scholarly journals Completeness theorem for a logic with imprecise and conditional probabilities

2005 ◽  
Vol 78 (92) ◽  
pp. 35-49 ◽  
Author(s):  
Zoran Ognjanovic ◽  
Zoran Markovic ◽  
Miodrag Raskovic
Keyword(s):  
Probability Space ◽  
Probabilistic Models ◽  
Completeness Theorem ◽  
Probability Logic ◽  
Conditional Probabilities ◽  
Possible World

We present a prepositional probability logic which allows making formulas that speak about imprecise and conditional probabilities. A class of Kripke-like probabilistic models is defined to give semantics to probabilistic formulas. Every possible world of such a model is equipped with a probability space. The corresponding probabilities may have nonstandard values. The proposition "the probability is close to r" means that there is an infinitesimal ?, such that the probability is equal to r ? ? (or r + ?). We provide an infinitary axiomatization and prove the corresponding extended completeness theorem.

scholarly journals A logic with higher order conditional probabilities

2007 ◽  
pp. 141-154 ◽  
Author(s):  
Zoran Ognjanovic ◽  
Nebojsa Ikodinovic
Keyword(s):  
Conditional Probability ◽  
Probability Space ◽  
Possible Worlds ◽  
Axiomatic System ◽  
Probability Logic ◽  
Conditional Probabilities ◽  
Unconditional Probability ◽  
Possible World ◽  
Intended Meaning ◽  
Probability Operators

We investigate probability logic with the conditional probability operators This logic, denoted LCP, allows making statements such as: P?s?, CP?s(? | ?) CP?0(? | ?) with the intended meaning "the probability of ? is at least s" "the conditional probability of ? given ? is at least s", "the conditional probability of ? given ? at most 0". A possible-world approach is proposed to give semantics to such formulas. Every world of a given set of worlds is equipped with a probability space and conditional probability is derived in the usual way: P(? | ?) = P(?^?)/P(?), P(?) > 0, by the (unconditional) probability measure that is defined on an algebra of subsets of possible worlds. Infinitary axiomatic system for our logic which is sound and complete with respect to the mentioned class of models is given. Decidability of the presented logic is proved.

scholarly journals QUANTUM TEAM LOGIC AND BELL’S INEQUALITIES

2015 ◽  
Vol 8 (4) ◽  
pp. 722-742 ◽  
Author(s):  
TAPANI HYTTINEN ◽  
GIANLUCA PAOLINI ◽  
JOUKO VÄÄNÄNEN
Keyword(s):  
Quantum Mechanics ◽  
Completeness Theorem ◽  
Probability Logic ◽  
Bell’S Inequalities ◽  
Dependence Logic ◽  
Logical Approach ◽  
Team Semantics ◽  
Bell's Inequalities

AbstractA logical approach to Bell’s Inequalities of quantum mechanics has been introduced by Abramsky and Hardy (Abramsky & Hardy, 2012). We point out that the logical Bell’s Inequalities of Abramsky & Hardy (2012) are provable in the probability logic of Fagin, Halpern and Megiddo (Fagin et al., 1990). Since it is now considered empirically established that quantum mechanics violates Bell’s Inequalities, we introduce a modified probability logic, that we call quantum team logic, in which Bell’s Inequalities are not provable, and prove a Completeness theorem for this logic. For this end we generalise the team semantics of dependence logic (Väänänen, 2007) first to probabilistic team semantics, and then to what we call quantum team semantics.

CANTIA Code for the Probabilistic Assessment of Inspection Strategies for Steam Generator Tubes

2008 ◽  
Author(s):  
Shripad T. Revankar ◽  
Brian Wolf ◽  
Jovica R. Riznic
Keyword(s):  
Steam Generator ◽  
Nuclear Power ◽  
Probabilistic Models ◽  
Crack Opening ◽  
Critical Flow ◽  
Probabilistic Assessment ◽  
Conditional Probabilities ◽  
Critical Flow Rate ◽  
Dose Limits ◽  
Steam Generator Tubes

The Canadian Nuclear Safety Commission (CNSC) recently developed the CANTIA methodology for probabilistic assessment of inspection strategies for steam generator tubes. Assessment of the conditional probabilities of tube failures, leak rates, and ultimately risk of exceeding licensing dose limits as an approach used to steam generator tube fitness-for-service assessment has been increasingly used in recent years throughout the nuclear power industry. The ANL/CANTIA code predictions were systematically studied to evaluate the code capability to predict the leak rates through the flawed steam generator tubes. In this evaluation the code models on the crack opening area, the probabilistic models and the critical flow rate models were studied and their applicability to available experimental data base was examined.

scholarly journals A first-order conditional probability logic with iterations

2013 ◽  
Vol 93 (107) ◽  
pp. 19-27 ◽  
Author(s):  
Milos Milosevic ◽  
Zoran Ognjanovic
Keyword(s):  
Conditional Probability ◽  
Completeness Theorem ◽  
Axiomatic System ◽  
Probability Logic ◽  
Intended Meaning ◽  
First Order

We investigate a first-order conditional probability logic with equality, which is, up to our knowledge, the first treatise of such logic. The logic, denoted LFPOIC=, allows making statements such as: CP?s(?, ?), and CP?s(?, ?), with the intended meaning that the conditional probability of ? given ? is at least (at most) s. The corresponding syntax, semantic, and axiomatic system are introduced, and Extended completeness theorem is proven.

Predicting the time of occurrence of decompression sickness

1992 ◽  
Vol 72 (4) ◽  
pp. 1541-1548 ◽  
Author(s):  
P. K. Weathersby ◽  
S. S. Survanshi ◽  
L. D. Homer ◽  
E. Parker ◽  
E. D. Thalmann
Keyword(s):  
Maximum Likelihood ◽  
Probabilistic Models ◽  
Decompression Sickness ◽  
Likelihood Estimation ◽  
Estimation Procedure ◽  
Model Parameters ◽  
Conditional Probabilities ◽  
Data Set ◽  
Additional Information ◽  
Use Of Knowledge

Probabilistic models and maximum likelihood estimation have been used to predict the occurrence of decompression sickness (DCS). We indicate a means of extending the maximum likelihood parameter estimation procedure to make use of knowledge of the time at which DCS occurs. Two models were compared in fitting a data set of nearly 1,000 exposures, in which greater than 50 cases of DCS have known times of symptom onset. The additional information provided by the time at which DCS occurred gave us better estimates of model parameters. It was also possible to discriminate between good models, which predict both the occurrence of DCS and the time at which symptoms occur, and poorer models, which may predict only the overall occurrence. The refined models may be useful in new applications for customizing decompression strategies during complex dives involving various times at several different depths. Conditional probabilities of DCS for such dives may be reckoned as the dive is taking place and the decompression strategy adjusted to circumstance. Some of the mechanistic implications and the assumptions needed for safe application of decompression strategies on the basis of conditional probabilities are discussed.

Algorithms for Generating XML Documents from Probabilistic XML

2012 ◽  
Vol 263-266 ◽  
pp. 1578-1583
Author(s):  
Yan Zhu ◽  
Hai Tao Ma
Keyword(s):  
Time Complexity ◽  
Probabilistic Models ◽  
Linear Time ◽  
Uncertain Information ◽  
Possible World ◽  
Xml Documents ◽  
Probabilistic Xml ◽  
Xml Document ◽  
High Flexibility ◽  
Independent Distribution

Uncertain relational data management has been investigated for a few years, but few works on uncertain XML. The natural structures with high flexibility make XML more appropriate for representing uncertain information. Based on the semantic of possible world and probabilistic models with independent distribution and mutual exclusive distribution nodes, the problem of how to generate instance from a probabilistic XML and calculate its probability was studied, which is one of the key problems of uncertain XML management. Moreover, an algorithm for a generating XML document from a probabilistic XML and calculating its probability are also proposed, which has linear time complexity. Finally, experiment results are made to show up the correct and efficiency of the algorithm.

Completeness theorem for biprobability models

1986 ◽  
Vol 51 (3) ◽  
pp. 586-590 ◽  
Author(s):  
Miodrag D. Rašković
Keyword(s):  
Completeness Theorem ◽  
Probability Measures ◽  
List Type ◽  
Probability Logic ◽  
Admissible Set ◽  
Absolutely Continuous ◽  
Image Position ◽  
Standard Probability ◽  
Natural Way

The aim of the paper is to prove the completeness theorem for biprobability models. This also solves Keisler's Problem 5.4 (see [4]).Let be a countable admissible set and ω ∈ . The logic is similar to the standard probability logic . The only difference is that two types of probability quantifiers and are allowed.A biprobability model is a structure (, μ1, μ2) where is a classical structure without operations and μ1, μ2 are two types of probability measures such that μ1 is absolutely continuous with respect to μ2, i.e. μ1 ≪ μ2.The quantifiers are interpreted in the natural way, i.e.for i = 1, 2. (The measure is the restriction of the completion of to the σ-algebra generated by the measurable rectangles and the diagonal sets Axioms and rules of inference are those of , as listed in [2] with the axiom B4 from [4], with the remark that both P1 and P2 can play the role of P, together with the following axioms:Axioms of continuity.1) .2) .Axiom of absolute continuity:where and Φn = {φ ∈ Φ: φ has n free variables}.

Completeness theorem for propositional probabilistic models whose measures have only finite ranges

2004 ◽  
Vol 43 (4) ◽  
pp. 557-563 ◽  
Author(s):  
Radosav Dordević ◽  
Miodrag Rašković ◽  
Zoran Ognjanović
Keyword(s):  
Probabilistic Models ◽  
Completeness Theorem

Divisible extension of probability

2020 ◽  
Vol 70 (6) ◽  
pp. 1445-1456
Author(s):  
Roman Frič ◽  
Peter Eliaš ◽  
Martin Papčo
Keyword(s):  
Category Theory ◽  
Probability Space ◽  
Boolean Logic ◽  
Conditional Probabilities ◽  
Real Numbers ◽  
Classical Probability ◽  
Random Event ◽  
Whole Numbers ◽  
Random Events ◽  
Probability Integral

AbstractWe outline the transition from classical probability space (Ω, A, p) to its "divisible" extension, where (as proposed by L. A. Zadeh) the σ-field A of Boolean random events is extended to the class 𝓜(A) of all measurable functions into [0,1] and the σ-additive probability measure p on A is extended to the probability integral ∫(·) dp on 𝓜(A). The resulting extension of (Ω, A,p) can be described as an epireflection reflecting A to 𝓜(A) and p to ∫(·) dp.The transition from A to 𝓜(A), resembling the transition from whole numbers to real numbers, is characterized by the extension of two-valued Boolean logic on A to multivalued Łukasiewicz logic on 𝓜(A) and the divisibility of random events: for each random event u ∈ 𝓜(A) and each positive natural number n we have u/n ∈ 𝓜(A) and ∫(u/n) dp = (1/n) ∫u dp.From the viewpoint of category theory, objects are of the form 𝓜(A), morphisms are observables from one object into another one and serve as channels through which stochastic information is conveyed.We study joint random experiments and asymmetrical stochastic dependence/independence of one constituent experiment on the other one. We present a canonical construction of conditional probability so that observables can be viewed as conditional probabilities.In the present paper we utilize various published results related to "quantum and fuzzy" generalizations of the classical theory, but our ultimate goal is to stress mathematical (categorical) aspects of the transition from classical to what we call divisible probability.

Management of Economics and State "Bottom"

2020 ◽  
Vol 17 (6) ◽  
pp. 76-91
Author(s):  
E. D. Solozhentsev
Keyword(s):  
Quality Of Life ◽  
Artificial Intelligence ◽  
Probabilistic Models ◽  
Human Life ◽  
Special Software ◽  
Human Quality ◽  
Relationship Of ◽  
The Relationship

The scientific problem of economics “Managing the quality of human life” is formulated on the basis of artificial intelligence, algebra of logic and logical-probabilistic calculus. Managing the quality of human life is represented by managing the processes of his treatment, training and decision making. Events in these processes and the corresponding logical variables relate to the behavior of a person, other persons and infrastructure. The processes of the quality of human life are modeled, analyzed and managed with the participation of the person himself. Scenarios and structural, logical and probabilistic models of managing the quality of human life are given. Special software for quality management is described. The relationship of human quality of life and the digital economy is examined. We consider the role of public opinion in the management of the “bottom” based on the synthesis of many studies on the management of the economics and the state. The bottom management is also feedback from the top management.

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