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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 ..."
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Cited by 273 (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...
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...
Decidability and expressiveness for firstorder logics of probability
 Information and Computation
, 1994
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Quantified Constraints under Perturbation
 ARTICLE SUBMITTED TO JOURNAL OF SYMBOLIC COMPUTATION
"... Quantified constraints (i.e., firstorder formulae over the real numbers) are often exposed to perturbations: Constants that come from measurements usually are only known up to certain precision, and numerical methods only compute with approximations of real numbers. In this paper we study the be ..."
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Cited by 16 (11 self)
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Quantified constraints (i.e., firstorder formulae over the real numbers) are often exposed to perturbations: Constants that come from measurements usually are only known up to certain precision, and numerical methods only compute with approximations of real numbers. In this paper we study the behavior of quantified constraints under perturbation by showing that one can formulate the problem of solving quantified constraints as a nested parametric optimization problem followed by one sign computation. Using the fact that minima and maxima are stable under perturbation, but the sign of a real number is stable only for nonzero inputs, we derive practically useful conditions for the stability of quantified constraints under perturbation.
A Logic for Default Reasoning About Probabilities
, 1998
"... A logic is defined that allows to express information about statistical probabilities and about degrees of belief in specific propositions. By interpreting the two types of probabilities in one common probability space, the semantics given are well suited to model the in uence of statistical informa ..."
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Cited by 12 (4 self)
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A logic is defined that allows to express information about statistical probabilities and about degrees of belief in specific propositions. By interpreting the two types of probabilities in one common probability space, the semantics given are well suited to model the in uence of statistical information on the formation of subjective beliefs. Cross entropy minimization is a key element in these semantics, the use of which is justified by showing that the resulting logic exhibits some very reasonable properties.
Lp, A Logic for Representing and Reasoning with Statistical Knowledge
, 1990
"... This paper presents a logical formalism for representing and reasoning with statistical knowledge. One of the key features of the formalism is its ability to deal with qualitative statistical information. It is argued that statistical knowledge, especially that of a qualitative nature, is an importa ..."
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Cited by 11 (0 self)
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This paper presents a logical formalism for representing and reasoning with statistical knowledge. One of the key features of the formalism is its ability to deal with qualitative statistical information. It is argued that statistical knowledge, especially that of a qualitative nature, is an important component of our world knowledge and that such knowledge is used in many different reasoning tasks. The work is further motivated by the observation that previous formalisms for representing probabilistic information are inadequate for representing statistical knowledge. The representation mechanism takes the form of a logic that is capable of representing a wide variety of statistical knowledge, and that possesses an intuitive formal semantics based on the simple notions of sets of objects and probabilities defined over those sets. Furthermore, a proof theory is developed and is shown to be sound and complete. The formalism offers a perspicuous and powerful representational tool for stat...
Quantifier Elimination for Neocompact Sets
 JOURNAL OF SYMBOLIC LOGIC
"... We shall prove quantifier elimination theorems for neocompact formulas, which define neocompact sets and are built from atomic formulas using finite disjunctions, infinite conjunctions, existential quantifiers, and bounded universal quantifiers. The neocompact sets were first introduced to provide a ..."
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Cited by 9 (8 self)
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We shall prove quantifier elimination theorems for neocompact formulas, which define neocompact sets and are built from atomic formulas using finite disjunctions, infinite conjunctions, existential quantifiers, and bounded universal quantifiers. The neocompact sets were first introduced to provide an easy alternative to nonstandard methods of proving existence theorems in probability theory, where they behave like compact sets. The quantifier elimination theorems in this paper can be applied in a general setting to show that the family of neocompact sets is countably compact. To provide the necessary setting we introduce the notion of a law structure. This notion was motivated by the probability law of a random variable. However, in this paper we discuss a variety of model theoretic examples of the notion in the light of our quantifier elimination results.
A Logic of Probability With Decidable ModelChecking
, 2001
"... A predicate logic of probability, close to logics of probability of Halpern and al., is introduced. We consider the following modelchecking problem: for a formula and a nite state Markov chain to decide whether this formula holds on the structure dened by this Markov chain. It is shown that the mod ..."
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Cited by 6 (0 self)
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A predicate logic of probability, close to logics of probability of Halpern and al., is introduced. We consider the following modelchecking problem: for a formula and a nite state Markov chain to decide whether this formula holds on the structure dened by this Markov chain. It is shown that the modelchecking problem is decidable for a rather large subclass of formulas of a secondorder monadic logic of probability. The validity problem of the propositional fragment of this logic of probability is shown to be decidable.
Formal Logics of Discovery and Hypothesis Formation By Machine
"... . The following are the aims of the paper: (1) To call the attention of the community of Discovery Science to certain existing formal systems for DS developed in Prague in 60's till 80's suitable for DS and unfortunately largely unknown. (2) To illustrate the use of the calculi in question ..."
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Cited by 3 (2 self)
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. The following are the aims of the paper: (1) To call the attention of the community of Discovery Science to certain existing formal systems for DS developed in Prague in 60's till 80's suitable for DS and unfortunately largely unknown. (2) To illustrate the use of the calculi in question on the example of the GUHA method of hypothesis generation by computer, subjecting this method to a critical evaluation in the context of contemporary data mining. (3) To stress the importance of Fuzzy Logic for DS and inform on the present state of mathematical foundations of Fuzzy Logic. (4) Finally, to present a running research program of developing calculi of symbolic fuzzy logic for DS and for a fuzzy GUHA method. 1 Introduction The term "logic of discovery" is admittedly not new: let us mention at least Popper's philosophical work [42], Buchanan's dissertation [4] analyzing the notion of a logic of discovery in relation to Artificial Intelligence and Plotkin's paper [41] with his notion of a ...