## Probabilistic Default Reasoning with Conditional Constraints (2000)

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Venue: | ANN. MATH. ARTIF. INTELL |

Citations: | 35 - 18 self |

### BibTeX

@ARTICLE{Lukasiewicz00probabilisticdefault,

author = {Thomas Lukasiewicz},

title = {Probabilistic Default Reasoning with Conditional Constraints},

journal = {ANN. MATH. ARTIF. INTELL},

year = {2000},

volume = {34},

pages = {200--2}

}

### Years of Citing Articles

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### Abstract

We present an approach to reasoning from statistical and subjective knowledge, which is based on a combination of probabilistic reasoning from conditional constraints with approaches to default reasoning from conditional knowledge bases. More precisely, we introduce the notions of -, lexicographic, and conditional entailment for conditional constraints, which are probabilistic generalizations of Pearl's entailment in system , Lehmann's lexicographic entailment, and Geffner's conditional entailment, respectively. We show that the new formalisms have nice properties. In particular, they show a similar behavior as referenceclass reasoning in a number of uncontroversial examples. The new formalisms, however, also avoid many drawbacks of reference-class reasoning. More precisely, they can handle complex scenarios and even purely probabilistic subjective knowledge as input. Moreover, conclusions are drawn in a global way from all the available knowledge as a whole. We then show that the new formalisms also have nice general nonmonotonic properties. In detail, the new notions of -, lexicographic, and conditional entailment have similar properties as their classical counterparts. In particular, they all satisfy the rationality postulates proposed by Kraus, Lehmann, and Magidor, and they have some general irrelevance and direct inference properties. Moreover, the new notions of - and lexicographic entailment satisfy the property of rational monotonicity. Furthermore, the new notions of -, lexicographic, and conditional entailment are proper generalizations of both their classical counterparts and the classical notion of logical entailment for conditional constraints. Finally, we provide algorithms for reasoning under the new formalisms, and we analyze its computational com...

### Citations

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Citation Context ...bed in Section 7.1. 8.1. COMPLEXITY CLASSES We assume some basic knowledge about P, NP, and co-NP. We now briefly describe some other complexity classes that will occur in our results. See especially =-=[31, 46, 68] for further background. The c-=-lass ���� (resp., ���� ) contains all decision problems that can be solved in deterministic (resp., nondeterministic) polynomial time with an oracle for ��� . The ���� cl... |

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Citation Context ...f revision deals with the problem of incorporating a new belief into a current state of belief. The most influential theory of belief revision is the one by Alchourrón, Gärdenfors, and Makinson (AGM=-=) [2]-=-. As pointed out lukasiewicz.tex; 1/09/2001; 20:56; p.41s42 Thomas Lukasiewicz by Boutilier [16] and Goldszmidt and Pearl [40], belief revision and default reasoning from conditional knowledge bases a... |

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Citation Context ...ins several different approaches to default reasoning and extensive work on the desired properties. The core of these properties are the rationality postulates proposed by Kraus, Lehmann, and Magidor =-=[47]-=-. These rationality postulates constitute a sound and complete axiom system for several classical model-theoretic entailment relations under uncertainty measures on worlds. In detail, they characteriz... |

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Citation Context ...bed in Section 7.1. 8.1. COMPLEXITY CLASSES We assume some basic knowledge about P, NP, and co-NP. We now briefly describe some other complexity classes that will occur in our results. See especially =-=[31, 46, 68] for further background. The c-=-lass ���� (resp., ���� ) contains all decision problems that can be solved in deterministic (resp., nondeterministic) polynomial time with an oracle for ��� . The ���� cl... |

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Citation Context ...oretic entailment relations under uncertainty measures on worlds. In detail, they characterize classical model-theoretic entailment under preferential structures [74, 47], infinitesimal probabilities =-=[1, 69]-=-, possibility measures [21], and world rankings [75, 39]. They also characterize an entailment relation based on conditional objects [22]. A survey of all these relationships is given in [9]. Recently... |

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Citation Context ...�� ���©��� � fly � penguin��� ����� ������� ��� bird � gentoo����� ��������� � penguin � gentoo������=-=� ����������� � Table II. Tight conclusions under reference-class reasoning. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) �¡s§©¨ � ����� ��� ����� � ¢¤£¦¥ ��� ¥���¨���������� ¢ ¥ ¢ ��¨ ��� ����-=-������� bird legs ¢ � bird ¥���¨���������� ¢ yellow ¥ ��¨ ��� ��������� ��� legs ¢ � ¥���¨�������... |

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Citation Context ...s, we conclude “Tweety flies with a probability of at most 0.05”. Reference-class reasoning, however, is not uncontroversial in the literature (see especially the work by Levi [57] and Bacchus et =-=al. [5, 6]-=-). In particular, it is often argued that even though reference-class reasoning deals very well with simple scenarios, it may produce unintuitive conclusions in more complex examples. Moreover, an eve... |

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Citation Context ...on worlds. In detail, they characterize classical model-theoretic entailment under preferential structures [74, 47], infinitesimal probabilities [1, 69], possibility measures [21], and world rankings =-=[75, 39]-=-. They also characterize an entailment relation based on conditional objects [22]. A survey of all these relationships is given in [9]. Recently, Friedman and Halpern [28] showed that many approaches ... |

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Citation Context ...f entailment has been introduced by Lehmann [54, 56]. This notion of entailment is equivalent to entailment in system ¥ by Pearl [70], to the least specific possibility entailment by Benferhat et al.=-= [8]-=-, and to a conditional (modal) logic-based entailment by Lamarre [53]. Finally, mainly in order to solve problems with property inheritance lukasiewicz.tex; 1/09/2001; 20:56; p.3s4 Thomas Lukasiewicz ... |

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Citation Context ...dszmidt 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 represent interval restrictions for ... |

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Citation Context ...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 enta... |

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Citation Context ...nsatisfiabilities). In the rest of this paper, we will focus on the second idea. 5. Probabilistic Default Reasoning In this section, we extend the classical notions of entailment in system ¥ by Pearl=-= [70, 40], Le-=-hmann’s lexicographic entailment [55], and Geffner’s conditional entailment [32, 33] to probabilistic default theories and knowledge lukasiewicz.tex; 1/09/2001; 20:56; p.22sProbabilistic Default R... |

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Citation Context ...ith probabilities. Spohn [75] briefly sketches the concept of probabilified OCFs, which associate with each proposition an ordered � pair � consisting of an ordinal and ����� a real in .=-= Blume et al. [13] define lex-=-icographic probability systems, which are sequences of probability measures. 10. Further Applications In this section, we discuss further applications of our new notions of § -, ����� -, an... |

81 | Plausibility measures and default reasoning
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Citation Context ...[21], and world rankings [75, 39]. They also characterize an entailment relation based on conditional objects [22]. A survey of all these relationships is given in [9]. Recently, Friedman and Halpern =-=[28]-=- showed that many approaches describe the same notion of inference, since they are all expressible as plausibility measures. Mainly to solve problems with irrelevant information, the notion of rationa... |

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Citation Context ...onal constraints, which can be roughly divided into approaches that use the model-theoretic notion of logical entailment [20, 25, 3, 45, 76, 24, 29, 43, 65, 62, 59] (which can be traced back to Boole =-=[14]) -=-and those that are based on entailment under de Finetti’s notion of coherence and its generalizations (see especially the work by Coletti [18] and Gilio [36]). However, we will see that neither logi... |

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Citation Context ...ty inheritance lukasiewicz.tex; 1/09/2001; 20:56; p.3s4 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... |

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Citation Context ... knowledge about a set of individuals and some subjective knowledge about a particular individual. Reference-class reasoning goes back to Reichenbach [73] and was further refined especially by Kyburg =-=[48, 49, 50, 51] -=-and Pollock [72]. See also the work by Loui [58] and Kyburg and Murtezao˘glu [52] for its implementations, and the work by Ng [66] for an important application in empirical deductive databases. The m... |

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Citation Context ...babilistic subjective input knowledge use especially minimum cross-entropy to combine statistical and subjective probability distributions (see in particular the work by Bacchus et al. [4] and Jaeger =-=[44]-=-). Other less closely related work concerns the idea of combining ranking functions with probabilities. Spohn [75] briefly sketches the concept of probabilified OCFs, which associate with each proposi... |

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Citation Context ...ities [1, 69], possibility measures [21], and world rankings [75, 39]. They also characterize an entailment relation based on conditional objects [22]. A survey of all these relationships is given in =-=[9]-=-. Recently, Friedman and Halpern [28] showed that many approaches describe the same notion of inference, since they are all expressible as plausibility measures. Mainly to solve problems with irreleva... |

70 |
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Citation Context ...onditional constraints [62]. There is extensive work on reasoning about conditional constraints, which can be roughly divided into approaches that use the model-theoretic notion of logical entailment =-=[20, 25, 3, 45, 76, 24, 29, 43, 65, 62, 59] (-=-which can be traced back to Boole [14]) and those that are based on entailment under de Finetti’s notion of coherence and its generalizations (see especially the work by Coletti [18] and Gilio [36])... |

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Citation Context ...onditional constraints [62]. There is extensive work on reasoning about conditional constraints, which can be roughly divided into approaches that use the model-theoretic notion of logical entailment =-=[20, 25, 3, 45, 76, 24, 29, 43, 65, 62, 59] (-=-which can be traced back to Boole [14]) and those that are based on entailment under de Finetti’s notion of coherence and its generalizations (see especially the work by Coletti [18] and Gilio [36])... |

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Citation Context ...hard in the literal-Horn case. � §������������ � 9. Related Work ��������� , a probaThe work closest in spirit to this paper is perhaps the one by Bacchu=-=s et al. [6] in which the random worlds method [41]-=- is used for reasoning from statistical lukasiewicz.tex; 1/09/2001; 20:56; p.39s40 Thomas Lukasiewicz knowledge and degrees of belief. However, even though Bacchus et al. have a much richer first-orde... |

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Citation Context ...der uncertainty measures on worlds. In detail, they characterize classical model-theoretic entailment under preferential structures [74, 47], infinitesimal probabilities [1, 69], possibility measures =-=[21]-=-, and world rankings [75, 39]. They also characterize an entailment relation based on conditional objects [22]. A survey of all these relationships is given in [9]. Recently, Friedman and Halpern [28]... |

44 |
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Citation Context ... The most influential theory of belief revision is the one by Alchourrón, Gärdenfors, and Makinson (AGM) [2]. As pointed out lukasiewicz.tex; 1/09/2001; 20:56; p.41s42 Thomas Lukasiewicz by Boutilie=-=r [16] -=-and Goldszmidt and Pearl [40], belief revision and default reasoning from conditional knowledge bases are closely related. In particular, Boutilier [16] showed that the ranking in system ¥ correspond... |

44 |
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Citation Context ... knowledge about a set of individuals and some subjective knowledge about a particular individual. Reference-class reasoning goes back to Reichenbach [73] and was further refined especially by Kyburg =-=[48, 49, 50, 51] -=-and Pollock [72]. See also the work by Loui [58] and Kyburg and Murtezao˘glu [52] for its implementations, and the work by Ng [66] for an important application in empirical deductive databases. The m... |

38 |
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Citation Context ... ������� � � � ������������������������������� � � ��� � ����� � � ����� ���=-=� � � ������������������� ��� ��� ��������������������� � Observe that � is literal-Horn. As shown by Georgakopoulos et al. [34], � is 3-colorable iff ����-=-��� is satisfiable. It is now easy to see that ����� is satisfiable iff � is � -consistent. In summary, � is 3-colorable iff � is � -consistent. ¡ . lukasiewicz.tex; 1/09/2001... |

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Citation Context ...lment 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 th-=-is paper, we combine the notion of logical entailment from conditional constraints with the notions of entailment in Pearl’s system ¥ , Lehmann’s lexicographic entailment, and Geffner’s conditi... |

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Citation Context ...eferential structures [74, 47], infinitesimal probabilities [1, 69], possibility measures [21], and world rankings [75, 39]. They also characterize an entailment relation based on conditional objects =-=[22]-=-. A survey of all these relationships is given in [9]. Recently, Friedman and Halpern [28] showed that many approaches describe the same notion of inference, since they are all expressible as plausibi... |

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Citation Context ... 29, 43, 65, 62, 59] (which can be traced back to Boole [14]) and those that are based on entailment under de Finetti’s notion of coherence and its generalizations (see especially the work by Colett=-=i [18]-=- and Gilio [36]). However, we will see that neither logical entailment nor the weaker notion of entailment under coherence show the behavior that is desired in reference-class reasoning. A more promis... |

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Citation Context ...onditional constraints [62]. There is extensive work on reasoning about conditional constraints, which can be roughly divided into approaches that use the model-theoretic notion of logical entailment =-=[20, 25, 3, 45, 76, 24, 29, 43, 65, 62, 59] (-=-which can be traced back to Boole [14]) and those that are based on entailment under de Finetti’s notion of coherence and its generalizations (see especially the work by Coletti [18] and Gilio [36])... |

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Citation Context ...is � -consistent, otherwise nil. It is based on Lemma 5.2 (b), and it is essentially a reformulation of an algorithm for deciding � -consistency in classical default reasoning by Goldszmidt and Pe=-=arl [38]. Algorithms-partition (essentially [38]) Input: Probabilistic default theory ��-=-����¢�������¨ . , if � � is -consistent, otherwise nil. Output:s-partition ¢�������������������¨ of � 1. if ����� then ret... |

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Citation Context ... knowledge about a set of individuals and some subjective knowledge about a particular individual. Reference-class reasoning goes back to Reichenbach [73] and was further refined especially by Kyburg =-=[48, 49, 50, 51] -=-and Pollock [72]. See also the work by Loui [58] and Kyburg and Murtezao˘glu [52] for its implementations, and the work by Ng [66] for an important application in empirical deductive databases. The m... |

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Citation Context ...ight - ����� ����� � -� ��� ������� consequence of � � � ��� � � � � ���������¡� , denoted , ��� iff . 4.3. R=-=ELATIONSHIP BETWEEN LOGICAL ENTAILMENT AND ENTAILMENT UNDER COHERENCE It has been shown in [12, 11] t-=-hat g-coherence is stronger than satisfiability, whereas g-coherent entailment is weaker than logical entailment. That is, every g-coherent finite set of strict logical constraints � and defaults is... |

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Citation Context ...ight - ����� ����� � -� ��� ������� consequence of � � � ��� � � � � ���������¡� , denoted , ��� iff . 4.3. R=-=ELATIONSHIP BETWEEN LOGICAL ENTAILMENT AND ENTAILMENT UNDER COHERENCE It has been shown in [12, 11] t-=-hat g-coherence is stronger than satisfiability, whereas g-coherent entailment is weaker than logical entailment. That is, every g-coherent finite set of strict logical constraints � and defaults is... |

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Citation Context ..., 59] (which can be traced back to Boole [14]) and those that are based on entailment under de Finetti’s notion of coherence and its generalizations (see especially the work by Coletti [18] and Gili=-=o [36]-=-). However, we will see that neither logical entailment nor the weaker notion of entailment under coherence show the behavior that is desired in reference-class reasoning. A more promising idea takes ... |

19 | Default reasoning from conditional knowledge bases: Complexity and tractable cases
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Citation Context ...II. Roughly speaking, the complexity of probabilistic default reasoning with conditional constraints is a combination of the complexity of classical default reasoning from conditional knowledge bases =-=[26] and with-=- the complexity of classical probabilistic reasoning with conditional constraints [64]. ¦ £ � � and ¤ is an atom. Table V. Complexity of CONSISTENCY. general case literal-Horn case � -consist... |

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Citation Context ... would be interesting to allow for variable strengths on statistical knowledge (as a generalization of system ¥�� by Goldszmidt and Pearl [40] or of the maximum entropy approach by Bourne and Par=-=sons [15]-=-). Another interesting question is whether there are tractable cases of reasoning under the new formalisms. In particular, can we gain tractability of deciding lukasiewicz.tex; 1/09/2001; 20:56; p.42s... |

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Citation Context ... for purely probabilistic subjective input knowledge use especially minimum cross-entropy to combine statistical and subjective probability distributions (see in particular the work by Bacchus et al. =-=[4]-=- and Jaeger [44]). Other less closely related work concerns the idea of combining ranking functions with probabilities. Spohn [75] briefly sketches the concept of probabilified OCFs, which associate w... |

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Citation Context ...conditional knowledge base is given by the set of all plausibly entailed defaults. Several different semantics for conditional knowledge bases have been proposed in the literature (see the surveys in =-=[23, 71, 30]-=-). The desired behavior has been characterized by benchmark examples and by general nonmonotonic properties. 2.2.2. Benchmark Examples We now describe some important benchmark examples for default rea... |

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Citation Context ...ted by probability distributions over a set of possible worlds. This probabilistic framework is well-accepted in the literature (see especially the survey on sentential probability logic by Hailperin =-=[42]-=-; see also the important work by Nilsson [67], Fagin et al. [27], and Frisch and Haddawy [29]). Some technical concepts are adopted from [59, 62]. We then define probabilistic default theories and pro... |

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Citation Context ...s, we conclude “Tweety flies with a probability of at most 0.05”. Reference-class reasoning, however, is not uncontroversial in the literature (see especially the work by Levi [57] and Bacchus et =-=al. [5, 6]-=-). In particular, it is often argued that even though reference-class reasoning deals very well with simple scenarios, it may produce unintuitive conclusions in more complex examples. Moreover, an eve... |

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