D. Poole. What the lottery paradox tells us about default reasoning. In Proceedings of the 1st International Conference on Principles of Knowledge Representation and Reasoning (KR'89), pages 333--340, 1989.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....classical negation, in which they present a connection between answer sets of a program and its corresponding default extensions. It is often said that the difficulty of Reiter s default logic arises when one considers default reasoning with disjunctive information. Using a popular example from [Poo89], when we consider default rules: lh usable :lh broken lh usable ; rh usable :rh broken rh usable with a disjunctive formula: lh broken rh broken; they have a single extension containing both lh usable and rh usable, which is unintuitive. From the viewpoint of disjunctive logic ....

Poole, D., What the Lottery Paradox Tells us about Default Reasoning, Proc. 1st Int. Conf. on Principles of Knowledge Representation and Reasoning, 333-340, 1989.

....the agent can progress from qualitative commonsense reasoning to more exact quantitative evidential reasoning: the same logic is used, it is just the amount and quality of information that changes. The fact that the conclusions are graded also means that the formalism avoids the lottery paradox [17, 18]. For example, we may know that birds typically have properties P 1 through P n , but also that no bird possesses all of these properties. Other mechanisms of default reasoning are paralyzed by this situation, as if they infer each of the P i they will also infer the conjunction of the P i . In ....

David Poole. What the lottery paradox tells us about default reasoning. In Ronald J. Brachman, Hector J. Levesque, and Raymond Reiter, editors, Proceedings of the First Conference on Principles of Knowledge Representation and Reasoning, pages 333--340. Morgan Kaufmann, San Mateo, California, 1989. 13

....belief sets of an agent reasoning with this theory. All its desirable properties notwithstanding, there are situations where default logic of Reiter produces counterintuitive results. In particular, this logic does not handle well incomplete information given in the form of disjunctive clauses [ Poole, 1989; Brewka, 1991a; Gelfond et al. 1991; Mikitiuk and Truszczy nski, 1993 ] To remedy this, several modifications of default logic were proposed: disjunctive default logic [ Gelfond et al. 1991 ] cumulative default logic [ Brewka, 1991a ] constrained default logic [ Schaub, 1992 ] and ....

D. Poole. What the lottery paradox tells us about default reasoning. In Proceedings of the 2nd conference on principles of knowledge representation and reasoning, KR '89, pages 333--340, San Mateo, CA., 1989. Morgan Kaufmann.

....on the notion of extensions a skeptical inference relation for DL can be introduced where a formula is considered derivable from a default theory T iff it is contained in all extensions of T . Unfortunately, Reiter s logic also has its drawbacks, as has been discussed in various papers, e.g. [13, 9, 2, 5]: 1. Existence of extensions is not guaranteed. 2. The consistency conditions (justifications) of defaults applied within one Reiter extension are not jointly consistent with the generated extension. Reiter s fixed point definition only guarantees that each consistency condition in isolation is ....

....of defaults applied within one Reiter extension are not jointly consistent with the generated extension. Reiter s fixed point definition only guarantees that each consistency condition in isolation is consistent with the extension. This leads to counterintuitive conclusions as discussed in [13]. 3. The use of inference rules as defaults makes it impossible to reason by cases in DL. Consider the following example (a) Italian:Likes Wine=Likes W ine (b) F rench:Likes Wine=Likes W ine (c) Italian F rench Intuitively one would expect to derive Likes W ine since no matter whether the ....

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Poole, David, What the Lottery Paradox Tells Us About Default Reasoning, Proceedings First International Conference on Principles of Knowledge Representation and Reasoning, Toronto (1989) 333-340.

....aspect of abductive reasoning is related to prediction. As a speci c form of prediction, it is about completing, or enriching, the initially speci ed, incomplete information. Example 3.2. Consider the popular, broken hand example originally discussed in the context of default reasoning [27]: We know either the left hand is broken or the right hand is broken, and in general, a hand is usable if not broken. The given information is incomplete as we don t know which hand is broken and which is not (perhaps both could have been broken) For the purpose of demonstrating the point of ....

D. Poole. What the lottery paradox tells us about default reasoning. In Proc. KR '89, pages 333-340, 1989.

....Databases ; EC US project DEUS EX MACHINA ; and a MURST grant (40 share) under the project Sistemi formali e strumenti per basi di dati evolute. 1 the attractions of disjunctive logic programming is its ability to naturally model incomplete knowledge [3, 37] Example 1. 1 (modified from [3, 47]) Consider the following situation: 1) we just saw Max with one broken arm but do not remember which; 2) we know that Max writes with his left hand, so he can write if his left arm is unbroken. The question is, Can Max write or not Because of the uncertainty deriving from our incomplete ....

Poole, D. (1989), What the lottery paradox tells us about default reasoning, in "Proc. of the First Int'l Conf. on Principles of Knowledge Representation and Reasoning," R. Brachman, H. Levesque and R. Reiter editors, pp. 333--340.

....Furthermore, he imposes a different condition on applicability. The main idea is that of defining commitment to the assumptions, namely if a default is applied with justification fi then another default with justification :fi cannot be applied. An illustrative example on commitment is given in [Poo89c]; we reconsider that problem in intuitionistic default logic in Section 5. A further variant of cumulative default logic which is not semi monotonic has been proposed by Giordano and Martelli in [GM94b] In addition, Gottlob and Mingyi 5 propose a finite characterization of CDL extensions in ....

.... 1i : T a; F: p oe p h1:1; 1i : T a; T: p;Fp h1:1; 1i : T a; F:p; Fp h1:1:1; 1i : T a; Tp open T a; Tp open FAILURE exit with D 0 and the open branches of T Gamma 0 Figure 4 Default Tableau for a Reiter s default theory in intuitionism Consider the example of the Broken arms, given in [Poo89c]. The pair hW; Di is as follows: W = broken(lef t) broken(right) and D = broken(x) broken(x) usable(x) broken(x) usable(x) Either broken(x) or :broken(x) hence either x = left or x = right has to be proved in order to apply the default. This default theory, in intuitionistic logic ....

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D. Poole. What the lottery paradox tells us about default reasoning. In Proceedings of the 1st International Conference on Principles of Knowledge Representation and Reasoning (KR'89), pages 333--340, 1989.

....where the closed world assumption (CWA) Rei78] can not be automatically assumed. In Section 4 we consider disjunctive logic programs where disjunctions are allowed in the heads of the rules of the program. We formalize two examples from the literature. In particular we consider an example from [Poo89] that was used to demonstrate the difficulties associated with representing disjunctive information in Reiter s default logic. We also discuss other semantics of general logic programs and review a method to compute the answer set semantics of a disjunctive logic program. In Section 5 we show the ....

....rule. 35 Pi 0 with the above three rules has the answer sets fp(a) ab(r; a) ab(r; b) p(c)g and fp(b) ab(r; a) ab(r; b) p(c)g. Its answers to the queries p(a) p(b) and p(c) is the same as Pi 2 s while its answer to the query p(a) p(b) is unknown. 2 The next example was used in [Poo89] to demonstrate difficulties with representing disjunctive information in Reiter s default logic. It is worth noting that it has a natural representation in the language of disjunctive programs. Example 4.2 Consider the following story [Poo89] Normally, a person s left arm is usable, but a ....

[Article contains additional citation context not shown here]

D. Poole. What the lottery paradox tells us about default reasoning. In R. Brachman, H. Levesque, and R. Reiter, editors, Proc. of the First Int'l Conf. on Principles of Knowledge Representation and Reasoning, pages 333--340, 1989. 86

....model all possible realities described by a default theory. All its desirable properties notwithstanding, there are situations where default logic of Reiter is not easily applicable. In particular, default logic does not handle well incomplete information given in the form of disjunctive clauses [9, 2, 4, 6]. To remedy this, several modifications of default logic were proposed: disjunctive default logic [4] cumulative default logic [2] constrained default logic [11] and rational default logic [6, 8] The first system introduces a new disjunction operator to handle effective disjunction. The ....

D. Poole. What the lottery paradox tells us about default reasoning. In Proceedings of the 2nd conference on principles of knowledge representation and reasoning, KR '89, pages 333--340, San Mateo, CA., 1989. Morgan Kaufmann.

....in all possible realities described by a default theory. All its desirable properties notwithstanding, there are situations where default logic of Reiter is not easily applicable. In particular, default logic does not handle well incomplete information given in the form of disjunctive clauses [9, 2, 4, 7]. To remedy this, several modifications of default logic were proposed: disjunctive default logic [4] cumulative default logic [2] constrained default logic [11] and rational default logic [7] The first system introduces a new disjunction operator to handle effective disjunction. The latter ....

D. Poole. What the lottery paradox tells us about default reasoning. In Proceedings of the 2nd conference on principles of knowledge representation and reasoning, KR '89, pages 333--340, San Mateo, CA., 1989. Morgan Kaufmann.

....aspect of abductive reasoning is related to prediction. As a specific form of prediction, it is about completing, or enriching, the initially specified, incomplete information. Example 3.2. Consider the popular, broken hand example originally discussed in the context of default reasoning [27]: We know either the left hand is broken or the right hand is broken, and in general, a hand is usable if not broken. The given information is incomplete as we don t know which hand is broken and which is not (perhaps both could have been broken) For the purpose of demonstrating the point of ....

D. Poole. What the lottery paradox tells us about default reasoning. In Proc. KR '89, pages 333--340, 1989.

....to statements in FOL, representing an agent s degree of belief in certain situations obtaining. This gives a way of thinking about the usual birds normally fly as a probabilistic statement. We illustrate this method in the resolution of an anomaly with standard default reasoning, due to Poole [9]. The paper is organized as follows. In section 2 we cover the basics of domain theory and information systems, introduce our non monotonic generalization, and state a representation theorem for default domains. In Section 3 we show how to interpret first order positive logic using default models. ....

....we will mostly be interested in finite default models for first order logic. 3.1 Syntax and Semantics In this workshop paper, we give just one application of constraints in first order default model theory. This consists of a probabilistic resolution of an anomaly in default logic, due to Poole [9]. We begin, though, with the syntax and semantics of first order logic itself. Here is the official syntax: t j f j R(v 1 ; v n ) j :R(v 1 ; vn ) j j j 9x j 8x where R(v 1 ; vn ) is an atomic formula and the v i are either variables in some set X, or ....

[Article contains additional citation context not shown here]

D. L. Poole. What the lottery paradox tells us about default reasoning. In Proceedings of First Annual Conference on Knowledge Representation. Morgan Kaufmann, 1989.

....where the closed world assumption (CWA) Rei78] can not be automatically assumed. In Section 4 we consider disjunctive logic programs where disjunctions are allowed in the heads of the rules of the program. We formalize two examples from the literature. In particular we consider an example from [Poo89] that was used to demonstrate the difficulties associated with representing disjunctive information in Reiter s default logic. We also discuss 1 Normative or normic statements [Scr59, Scr63] frequently involve terms such as naturally , normally , typically , tendency , ought , should and ....

....of the first rule. 5 0 with the above three rules has the answer sets fp(a) ab(r; a) ab(r; b) p(c)g and fp(b) ab(r; a) ab(r; b) p(c)g. Its answers to the queries p(a) p(b) and p(c) is the same as 5 2 s while its answer to the query p(a) p(b) is unknown. 2 The next example was used in [Poo89] to demonstrate difficulties with representing disjunctive information in Reiter s default logic. It is worth noting that it has a natural representation in the language of disjunctive programs. Example 4.2 Consider the following story [Poo89] Normally, a person s left arm is usable, but a ....

[Article contains additional citation context not shown here]

D. Poole. What the lottery paradox tells us about default reasoning. In R. Brachman, H. Levesque, and R. Reiter, editors, Proc. of the First Int'l Conf. on Principles of Knowledge Representation and Reasoning, pages 333--340, 1989.

....effort to make default constraints into hard ones. But these systems of defaults can be giving a lot of non information and even false information in a probabilistic sense. They, and the default constraints themselves, should be undergoing revision. We illustrate this by considering an anomaly of Poole (Poole 1989), related to the so called lottery paradox (Jr. 1961) and we consider a more complex case involving the well known (folk ) Nixon Diamond. To this end, we introduce a notion both of non monotonic consequence j which can be used to state default constraints, and, in the finite case, a ....

....to statements in FOL, representing an agent s degree of belief in certain situations obtaining. This gives a way of thinking about the usual birds normally fly as a pseudo probabilistic statement. We illustrate this method in the resolution of an anomaly with standard default reasoning, due to Poole (Poole 1989. The paper is organized as follows. In section 3 we cover the basics of domain theory and information systems, introduce our non monotonic generalization, and state a representation theorem for default domains. In Section 4 we show how to interpret first order positive logic using default ....

[Article contains additional citation context not shown here]

Poole, D. L. 1989. What the Lottery Paradox Tells us about Default Reasoning.

....classification noise with error rate j 1=2 [Kearns, 1993] world knowledge and some subset of the default assumptions, and at the same time support our intuition about a plausible conclusion. Attempts to represent and reason with defaults have encountered many problems (e.g. Neufeld, 1989; Poole, 1989; Geffner, 1990] In many cases, reasoning with acceptable defaults lead to unacceptable conclusions. Problems occur whenever defaults interact, and can be characterized frequently as problems of distinguishing good defaults from bad ones. But, reasons for deciding between good and bad ....

D. Poole. What the lottery paradox tells us about default reasoning. In Proceedingsof the International Conference on the Principles of Knowledge Representation and Reasoning, pages 333--340, 1989.

....information to an element of a domain, or to a disjunction of such elements. One might expect that a standard version of default feature logic would do the job. This does not work, though. It turns out that we run into the or problem considered by workers in default logic (see, e.g. Poole [19], or Gelfond et al. 6] For example, consider the default rules p : q q and r : q q : If we start with the formula p r then neither precondition of the rules can be established (by proof) so neither is applicable; but it seems that we should be able to add the information q (conjunctively) ....

....Diamond it seems that such extensions are what is wanted. More generally, we have indicated through the use of update defaults a universal way of adding disjunctive information to an element or elements of a domain which does not suffer from the reasoning by cases problem mentioned by Poole [19]. We plan to apply this idea in other Scott domains besides the domain of feature structures. A primary candidate is the domain theoretic model theory for first order logic studied in [22] As problems needing further research, we should mention that our semantics for defaults yields a notion of ....

D. L. Poole. What the lottery paradox tells us about default reasoning. In Proceedings of First Annual Conference on Knowledge Representation. Morgan Kaufmann, 1989.

....to be the major cause. Similar problems exist in approaches to common sense reasoning that are not model based. As an example, I shall discuss the FOG system for order of magnitude reasoning (Raiman, 1986) The problem which surfaces in this context is intimately related to the lottery paradox (Poole, 1989), which, in turn, can be interpreted as an instance of the complications to the naive negation as failure rule in the face of disjunctive logic formulas (Reiter, 1978; Gelfond et al. 1989) All in all, the problem seems to be pervasive. Though the contexts of the approaches investigated in this ....

.... breaks down (the transitivity of Vo becomes a useless inference rule) in the context of the obligation to reduce the infinite resolution of the distance between real numbers to the binary resolution of a predicate (Vo) In this respect, the problem with FOG is very similar to the lottery paradox (Poole, 1989): If we perform assumption based reasoning about the changes of winning the prize in a lottery, using the convention that we assume that we win the prize if the chance of doing so exceeds 0:5, we may safely assume that buying one from 10; 000 available tickets will not render us the prize. We may ....

Poole D. (1989). What the lottery paradox tells us about default reasoning. In: Levesgue H.J., Brachman R.J., Reiter R. (Eds.), Proceedings First International Conference on Principles of Knowledge Representation and Reasoning , pp. 189-197.

....semantics of disjunctive and extended disjunctive databases. 1 INTRODUCTION In this paper we generalize the theory of default reasoning developed by Reiter [Rei80] The generalization is motivated by a difficulty encountered in attempts to use defaults in the presence of disjunctive information [Poo89]. The difficulty has to do with the difference between a default theory with two extensions one containing a sentence ff, the other a sentence fi and the theory with a single extension, containing the disjunction ff fi. This difficulty was also observed by Lin and Shoham in [LS90] They ....

....conversely. Formally, we have the following theorem Theorem 2.3 A set of sentences E is an extension for a default theory D if and only if E is the minimal set E 0 closed under provability in propositional calculus and under the rules from D E . 3 POOLE S EXAMPLE The following example from [Poo89] illustrates a difficulty that arises in some attempts to use Reiter s formalism in the presence of disjunctive information. By default, people s left arms are usable, but a person with a broken left arm is an exception, and similarly for the right arms. One way to express this in Reiter s ....

D. Poole. What the lottery paradox tells us about default reasoning. In Principles of Knowledge Representation and Reasoning, pages 333--340, San Mateo, CA., 1989. Morgan Kaufmann.

....An interesting aspect of abductive reasoning is about prediction. As a specific form of prediction, it is about completing, or enriching, the initially specified, incomplete information. Example 2. Consider the popular, broken hand example originally discussed in the context of default reasoning [23]: We know either the left hand is broken or the right hand is broken, and in general, a hand is usable if not broken. The given information is incomplete as we don t know which hand is broken and which is not (perhaps both could have been broken) For the purpose of demonstrating the point of ....

D. Poole. What the lottery paradox tells us about default reasoning. In Proc. KR '89, pages 333--340, 1989.

....[ Reiter, 1980 ] that normal defaults are expressive enough for most common sense applications. However, on closer inspection, default rules that appear intuitively reasonable in isolation can give rise to unintuitive results when taken together [ Reiter and Criscuolo, 1981; Lukaszewicz, 1985; Poole, 1989 ] The facts and default rules may interact in unexpected ways, and result in no extension or unwanted multiple extensions. We argue that this problem is to be expected in the default logic framework. Consider the following canonical normal default theory. Example 3 Delta = hD; F i, where D = ....

David Poole. What the lottery paradox tells us about default reasoning. In Proceedings of the First International Conference on Principles of Knowledge Representation and Reasoning, pages 333--340, 1989.

....to a dialectical view of prediction which can be exploited to implement membership in every extension. 2. 4 Breaking Conventions If we equate defaults with conventions, as exemplified in Autoepistemic logic [Moore85] it is reasonable that multiple extensions indicate a bug in the knowledge base [Poole89] The convention view of a default says that if there is an exception to a default it must be explicitly listed. If there are multiple extensions, we should debug the knowledge base rather than solve the multiple extension problem. If we can explain p and explain q, where p and q are mutually ....

D. Poole, "What the lottery paradox tells us about default reasoning ", to appear First International Conference on the Principles of Knowledge Representation and Reasoning, Toronto, May 1989.

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D. Poole. What the lottery paradox tells us about default reasoning. In Proceedings of the 1st International Conference on Principles of Knowledge Representation and Reasoning (KR'89), pages 333--340, 1989.

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D.L. Poole. What the lottery paradox tells us about default reasoning (extended abstract). In Proceedings of the First International Conference on the Principles of Knowledge Representation and Reasoning, Toronto, Ont., 1989.

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D.L. Poole. What the lottery paradox tells us about default reasoning (extended abstract). In Proceedings of the First International Conference on the Principles of Knowledge Representation and Reasoning, Toronto, Ont., 1989.

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D. Poole. What the lottery paradox tells us about default reasoning. Proc. KR'89, pp. 333-340, 1989.

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