Cumulative default logic: Finite characterization, algorithms, and complexity

Investigations of default logic have been so far mostly concerned with the notion of an extension of a default theory. It turns out, however, that default logic is much richer. Namely, there are other natural classes of objects that might be associated with default reasoning. We study two such classes of objects with emphasis on their relations with modal nonmonotonic formalisms. First, we introduce the concept of a weak extension and study its properties. It has long been suspected that there are close connections between default and autoepistemic logics. The notion of weak extension allows us to precisely describe the relationship between these two formalisms. In particular, we show that default logic with weak extensions is essentially equivalent to autoepistemic logic, that is, nonmonotonic logic KD45. In the paper we also study the notion of a set of formulas closed under a default theory. These objects are shown to correspond to stable theories and to modal logic S5. In particular, we show that skeptical reasoning with sets closed under default theories is closely related with provability in S5. As an application of our results we determine the complexity of reasoning with weak extensions and sets closed under default theories.

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Binary decompositions of probability densities and random-bit simulation

Monte Carlo Methods and Applications ◽

10.1515/mcma-2020-2063 ◽

2020 ◽

Vol 26 (2) ◽

pp. 163-169

Author(s):

Vladimir Nekrutkin

Keyword(s):

New York ◽

Distribution Function ◽

Random Number ◽

Discrete Distribution ◽

Random Number Generation ◽

Academic Press ◽

Bernoulli Trials ◽

Algorithms And Complexity ◽

Binary Decomposition ◽

Probability Densities

AbstractThis paper is devoted to random-bit simulation of probability densities, supported on {[0,1]}. The term “random-bit” means that the source of randomness for simulation is a sequence of symmetrical Bernoulli trials. In contrast to the pioneer paper [D. E. Knuth and A. C. Yao, The complexity of nonuniform random number generation, Algorithms and Complexity, Academic Press, New York 1976, 357–428], the proposed method demands the knowledge of the probability density under simulation, and not the values of the corresponding distribution function. The method is based on the so-called binary decomposition of the density and comes down to simulation of a special discrete distribution to get several principal bits of output, while further bits of output are produced by “flipping a coin”. The complexity of the method is studied and several examples are presented.

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Algorithms and Complexity

10.1007/978-3-030-75242-2 ◽

2021 ◽

Keyword(s):

Algorithms And Complexity

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Default Logic Models of Certain Inheritance Systems with Exceptions

Fundamenta Informaticae ◽

10.3233/fi-1993-182-404 ◽

1993 ◽

Vol 18 (2-4) ◽

pp. 129-149

Author(s):

Serge Garlatti

Keyword(s):

Hierarchical Structure ◽

Formal Semantics ◽

Sufficient Conditions ◽

Nonmonotonic Reasoning ◽

Default Logic ◽

Representation Systems ◽

The Hierarchical Structure ◽

Necessary And Sufficient ◽

Structure Of Knowledge ◽

Made In

Representation systems based on inheritance networks are founded on the hierarchical structure of knowledge. Such representation is composed of a set of objects and a set of is-a links between nodes. Objects are generally defined by means of a set of properties. An inheritance mechanism enables us to share properties across the hierarchy, called an inheritance graph. It is often difficult, even impossible to define classes by means of a set of necessary and sufficient conditions. For this reason, exceptions must be allowed and they induce nonmonotonic reasoning. Many researchers have used default logic to give them formal semantics and to define sound inferences. In this paper, we propose a survey of the different models of nonmonotonic inheritance systems by means of default logic. A comparison between default theories and inheritance mechanisms is made. In conclusion, the ability of default logic to take some inheritance mechanisms into account is discussed.

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