced by Pinkas [?], associates to each formula of a knowledge base the price to pay if this formula is violated. Penalties may be used as a criterion for selecting preferred consistent subsets in an inconsistent knowledge base, thus inducing a nonmonotonic inference relation. A ..."
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Cited by 12 (4 self)
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Penalty logic, introduced by Pinkas [?], associates to each formula of a knowledge base the price to pay if this formula is violated. Penalties may be used as a criterion for selecting preferred consistent subsets in an inconsistent knowledge base, thus inducing a nonmonotonic inference relation. A precise formalization and the main properties of penalty logic and of its associated nonmonotonic inference relation are given in the first part. We also show that penalty logic and DempsterShafer theory are related, especially in the infinitesimal case. 1 Introduction The problem of inconsistency handling appears when the available knowledge base  KB for short  (here a set of propositional formulas) is inconsistent. Most approaches come up with the inconsistency by selecting among the consistent subsets of KB some preferred subsets; the selection criterion generally makes use of uncertainty considerations, sometimes by using explicitly uncertainty measures (such as Wilson [?], Benfer...
Basic Infobase Change
 Advanced Topics in Artificial Intelligence
, 1999
"... . Generalisations of theory change involving arbitrary sets of w#s instead of belief sets have become known as base change. In one view, a base should be thought of as providing more structure to its generated belief set, and can be used to determine the theory change operation associated with a bas ..."
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Cited by 8 (3 self)
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. Generalisations of theory change involving arbitrary sets of w#s instead of belief sets have become known as base change. In one view, a base should be thought of as providing more structure to its generated belief set, and can be used to determine the theory change operation associated with a base change operation. In this paper we extend a proposal along these lines by Meyer et al. [12]. We take an infobase as a finite sequence of w#s, with each element in the sequence being seen as an independently obtained bit of information, and define appropriate infobase change operations. The associated theory change operations satisfy the AGM postulates for theory change [1]. Since an infobase change operation produces a new infobase, it allows for iterated infobase change. We measure iterated infobase change against the postulates proposed by Darwiche et al. [2, 3] and Lehmann [10]. 1