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39
A Logic for Reasoning about Probabilities
 Information and Computation
, 1990
"... We consider a language for reasoning about probability which allows us to make statements such as “the probability of E, is less than f ” and “the probability of E, is at least twice the probability of E,, ” where E, and EZ are arbitrary events. We consider the case where all events are measurable ( ..."
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Cited by 214 (19 self)
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We consider a language for reasoning about probability which allows us to make statements such as “the probability of E, is less than f ” and “the probability of E, is at least twice the probability of E,, ” where E, and EZ are arbitrary events. We consider the case where all events are measurable (i.e., represent measurable sets) and the more general case, which is also of interest in practice, where they may not be measurable. The measurable case is essentially a formalization of (the propositional fragment of) Nilsson’s probabilistic logic. As we show elsewhere, the general (nonmeasurable) case corresponds precisely to replacing probability measures by DempsterShafer belief functions. In both cases, we provide a complete axiomatization and show that the problem of deciding satistiability is NPcomplete, no worse than that of propositional logic. As a tool for proving our complete axiomatizations, we give a complete axiomatization for reasoning about Boolean combinations of linear inequalities, which is of independent interest. This proof and others make crucial use of results from the theory of linear programming. We then extend the language to allow reasoning about conditional probability and show that the resulting logic is decidable and completely axiomatizable, by making use of the theory of real closed fields. ( 1990 Academic Press. Inc 1.
Lifted firstorder probabilistic inference
 In Proceedings of IJCAI05, 19th International Joint Conference on Artificial Intelligence
, 2005
"... Most probabilistic inference algorithms are specified and processed on a propositional level. In the last decade, many proposals for algorithms accepting firstorder specifications have been presented, but in the inference stage they still operate on a mostly propositional representation level. [Poo ..."
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Cited by 88 (7 self)
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Most probabilistic inference algorithms are specified and processed on a propositional level. In the last decade, many proposals for algorithms accepting firstorder specifications have been presented, but in the inference stage they still operate on a mostly propositional representation level. [Poole, 2003] presented a method to perform inference directly on the firstorder level, but this method is limited to special cases. In this paper we present the first exact inference algorithm that operates directly on a firstorder level, and that can be applied to any firstorder model (specified in a language that generalizes undirected graphical models). Our experiments show superior performance in comparison with propositional exact inference. 1
PSHOQ(D): A Probabilistic Extension of SHOQ(D) for Probabilistic Ontologies in the Semantic Web
, 2002
"... Ontologies play a central role in the development of the semantic web, as they provide precise definitions of shared terms in web resources. One important web ontology language is DAML+OIL; it has a formal semantics and a reasoning support through a mapping to the expressive description logic SHOQ ..."
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Cited by 83 (14 self)
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Ontologies play a central role in the development of the semantic web, as they provide precise definitions of shared terms in web resources. One important web ontology language is DAML+OIL; it has a formal semantics and a reasoning support through a mapping to the expressive description logic SHOQ(D) with the addition of inverse roles. In this paper, we present a probabilistic extension of SHOQ(D), called PSHOQ(D), to allow for dealing with probabilistic ontologies in the semantic web. The description logic PSHOQ(D) is based on the notion of probabilistic lexicographic entailment from probabilistic default reasoning. It allows to express rich probabilistic knowledge about concepts and instances, as well as default knowledge about concepts. We also present sound and complete reasoning techniques for PSHOQ(D), which are based on reductions to classical reasoning in SHOQ(D) and to linear programming, and which show in particular that reasoning in PSHOQ(D) is decidable.
Managing Uncertainty and Vagueness in Description Logics for the Semantic Web
, 2007
"... Ontologies play a crucial role in the development of the Semantic Web as a means for defining shared terms in web resources. They are formulated in web ontology languages, which are based on expressive description logics. Significant research efforts in the semantic web community are recently direct ..."
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Cited by 58 (7 self)
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Ontologies play a crucial role in the development of the Semantic Web as a means for defining shared terms in web resources. They are formulated in web ontology languages, which are based on expressive description logics. Significant research efforts in the semantic web community are recently directed towards representing and reasoning with uncertainty and vagueness in ontologies for the Semantic Web. In this paper, we give an overview of approaches in this context to managing probabilistic uncertainty, possibilistic uncertainty, and vagueness in expressive description logics for the Semantic Web.
Probabilistic Logic Programming
 In Proc. of the 13th European Conf. on Artificial Intelligence (ECAI98
, 1998
"... . We present a new approach to probabilistic logic programs with a possible worlds semantics. Classical program clauses are extended by a subinterval of [0; 1] that describes the range for the conditional probability of the head of a clause given its body. We show that deduction in the defined proba ..."
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Cited by 45 (11 self)
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. We present a new approach to probabilistic logic programs with a possible worlds semantics. Classical program clauses are extended by a subinterval of [0; 1] that describes the range for the conditional probability of the head of a clause given its body. We show that deduction in the defined probabilistic logic programs is computationally more complex than deduction in classical logic programs. More precisely, restricted deduction problems that are Pcomplete for classical logic programs are already NPhard for probabilistic logic programs. We then elaborate a linear programming approach to probabilistic deduction that is efficient in interesting special cases. In the best case, the generated linear programs have a number of variables that is linear in the number of ground instances of purely probabilistic clauses in a probabilistic logic program. 1 INTRODUCTION There is already a quite extensive literature on probabilistic propositional logics and their various dialects. The most fa...
Probabilistic Deduction with Conditional Constraints over Basic Events
 J. Artif. Intell. Res
, 1999
"... We study the problem of probabilistic deduction with conditional constraints over basic events. We show that globally complete probabilistic deduction with conditional constraints over basic events is NPhard. We then concentrate on the special case of probabilistic deduction in conditional constrai ..."
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Cited by 44 (30 self)
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We study the problem of probabilistic deduction with conditional constraints over basic events. We show that globally complete probabilistic deduction with conditional constraints over basic events is NPhard. We then concentrate on the special case of probabilistic deduction in conditional constraint trees. We elaborate very efficient techniques for globally complete probabilistic deduction. In detail, for conditional constraint trees with point probabilities, we present a local approach to globally complete probabilistic deduction, which runs in linear time in the size of the conditional constraint trees. For conditional constraint trees with interval probabilities, we show that globally complete probabilistic deduction can be done in a global approach by solving nonlinear programs. We show how these nonlinear programs can be transformed into equivalent linear programs, which are solvable in polynomial time in the size of the conditional constraint trees. 1. Introduction Dealing wit...
ProblemFocused Incremental Elicitation of MultiAttribute Utility Models
, 1997
"... Decision theory has become widely accepted in the AI community as a useful framework for planning and decision making. Applying the framework typically requires elicitation of some form of probability and utility information. While much work in AI has focused on providing representations and tools f ..."
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Cited by 41 (3 self)
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Decision theory has become widely accepted in the AI community as a useful framework for planning and decision making. Applying the framework typically requires elicitation of some form of probability and utility information. While much work in AI has focused on providing representations and tools for elicitation of probabilities, relatively little work has addressed the elicitation of utility models. This imbalance is not particularly justified considering that probability models are relatively stable across problem instances, while utility models may be different for each instance. Spending large amounts of time on elicitation can be undesirable for interactive systems used in lowstakes decision making and in timecritical decision making. In this paper we investigate the issues of reasoning with incomplete utility models. We identify patterns of problem instances where plans can be proved to be suboptimal if the (unknown) utility function satisfies certain conditions. We present an...
Probabilistic Default Reasoning with Conditional Constraints
 ANN. MATH. ARTIF. INTELL
, 2000
"... 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, ..."
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Cited by 35 (20 self)
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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 referenceclass 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...
Active Logics: A Unified Formal Approach to Episodic Reasoning
"... Artificial intelligence research falls roughly into two categories: formal and implementational. This division is not completely firm: there are implementational studies based on (formal or informal) theories (e.g., CYC, SOAR, OSCAR), and there are theories framed with an eye toward implementabili ..."
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Cited by 35 (2 self)
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Artificial intelligence research falls roughly into two categories: formal and implementational. This division is not completely firm: there are implementational studies based on (formal or informal) theories (e.g., CYC, SOAR, OSCAR), and there are theories framed with an eye toward implementability (e.g., predicate circumscription). Nevertheless, formal /theoretical work tends to focus on very narrow problems (and even on very special cases of very narrow problems) while trying to get them "right" in a very strict sense, while implementational work tends to aim at fairly broad ranges of behavior but often at the expense of any kind of overall conceptually unifying framework that informs understanding. It is sometimes urged that this gap is intrinsic to the topic: intelligence is not a unitary thing for which there will be a unifying theory, but rather a "society" of subintelligences whose overall behavior cannot be reduced to useful characterizing and predictive principles.
Probabilistic Logic under Coherence: Complexity and Algorithms
 In Proceedings ISIPTA01
, 2001
"... We study probabilistic logic under the viewpoint of the coherence principle of de Finetti. In detail, we explore the relationship between coherencebased and classical modeltheoretic probabilistic logic. Interestingly, we show that the notions of gcoherence and of gcoherent entailment can be expre ..."
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Cited by 22 (11 self)
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We study probabilistic logic under the viewpoint of the coherence principle of de Finetti. In detail, we explore the relationship between coherencebased and classical modeltheoretic probabilistic logic. Interestingly, we show that the notions of gcoherence and of gcoherent entailment can be expressed by combining notions in modeltheoretic probabilistic logic with concepts from default reasoning. Using these results, we analyze the computational complexity of probabilistic reasoning under coherence. Moreover, we present new algorithms for deciding gcoherence and for computing tight gcoherent intervals, which reduce these tasks to standard reasoning tasks in modeltheoretic probabilistic logic. Thus, efficient techniques for modeltheoretic probabilistic reasoning can immediately be applied for probabilistic reasoning under coherence, for example, column generation techniques. We then describe two other interesting techniques for efficient modeltheoretic probabilistic reasoning in the conjunctive case.