Lehmann, D. 1995. Another perspective on default reasoning. Annals of Mathematics and Artificial Intelligence 15:61--82.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... (modal) logic based entailment by Lamarre [35] Finally, mainly to solve problems with property inheritance from classes to exceptional subclasses, the maximum entropy approach to default entailment was proposed by Goldszmidt et al. 30] lexicographic entailment was introduced by Lehmann [37] and Benferhat et al. 4] conditional entailment was proposed by Geffner [25, 26] and an infinitesimal belief function approach was suggested by Benferhat et al. 7] In this paper, we define probabilistic generalizations of entailment and Lehmann s lexicographic entailment that lie between ....

....penguins. The following lemma describes an alternative way of defining the notion of consequence. x(d 0 , 3 58 87 9 be a conditional constraint. o( #wi= h o( #wi=k) Dwi= F , 3. 3 Lexicographic Entailment We next extend Lehmann s lexicographic entailment [37] . Note that even though we do not use probability rankings here, the new notion of entailment can be easily expressed by a unique single probability ranking. We use the partition 4 of to define a lexicographic preference relation on probabilistic interpretations ....

D. Lehmann. Another perspective on default reasoning. Ann. Math. Artif. Intell., 15(1):61--82, 1995.

....that a companion paper [LUK 02b] continues this line of research on the relationship between coherence based and model theoretic probabilistic reasoning. In the spirit of [LUK 02a] it introduces probabilistic generalizations of Pearl s entailment [PEA 90] and Lehmann s lexicographic entailment [LEH 95] which lie between g coherent entailment and logical entailment. As for the relationship between probabilistic reasoning under coherence and default , we showed that g coherence and g coherent entailment , respectively. This result gives a clear picture of the main properties of ....

LEHMANN D., "Another perspective on default reasoning", Ann. Math. Artif. Intell., vol. 15, num. 1, 1995, p. 61--82.

.... generic and assertional (i.e. Abox) probabilistic knowledge [22,23] P 0(2 ,3 4 allows to express default knowledge as a special case of generic probabilistic knowledge, where reasoning with such default knowledge is based on the sophisticated notion of lexicographic entailment by Lehmann [20]. We present sound and complete techniques for probabilistic reasoning in the description logic P 0(2 , 0 , which are based on reductions to classical reasoning and to linear programming, and which show in particular that reasoning in P 5 0( 0 is decidable. Note that due to the ....

D. Lehmann. Another perspective on default reasoning. Ann. Math. Artif. Intell., 15(1):61-- 82, 1995.

.... (modal) logic based entailment by Lamarre [42] Finally, mainly to solve problems with property inheritance from classes to exceptional subclasses, the maximum entropy approach to default entailment was proposed by Goldszmidt et al. 34] lexicographic entailment was introduced by Lehmann [44] and Benferhat et al. 4] conditional entailment was proposed by Geffner [28, 30] and an infinitesimal belief function approach was suggested by Benferhat et al. 7] In this paper, we present probabilistic generalizations of Pearl s entailment in System [55] and of Lehmann s lexicographic ....

.... et al. 4] conditional entailment was proposed by Geffner [28, 30] and an infinitesimal belief function approach was suggested by Benferhat et al. 7] In this paper, we present probabilistic generalizations of Pearl s entailment in System [55] and of Lehmann s lexicographic entailment [44] that lie between entailment in model theoretic probabilistic logic and entailment in probabilistic logic under coherence. Roughly, the main difference between model theoretic probabilistic entailment and probabilistic entailment under coherence is that the former realizes an inheritance of ....

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D. Lehmann. Another perspective on default reasoning. Ann. Math. Artif. Intell., 15(1):61--82, 1995.

.... INFSYS RESEARCH REPORT 1843 03 02, JANUARY 2003 NONMONOTONIC PROBABILISTIC REASONING UNDER VARIABLESTRENGTH INHERITANCE WITH OVERRIDING 20 JANUARY 2003 Abstract.We present new probabilistic generalizations of Pearl s entailment in System # [50, 32] and Lehmann s lexicographic entailment [39], called entailment, which are parameterized through a value##### that describes the strength of the inheritance of purely probabilistic knowledge. In the special cases of##### and##### , the notions of entailment coincide with the probabilistic notions of entailment ....

.... (modal) INFSYS RR 1843 03 02 logic based entailment by Lamarre [37] Finally, mainly to solve problems with property inheritance from classes to exceptional subclasses, the maximum entropy approach was proposed by Goldszmidt et al. 30] lexicographic entailment was introduced by Lehmann [39] and Benferhat et al. 3] conditional entailment was proposed by Geffner [25, 26] and a belief function approach was suggested by Benferhat et al. 6] The main ideas behind the probabilistic generalizations of sophisticated default reasoning formalisms in [45, 46] can be summarized as ....

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D. Lehmann. Another perspective on default reasoning. Ann. Math. Artif. Intell., 15(1):61--82, 1995.

.... to probabilistic default reasoning [21, 22] The description logic P 1 2( also allows to express default knowledge as a special case of generic probabilistic knowledge, where reasoning with such default knowledge is based on the sophisticated notion of lexicographic entailment by Lehmann [19]. There are several related approaches to probabilistic description logics in the literature [8, 17, 18] which can be classified according to the supported forms of probabilistic knowledge and the underlying probabilistic reasoning formalism. Heinsohn [8] presents a probabilistic extension of the ....

.... assertional probabilistic knowledge [21, 22] The description logic P 1 2( allows to express default knowledge as a special case of generic probabilistic knowledge, where reasoning with such default knowledge is based on the sophisticated notion of lexicographic entailment by Lehmann [19]. We present sound and complete techniques for probabilistic reasoning in P ) which are based on reductions to classical reasoning in and to linear programming, and which show in particular that reasoning in P 1 2( is decidable. Note that due to the presence of ....

D. Lehmann. Another perspective on default reasoning. Ann. Math. Artif. Intell., 15(1):61--82, 1995.

....that a companion paper [30] continues this line of research on the relationship between coherencebased and model theoretic probabilistic reasoning. In the spirit of [31] it introduces probabilistic generalizations of Pearl s entailment in System ( 35] and Lehmann s lexicographic entailment [26], which lie between g coherent entailment and logical entailment. As for the relationship between probabilistic reasoning under coherence and default reasoning in System , we showed that g coherence and g coherent entailment are generalizations of consistency and entailment , respectively. ....

D. Lehmann. Another perspective on default reasoning. Ann. Math. Artif. Intell., 15(1):61--82, 1995.

.... (modal) logic based entailment by Lamarre [35] Finally, mainly to solve problems with property inheritance from classes to exceptional subclasses, the maximum entropy approach to default entailment was proposed by Goldszmidt et al. 30] lexicographic entailment was introduced by Lehmann [37] and Benferhat et al. 4] conditional entailment was proposed by Geffner [25, 26] and an infinitesimal belief function approach was suggested by Benferhat et al. 7] INFSYS RR 1843 02 02 3 In this paper, we define probabilistic generalizations of entailment in Pearl s System Z and Lehmann s ....

....way of defining the notion of z consequence. Lemma 3.6 Let KB be a g coherent probabilistic knowledge base, and let C = j ) l; u] be a conditional C iff ( 0 C) 0 :C) INFSYS RR 1843 02 02 9 3. 3 Lexicographic Entailment We next extend Lehmann s lexicographic entailment [37] to g coherent probabilistic knowledge bases KB = L; P ) Note that even though we do not use probability rankings here, the new notion of entailment can be easily expressed by a unique single probability ranking. We use the z partition (P 0 ; P k ) of KB to define a lexicographic ....

D. Lehmann. Another perspective on default reasoning. Ann. Math. Artif. Intell., 15(1):61--82, 1995.

....in order to solve problems with property inheritance lukasiewicz.tex; 1 09 2001; 20:56; p. 3 4 Thomas Lukasiewicz from classes to exceptional subclasses, the maximum entropy approach to default entailment was proposed by Goldszmidt et al. 37] lexicographic entailment was introduced by Lehmann [55] and Benferhat et al. 7] conditional entailment was proposed by Geffner [32, 33] and an infinitesimal belief function approach was suggested by Benferhat et al. 10] In this paper, we combine the notion of logical entailment from conditional constraints with the notions of entailment in ....

....from conditional knowledge bases. In detail, we present the notions of , lexicographic, and conditional entailment for conditional constraints, which combine logical entailment from conditional constraints with entailment in Pearl s system [70] Lehmann s lexicographic entailment [55], and Geffner s conditional entailment [32, 33] respectively. We show that the new formalisms have a similar behavior as referenceclass reasoning in a number of benchmark examples. Moreover, they can handle complex scenarios and even purely probabilistic subjective knowledge as input. ....

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Lehmann, D.: 1995, `Another perspective on default reasoning'. Ann. Math. Artif. Intell. 15(1), 61--82.

....of acb3 ) subject to all minimal models aeb of with acb3 # Hi . INFSYS RR 1843 01 05 9 4. 3 Lexicographic Entailment We next recall ] entailment for probabilistic default theories from [22, 23] which is a proper generalization of Lehmann s lexicographic entailment [19]. In the sequel, consider a consistent probabilistic default theory Y # 0 . We use the partition 8 on 20038 of to define a lexicographic preference relation on probabilistic interpretations as follows. For probabilistic interpretations acb and acb , we say acb is ] ....

D. Lehmann. Another perspective on default reasoning. Ann. Math. Artif. Intell., 15(1):61-- 82, 1995.

....the infimum (resp. supremum) of 4 subject to all minimal models of th with a [ b YAm . 4. 3 LEXICOGRAPHIC ENTAILMENT We next recall (4 entailment for probabilistic default theories [21, 22] which is a proper generalization of Lehmann s lexicographic entailment [18]. In the sequel, consider a consistent probabilistic default theory O M T . We use the partition 4 of to define a lexicographic preference relation on probabilistic interpretations as follows. For probabilistic interpretations and , we say is (4 ....

D. Lehmann. Another perspective on default reasoning. Ann. Math. Artif. Intell., 15(1):61--82, 1995.

....(resp. supremum) of Pr( j ) subject to all z minimal models Pr of S [ fEg with Pr( 0. INFSYS RR 1843 01 05 9 4. 3 Lexicographic Entailment We next recall lex entailment for probabilistic default theories from [22, 23] which is a proper generalization of Lehmann s lexicographic entailment [19]. In the sequel, consider a consistent probabilistic default theory T = S; D) We use the z partition (D 0 ; D k ) of D to define a lexicographic preference relation on probabilistic interpretations as follows. For probabilistic interpretations Pr and Pr 0 , we say Pr is ....

D. Lehmann. Another perspective on default reasoning. Ann. Math. Artif. Intell., 15(1):61-- 82, 1995.

.... entailment by Lamarre [48] Finally, mainly in order to solve problems with property inheritance from classes to exceptional subclasses, the maximum entropy approach to default entailment was proposed by Goldszmidt et al. 32] the notion of lexicographic entailment was introduced by Lehmann [50] and Benferhat et al. 5] the notion of conditional entailment was proposed by Geffner [28, 29] and an infinitesimal belief function approach was suggested by Benferhat et al. 8] In this paper, we combine the notion of logical entailment from conditional constraints with the notions of ....

....reasoning from conditional knowledge bases. In detail, we introduce the notions of z , lexicographic, and conditional entailment for conditional constraints, which combine logical entailment from conditional constraints with Pearl s entailment in system Z [65] Lehmann s lexicographic entailment [50], and Geffner s conditional entailment [28, 29] respectively. We show that the new formalisms have a similar behavior as reference class reasoning in a number of benchmark examples. Moreover, they can handle complex scenarios and even purely probabilistic subjective knowledge as input. ....

[Article contains additional citation context not shown here]

D. Lehmann. Another perspective on default reasoning. Ann. Math. Artif. Intell., 15(1):61--82, 1995.

.... mainly to solve problems with property inheritance from classes to exceptional subclasses, the maximum entropy approach was proposed by Goldszmidt et al. 13] and recently generalized to variable strength defaults by Bourne and Parsons [7] lexicographic entailment was introduced by Lehmann [19] and Benferhat et al. 3] and conditional entailment was proposed by Geffner [11, 12] However, while the semantic aspects of these formalisms are quite well understood, about their computational properties only partial results have been known so far. In previous work [9] we were filling some ....

....particular, we show that taxonomic hierarchies that are augmented by default knowledge can be expressed in the feedback free Horn case. We show that in the feedback free Horn case, default reasoning under z entailment [13] z s entailment [7] lex entailment [3] and lex p entailment [19] is tractable. To our knowledge, no or only limited tractable cases [8] for these notions of entailment from conditional knowledge bases have been identified so far. Our tractability results for the feedback free Horn case are complemented by our proof that without a similar restriction on ....

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D. Lehmann. Another perspective on default reasoning. Ann. Math. Artif. Intell., 15(1):61-- 82, 1995.

.... entailment by Lamarre (1992) Finally, mainly in order to solve problems with property inheritance from classes to exceptional subclasses, the maximum entropy approach to default entailment was proposed by Goldszmidt et al. 1993) the notion of lexicographic entailment was introduced by Lehmann (1995) and Benferhat et al. 1993) the notion of conditional entailment was proposed by Geffner (Geffner 1992; Geffner Pearl 1992) and an infinitesimal belief function approach was suggested by Benferhat et al. 1995) Coming back to our introductory example, we realize that G1 G3 and E1 E2 ....

....default reasoning with conditional constraints. We introduce the notions of z entailment, lexicographic entailment, and conditional entailment for conditional constraints, which are a combination of the classical notions of entailment in system Z (Pearl 1990) Lehmann s lexicographic entailment (Lehmann 1995), and Geffner s conditional entailment (Geffner 1992; Geffner Pearl 1992) respectively, with the classical notion of logical entailment for conditional constraints. We give some examples that analyze the nonmonotonic properties of the new notions of entailment for default reasoning with ....

[Article contains additional citation context not shown here]

Lehmann, D. 1995. Another perspective on default reasoning. Ann. Math. Artif. Intell. 15(1):61--82.

.... entailment by Lamarre [29] Finally, mainly in order to solve problems with property inheritance from classes to exceptional subclasses, the maximum entropy approach to default entailment was proposed by Goldszmidt et al. 21] the notion of lexicographic entailment was introduced by Lehmann [31] and Benferhat et al. 4] the notion of conditional entailment was proposed by Geffner [18, 19] and an infinitesimal belief function approach was suggested by Benferhat et al. 7] Coming back to our introductory example, we realize that G1 G3 and E1 E2 express interval restrictions for ....

....to the propositional setting. The main contributions of this paper can be summarized as follows: We present notions of nonmonotonic entailment for probabilistic default reasoning. More precisely, we generalize the notions of entailment in system Z [41] Lehmann s lexicographic entailment [31], and Geffner s conditional entailment [18, 19] to strict and defeasible conditional constraints. The new notions of entailment allow desirable conclusions from certain kinds of unsatisfiable sets of conditional constraints, and can thus be used for handling inconsistent probabilistic knowledge ....

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D. Lehmann. Another perspective on default reasoning. Ann. Math. Artif. Intell., 15(1):61--82, 1995.

....summarizes the relation between LCD and system Z. Theorem 4. The consequence relations lcd and Z are incomparable. Another solution to get non monotonic inferences that go beyond those of system P is to use the so called lexicographic approach (Dubois et al. 1991, Benferhat et al. 1993, Lehmann, 1993). One well know drawback of this approach is that it is syntax sensitive: repetitions of the same default in D may change the result. We show that LCD is not sensitive to duplications. Example 7. Consider a variant of the Quaker Republican problem where the rule Generally, Quaker are pacifists ....

Lehmann, D. (1993) Another perspective on Default Reasoning. Technical report, Hebrew University.

....way. So, both 1 and pen are preferential relations, and the thesis follows by noting that they are based on the same order. y e Bel Fct July 27, 1999 27 5. 3 The lexicographic system The second case that we consider is the lexicographic system defined by Benferhat et al. 1993) and Lehmann (1993), and set by Dubois et al. 1992) in the framework of possibilistic logic. A similar approach has also been considered in diagnosis by De Kleer (1990) and Lang (1994) The main idea is to again start from a stratification D 1 , D k of D, and regard all defaults in the i th layer as being ....

Lehmann, D. (1993) Another perspective on Default Reasoning. Technical report, Hebrew University.

.... inheritance from classes to exceptional subclasses, the max imum entropy approach to default entailment was proposed by Goldszmidt et al. 24] and recently generalized to variable strength defaults by Bourne and Parsons [7] the notion of lexicographic entailment was introduced by Lehmann [38] and Benferhat et al. 3] the notion of conditional entailment was proposed by Geffner [20, 22] and, an infinitesimal belief function approach was suggested by Benferhat et al. 6] While the semantic properties and interrelationships of the various formalisms are quite well understood, their ....

.... for which only very rough or even no complexity results have been derived so far, namely proper entailment [25] maximum entropy entailment [24] together with its variable strength extension [7] and Geffner s conditional entailment [20] We reconsider Lehmann s lexicographic entailment [38], and correct an error of [12] in the derivation of the complexity of generalized lexicographic entailment [3] We analyze the effect of compilation for ranking based approaches, in terms of off line computation of the ranking implicitly associated with the defaults in the knowledge base, such ....

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D. Lehmann. Another perspective on default reasoning. Ann. Math. Artif. Intell., 15(1):61--82, 1995.

....relations obtained in the second way. The relations we get are already known in the literature: in the first case, we get the so called penalty logic (Pinkas, 1992; Dupin et al. 1994) in the second case, we get the lexicographic approach (Dubois et al. 1992; Benferhat et al. 1993; Lehmann, 1993); and the last case leads to the preferred sub theories of Brewka (1989) Consider a default base D, and let D 0 , D 1 , D n be the partition of D given by System Z. As we did for LCD, we associate each default d in D with the following simple support function m d ( f d ] 1 e d ....

....of D i satisfied by w. Proposition 1. pl(w) p l(w ) if and only if there exist a positive number 1i n such that: j i, w] j = w ] j , and . w] i [w ] i This ordering corresponds to the so called Lexicographical ordering defined in (Dubois et al. 1992; Benferhat et al. 1993; Lehmann, 1993), and hence the consequence relation obtained by using Dempster s combination and the e d s defined above is the same as the one obtained by the lexicographic approaches. This ordering has also been considered in diagnosis by De Kleer (1990) and Lang (1994) For the last case, let d ij be the ....

Lehmann, D. (1993) Another perspective on Default Reasoning. Technical report, Hebrew University.

....meet base revision, we can extend the cardinality maximizing scheme by introducing priority classes. Such a scheme has been proposed for inference from inconsistent belief bases (which is equivalent to revising a base with ) Benferhat et al. 1993 ] and for reasoning in a nonmonotonic logic [ Lehmann, 1993 ] Defining the lexicographic preference relation OEOE on subsets of a belief base A as BOEOE C def ( 9i kA i Bk kA i Ck 8j i : kA j Bk = kA j Ck; 36) the remainder set for the new scheme is defined as follows: A ) def = fB Aj B 6j= 8C A BOEOE C ) C j= g: 37) ....

Daniel Lehmann. Another perspective on default reasoning. Technical report, Hebrew University, Jerusalem, Israel, 1993.

.... In order to discriminate between the consistent subsets of KB, the selection criterion generally makes use of uncertainty considerations by using explicitly uncertainty measures [Wilson93, Benferhat al95] or by using measures expressing priorities [Rescher64, Brewka89, Nebel91, Benferhat al93, Lehmann92] for a more detailed description of this kind of approach see [Cayrol Lagasquie95] These approaches consist in ranking the KB into priority levels (these levels are often supposed given by an expert (except in Pearl s system Z [Pearl90] where an automatic technique to rank the formulas is ....

D. Lehmann. 1992. Another perspective on default reasoning. Tec. Report.

....et al. 1992a) in the setting of possibilistic logic under the name of strongly maximal consistent subbases. Baral et al. 1992) have also used a similar approach to combine belief bases. The definition of Lex( has been proposed in another form in (Dubois et al. 1992a) and also independently in (Lehmann, 1993). The idea of selecting a subset of the set of maximally consistent subbases of using a cardinality criterion was used independently in diagnostic problems (De Kleer, 1990; Lang, 1994) It corresponds to the property of parsimony advocated in (Reggia et al. 1985) Once Incl( and Lex( are ....

D. Lehmann (1993) Another perspective on default reasoning.

....inferences from a default base. However, all the considered approaches suffer from the same limitations regarding the drowning problem. Some systematic methods for coping with this problem have been proposed, based on refined inconsistency handling (Geffner [27] Benferhat et al. 3] 6] Lehmann [37]) But they always face counter examples and have high complexity [40] 10] A more promising way of overcoming this problem may be to add information about conditional independence of the form in the context a, accepting b has no influence on accepting g . This kind of information can be ....

D. Lehmann,Another perspective on default reasoning. Institute of Computer Science, Hebrew University of Jerusalem, Israel, 1993.

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Lehmann, D. 1995. Another perspective on default reasoning. Annals of Mathematics and Artificial Intelligence 15:61--82.

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