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41
An Analysis of FirstOrder Logics of Probability
 Artificial Intelligence
, 1990
"... : We consider two approaches to giving semantics to firstorder logics of probability. The first approach puts a probability on the domain, and is appropriate for giving semantics to formulas involving statistical information such as "The probability that a randomly chosen bird flies is greater than ..."
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Cited by 271 (18 self)
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: We consider two approaches to giving semantics to firstorder logics of probability. The first approach puts a probability on the domain, and is appropriate for giving semantics to formulas involving statistical information such as "The probability that a randomly chosen bird flies is greater than .9." The second approach puts a probability on possible worlds, and is appropriate for giving semantics to formulas describing degrees of belief, such as "The probability that Tweety (a particular bird) flies is greater than .9." We show that the two approaches can be easily combined, allowing us to reason in a straightforward way about statistical information and degrees of belief. We then consider axiomatizing these logics. In general, it can be shown that no complete axiomatization is possible. We provide axiom systems that are sound and complete in cases where a complete axiomatization is possible, showing that they do allow us to capture a great deal of interesting reasoning about prob...
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 215 (21 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.
Probabilistic Logic Programming
, 1992
"... Of all scientific investigations into reasoning with uncertainty and chance, probability theory is perhaps the best understood paradigm. Nevertheless, all studies conducted thus far into the semantics of quantitative logic programming (cf. van Emden [51], Fitting [18, 19, 20], Blair and Subrahmanian ..."
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Cited by 133 (7 self)
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Of all scientific investigations into reasoning with uncertainty and chance, probability theory is perhaps the best understood paradigm. Nevertheless, all studies conducted thus far into the semantics of quantitative logic programming (cf. van Emden [51], Fitting [18, 19, 20], Blair and Subrahmanian [5, 6, 49, 50], Kifer et al [29, 30, 31]) have restricted themselves to nonprobabilistic semantical characterizations. In this paper, we take a few steps towards rectifying this situation. We define a logic programming language that is syntactically similar to the annotated logics of [5, 6], but in which the truth values are interpreted probabilistically. A probabilistic model theory and fixpoint theory is developed for such programs. This probabilistic model theory satisfies the requirements proposed by Fenstad [16] for a function to be called probabilistic. The logical treatment of probabilities is complicated by two facts: first, that the connectives cannot be interpreted truth function...
Firstorder probabilistic models for coreference resolution
 In HLT/NAACL
, 2007
"... Traditional noun phrase coreference resolution systems represent features only of pairs of noun phrases. In this paper, we propose a machine learning method that enables features over sets of noun phrases, resulting in a firstorder probabilistic model for coreference. We outline a set of approximat ..."
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Cited by 57 (17 self)
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Traditional noun phrase coreference resolution systems represent features only of pairs of noun phrases. In this paper, we propose a machine learning method that enables features over sets of noun phrases, resulting in a firstorder probabilistic model for coreference. We outline a set of approximations that make this approach practical, and apply our method to the ACE coreference dataset, achieving a 45 % error reduction over a comparable method that only considers features of pairs of noun phrases. This result demonstrates an example of how a firstorder logic representation can be incorporated into a probabilistic model and scaled efficiently. 1
Decidability and Expressiveness for FirstOrder Logics of Probability
 Information and Computation
, 1989
"... We consider decidability and expressiveness issues for two firstorder logics of probability. In one, the probability is on possible worlds, while in the other, it is on the domain. It turns out that in both cases it takes very little to make reasoning about probability highly undecidable. We show t ..."
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Cited by 40 (6 self)
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We consider decidability and expressiveness issues for two firstorder logics of probability. In one, the probability is on possible worlds, while in the other, it is on the domain. It turns out that in both cases it takes very little to make reasoning about probability highly undecidable. We show that when the probability is on the domain, if the language contains only unary predicates then the validity problem is decidable. However, if the language contains even one binary predicate, the validity problem is \Pi 2 1 complete, as hard as elementary analysis with free predicate and function symbols. With equality in the language, even with no other symbol, the validity problem is at least as hard as that for elementary analysis, \Pi 1 1 hard. Thus, the logic cannot be axiomatized in either case. When we put the probability on the set of possible worlds, the validity problem is \Pi 2 1 complete with as little as one unary predicate in the language, even without equality. With equalit...
On A Theory of Probabilistic Deductive Databases
 THEORY AND PRACTICE OF LOGIC PROGRAMMING
, 2001
"... We propose a framework for modeling uncertainty where both belief and doubt can be given independent, firstclass status. We adopt probability theory as the mathematical formalism for manipulating uncertainty. An agent can express the uncertainty in her knowledge about a piece of information in the ..."
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Cited by 27 (0 self)
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We propose a framework for modeling uncertainty where both belief and doubt can be given independent, firstclass status. We adopt probability theory as the mathematical formalism for manipulating uncertainty. An agent can express the uncertainty in her knowledge about a piece of information in the form of a confidence level, consisting of a pair of intervals of probability, one for each of her belief and doubt. The space of confidence levels naturally leads to the notion of a trilattice, similar in spirit to Fitting's bilattices. Intuitively, the points in such a trilattice can be ordered according to truth, information, or precision. We develop a framework for probabilistic deductive databases by associating confidence levels with the facts and rules of a classical deductive database. While the trilattice structure offers a variety of choices for defining the semantics of probabilistic deductive databases, our choice of semantics is based on the truthordering, which we find to be closest to the classical framework for deductive databases. In addition to proposing a declarative semantics based on valuations and an equivalent semantics based on fixpoint theory, we also propose a proof procedure and prove it sound and complete. We show that while classical Datalog query programs have a polynomial time data complexity, certain query programs in the probabilistic deductive database framework do not even terminate on some input databases. We identify a large natural class of query programs of practical interest in our framework, and show that programs in this class possess polynomial time data complexity, i.e. not only do they terminate on every input database, they are guaranteed to do so in a number of steps polynomial in the input database size.
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
A Data Model and Algebra for Probabilistic Complex Values
 Annals of Mathematics and Artificial Intelligence
, 2000
"... We present a probabilistic data model for complex values. More precisely, we introduce probabilistic complex value relations, which combine the concept of probabilistic relations with the idea of complex values in a uniform framework. We elaborate a modeltheoretic definition of probabilistic combina ..."
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Cited by 18 (4 self)
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We present a probabilistic data model for complex values. More precisely, we introduce probabilistic complex value relations, which combine the concept of probabilistic relations with the idea of complex values in a uniform framework. We elaborate a modeltheoretic definition of probabilistic combination strategies, which has a rigorous foundation on probability theory. We then define an algebra for querying database instances, which comprises the operations of selection, projection, renaming, join, Cartesian product, union, intersection, and difference. We prove that our data model and algebra for probabilistic complex values generalizes the classical relational data model and algebra. Moreover, we show that under certain assumptions, all our algebraic operations are tractable. We finally show that most of the query equivalences of classical relational algebra carry over to our algebra on probabilistic complex value relations. Hence, query optimization techniques for classical relational algebra can easily be applied to optimize queries on probabilistic complex value relations. Keywords: Complex value databases, probabilistic databases, data model, relational algebra, query languages. AMS Subject classification: Primary 68P15, 68P20; Secondary 68T30, 68T37 1.
On probability distributions over possible worlds
 in Uncertainty in Arti Intelligence 4
, 1990
"... In Probabilistic Logic Nilsson uses the device of a probability distribution over the set of possible worlds to assign probabilities to the sentences of a logical language. In his paper Nilsson concentrated on inference and associated computational issues. This paper, on the other hand, examines the ..."
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Cited by 17 (2 self)
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In Probabilistic Logic Nilsson uses the device of a probability distribution over the set of possible worlds to assign probabilities to the sentences of a logical language. In his paper Nilsson concentrated on inference and associated computational issues. This paper, on the other hand, examines the probabilistic semantics in more detail, particularly for the case of first order languages, and attempts to explain some of the features and limitations of this form of probability logic. It is pointed out that the device of assigning probabilities to logical sentences has certain expressive limitations. In particular, statistical assertions are not easily expressed by such a device. This leads to certain difficulties with attempts to give probabilistic semantics to default reasoning using probabilities assigned to logical sentences. *This research was supported by a PostDoctoral fellowship funded by the U.S. Army Signals Warfare Laboratory. ~I)S11OTICI.A SATFML * A Apprc v d for p':r'c 1 9213691 D~lstribution 1ndted iIUUUUUUII
Probabilistic ABox reasoning: Preliminary results
 In Proceedings DL2005
, 2005
"... Most probabilistic extensions of description logics focus on the terminological apparatus. While some allow for expressing probabilistic knowledge about concept assertions, systems which can express probabilistic knowledge about role assertions have received very little attention as yet. We present ..."
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Cited by 16 (1 self)
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Most probabilistic extensions of description logics focus on the terminological apparatus. While some allow for expressing probabilistic knowledge about concept assertions, systems which can express probabilistic knowledge about role assertions have received very little attention as yet. We present a system PALC which allows us to express degrees of belief in concept and role assertions for individuals. We introduce syntax and semantics for PALC and we define the corresponding reasoning problem. An independence assumption regarding the assertions for different individuals yields additional constraints on the possible interpretations. This considerably reduces the solution space of the PALC reasoning problem. 1