H. Gaifman. Concerning measures in first-order calculi. Israel Journal of Mathematics, 2:1--18, 1964.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....logic programming tasks that we especially focus on in this paper. 2. 1 Probabilistic Background We now briefly describe how first order logics of probability are given a semantics in which probabilities are defined over a set of possible worlds (cf. especially the work by Carnap [4] Gaifman [18], Scott and Krauss [74] and Halpern [23] We restrict our considerations to a language of first order Boolean combinations of conditional constraints that are implicitly universally quantified and that are interpreted by probabilities over a set of Herbrand interpretations. be a first order ....

H. Gaifman. Concerning measures in first order calculi. Israel J. Math., 2:1--18, 1964.

....and Algebra for Probabilistic Complex Values 5 2. Probabilistic Background In this section, we describe the probabilistic background of our approach to probabilistic complex value databases. We assume a semantics in which probabilities are defined over a set of possible worlds (see especially [4,14,27,16]) where we adopt some technical notions from [22,23] A major goal of this section is to give a model theoretic definition of probabilistic conjunction, disjunction, and difference strategies, which have been introduced by an axiomatic characterization in [20] Given the probability ranges of two ....

H. Gaifman. Concerning measures in first order calculi. Israel Journal of Mathematics, 2:1--18, 1964.

....background of this work. We then define probabilistic logic programs and give some illustrative examples. 2. 1 Probabilistic Background We now briefly recall how first order logics of probability are given a semantics in which probabilities are defined over a set of possible worlds (cf. especially [3, 8, 35, 14]) We restrict our considerations to a language of first order Boolean combinations of conditional constraints that are implicitly universally quantified and that are interpreted by probabilities over a set of Herbrand interpretations. Let be a first order vocabulary that contains a finite set ....

H. Gaifman. Concerning measures in first order calculi. Israel J. Math., 2:1--18, 1964.

....inheritance with overriding. Note that all proofs are given in the extended paper [23] 2 PRELIMINARIES 2. 1 PROBABILISTIC BACKGROUND We briefly recall how first order logics of probability are given a semantics in which probabilities are defined over a set of possible worlds (cf. especially [3, 8, 35, 13]) We restrict our considerations to a language of first order Boolean combinations of conditional constraints that are implicitly universally quantified and that are interpreted by probabilities over a set of Herbrand interpretations. Let be a first order vocabulary that contains a finite set ....

H. Gaifman. Concerning measures in first order calculi. Israel J. Math., 2:1--18, 1964.

....and results from [13] 2.1 Probabilistic Background We now briefly summarize how (a quantifier free fragment of) classical first order logics can be given a probabilistic semantics in which probabilities are defined over a set of possible worlds. We essentially follow the approaches in [3, 8, 9]. Let be a first order vocabulary that contains a set of function symbols and a set of predicate symbols (as usual, constant symbols are function symbols of arity zero) Let be a set of variables. We define terms by induction as follows. A term is a variable from or an expression of the ....

H. Gaifman. Concerning measures in first order calculi. Israel J. Math., 2:1--18, 1964.

....and results from [13] 2.1 Probabilistic Background We now briefly summarize how (a quantifier free fragment of) classical first order logics can be given a probabilistic semantics in which probabilities are defined over a set of possible worlds. We essentially follow the approaches in [3, 8, 9]. Let be a first order vocabulary that contains a set of function symbols and a set of predicate symbols (as usual, constant symbols are function symbols of arity zero) Let X be a set of variables. We define terms by induction as follows. A term is a variable from X or an expression of the ....

H. Gaifman. Concerning measures in first order calculi. Israel J. Math., 2:1--18, 1964.

....background of this work. We then define probabilistic logic programs and give some illustrative examples. 2. 1 Probabilistic Background We now briefly recall how first order logics of probability are given a semantics in which probabilities are defined over a set of possible worlds (cf. especially [3, 8, 35, 14]) We restrict our considerations to a language of first order Boolean combinations of conditional constraints that are implicitly universally quantified and that are interpreted by probabilities over a set of Herbrand interpretations. Let be a first order vocabulary that contains a finite set ....

H. Gaifman. Concerning measures in first order calculi. Israel J. Math., 2:1--18, 1964.

....using Cantor s backand forth argument. Given any two countable models A and B of T , the idea of the back and forth argument is to build up an isomorphism, step by step, between A and B by using the extension axioms. The first published proof that T is 0 categorical is due to Gaifman [Gai64], who also uses the back and forth argument. Let R be the unique (up to isomorphism) countable graph that satisfies T . This graph was studied by Rado [Rad64] and is sometimes called the Rado graph. It is also sometimes called the random countable graph, since with probability 1, this graph is ....

....called the Rado graph. It is also sometimes called the random countable graph, since with probability 1, this graph is generated by a random process where we start with a countable set of nodes, and where each possible edge appears with probability 1 2 , independently of the other edges [ER63, Gai64] (see also [ES74, pp. 98 99] A useful feature of the random countable graph (that will be exploited later) is that every countable graph is embeddable in it. Thus, let us say that a graph H with universe H is a subgraph of the graph G if the edges of H are precisely those edges of G where both ....

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H. Gaifman. Concerning measures in first-order calculi. Israel Journal of Mathematics, 2:1--18, 1964.

....random finite structures, and that the rate of convergence is exponential. Lemma 3.2 For every extension axiom oe s;t , and all sufficiently large n n (oe s;t ) 1 Gamma c Gamman for some constant c 1. Using a common model theoretic technique known as a back and forth argument, Gaifman [14] showed that the collection T of all extension axioms over a vocabulary is an 0 categorical theory. This means that up to isomorphism, T has precisely one countably infinite model R, which is often called the countable random structure over the vocabulary . When we deal with graphs R is also ....

H. Gaifman, Concerning measures in first-order calculi, Israel Journal of Mathematics, 2 (1964), pp. 1--18.

....the fraction of the structures with universe f0; n Gamma 1g that satisfy . The 0 1 law for first order logic, proved independently by Fagin [2] and Glebskii et al. 4] states that for first order sentences the limit ( lim n 1 n ( always exists and is either 0 or 1. Gaifman [3] considered countable structures with universe and finite relational vocabulary. He proved that with respect to the Lebesgue measure every property of countable structures that is closed under isomorphisms has probability either 0 or 1. The proof is based on the fact that the set of extension ....

....4.2. For every m; s(x 1 ; xm ) and t(x 1 ; xm 1 ) where t is an extension of s, we define the m extension axiom ae s;t = 8x 1 : 8xm (s(x 1 ; xm ) 9xm 1 t(x 1 ; xm 1 ) Let T be the set of all extension axioms. Using a back and forth argument Gaifman [3] showed that T is an categorical theory, i.e. that T has up to isomorphism exactly one countable model R, called the countable random structure over oe. As a consequence, T is complete. The completeness of T and the fact that every extension axiom has asymptotic probability 1 imply the 0 1 law ....

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H. Gaifman. Concerning measures in first-order calculi. Israel Journal of Mathematics, 2:1--18, 1964.

....technical results have been moved to the appendix. 2 Probabilistic Background In this section, we describe the probabilistic background of our approach to probabilistic complex value databases. We assume a semantics in which probabilities are defined over a set of possible worlds (see especially [4, 12, 23, 14]) where we adopt some technical notions from [19, 20] A major goal of this section is to give a model theoretic definition of probabilistic conjunction, disjunction, and difference strategies, which have been introduced by an axiomatic characterization in [17] Given the probability ranges of ....

H. Gaifman. Concerning measures in first order calculi. Israel Journal of Mathematics, 2:1--18, 1964.

....constant symbol to L(M) for each element of K. Any randomization of M will determine a probability measure P on the set of sentences L(K; M) the sentence ( X) gets probability P ( X) Probability measures on sets of sentences were studied in a model theoretic setting by Gaifman in [G] and Scott and Krauss in [SK] In the paper [SK] the idea of developing probability theory by assigning probabilities to sentences is suggested as an alternative to the classical Kolmogorov approach using a probability space( Omega ; B; P ) In the present work, we maintain a flexible position ....

....extends (K 1 ; B 1 ; R 1 ) The Scalar Axioms hold in (K; B; R) because R j= Th(R) The construction of (K; B; R) from (K 2 ; B 2 ; R 2 ) preserves the other axioms, and therefore (K; B; R) is a model of S. 2 Theorem 6. 9 is closely related to Gaifman s completeness theorem for measure models in [G]. Gaifman proved that for every set K 1 of new constant symbols and every real valued probability measure P 1 on the set of sentences L(K 1 ; M) there is a set K oe K 1 and a real valued probability measure P P 1 on the set of sentences L(K; M) which satisfies the Fullness Axiom. This can be ....

H. Gaifman. Concerning Measures on First Order Calculi. Israel J. Math. 2 (1964), pp. 1-18. 34

....give a formal definition of the main problems analyzed in this work. 2. 1 Probabilistic Background We now briefly describe how first order logics of probability are given a semantics in which probabilities are defined over a set of possible worlds (see especially the work by Carnap [8] Gaifman [21], Scott and Krauss [65] and Halpern [29] We restrict our considerations to a function free language of first order conditional constraints that are implicitly universally quantified and that are interpreted by probabilities over a set of Herbrand interpretations. Let be a first order ....

H. Gaifman. Concerning measures in first order calculi. Israel J. Math., 2:1--18, 1964.

....as a quantifier free formula that is the conjunction of all members of t(x) With each pair of types s and t such that s extends t we associate a first order extension axiom which states that (8x) t(x) 9z)s(x; z) Let T be the set of all extension axioms. The theory T was studied by Gaifman [Gai64], who showed, using a back and forth argument, that every two countable models of T are isomorphic (i.e. T is an categorical theory) The extension axioms can also be used to show that the unique (up to isomorphism) countable model A of T is universal for all countable structures over R, i.e. ....

Gaifman, H.: Concerning measures in first-order calculi. Israel J. Math. 2(1964). pp. 1--18.

....Indeed, there has been substantial amount of research into probabilistic logics ever since Boole [8] Carnap [9] is a seminal work on probabilistic logic. Fagin, Halpern, and Meggido [13] study the satisfiablity of systems of probabilistic inequalities from a model theoretic perspective. Gaifman [19] extends probability theory by borrowing notions and techniques from logic. Nilsson [38] uses a possible worlds approach to give model theoretic semantics for probabilistic logic. Hailperin s [22] notion of probabilistic entailment is similar to that of Nilsson. These works are mainly concerned ....

Gaifman, H. Concerning measures in first order calculi. Israel J. of Math., 2:1--17, 1964.

....mechanism for evidential reasoning based on Dempster s rules of combination [38] the semantics of his approach has not been studied so far. There have also been many proposals on relating probability with logic, such as those by Bacchus [2] Fagin, Halpern and Megiddo[13] Fenstad [14] Gaifman [17], Hailperin [20] Nilsson [34] and Scott [37] Fagin, Halpern and Megiddo in [13] propose a model theoretic basis for reasoning about systems of probabilistic inequalities. Their primary focus is on the satisfiability of such systems. Fenstad in [14] investigates how probability theory can be ....

....[37] Fagin, Halpern and Megiddo in [13] propose a model theoretic basis for reasoning about systems of probabilistic inequalities. Their primary focus is on the satisfiability of such systems. Fenstad in [14] investigates how probability theory can be understood by using model theory. Gaifman [17] and Scott [37] on the other hand aim at extending the mathematical theory of probability by borrowing notions and methods from logic. While Gaifman s theory deals with finitary languages, Scott extends Gaifman s formalism to deal with infinitary formulas involving countable disjunctions and ....

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H. Gaifman. (1964) Concerning Measures in First Order Calculi, Israel J. of Math., 2, pps 1--17.

....proved that on the class C of all finite structures of vocabulary oe, 1 for any extension axiom . The theory of all extensions axioms, denoted T , was proven to be categorical (that is, every two countable models are isomorphic) by Ehrenfeucht and Ryll Nardzeweski [Ry] and by Gaifman [Ga]. Hence, A, which is a model for T , is unique up to isomorphism. This structure is called the random countable structure (or the random countable graph if oe has just one binary relation symbol) since it is generated, with probability 1, by a random process in which we start with a countable ....

H. Gaifman, "Concerning Measures in First-Order Calculi", Israel J. Math. 2 (1964), 1--18.

.... lose track of precisely which events it is that are being assigned a probability, and how that probability should be assigned (see [HT93] for a discussion of the situation in the context of distributed systems) There is a fairly extensive literature on reasoning about probability (see for example [Bac90, Car50, Gai64, GKP88, GF87, HF87, Hoo78, Kav89, Kei85, Luk70, Nil86, Nut87, Sha76] and the references in [Nil86] but remarkably few attempts at constructing a logic to reason explicitly about probabilities. We start by considering a language that allows linear inequalities involving probabilities. Thus, typical formulas include 3w( 1 and w( 2w( We consider two ....

H. Gaifman. Concerning measures in first order calculi. Israel Journal of Mathematics, 2:1--18, 1964.

....In this paper and in [GHK93] we address the problem of computing conditional probabilities in the first order case. In a companion paper [GHK94] we consider the case of statistical knowledge. The general problem of assigning probabilities to first order sentences has been well studied (cf. [Gai60, Gai64]) In this paper, we investigate two specific formalisms for computing probabilities, based on the same basic approach. The approach is based on an old idea, that goes back to Laplace [Lap20] It is essentially an application of what has been called the principle of insufficient reason [Kri86] or ....

H. Gaifman. Concerning measures in first order calculi. Israel Journal of Mathematics, 2:1--18, 1964.

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H. Gaifman. Concerning measures in first-order calculi. Israel Journal of Mathematics, 2:1--18, 1964.

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H. Gaifman. Concerning measures in first-order calculi. Israel Journal of Mathematics, 2:1--18, 1964.

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H. Gaifman. Concerning measures in first order calculi. Israel Journal of Mathematics, 2:1--18, 1964.

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H. Gaifman. Concerning measures in first-order calculi. Israel Journal of Mathematics, 2:1--18, 1964.

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Gaifman, H. (1964). Concerning measures in first-order calculi. Israel Journal of Mathematics, 2:1--18.

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H. Gaifman. (1964) Concerning Measures in First Order Calculi, Israel J. of Math., 2, pps 1--17.

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