Reasoning about Minimal Belief and Negation as Failure

Journal of Artificial Intelligence Research ◽

10.1613/jair.637 ◽

1999 ◽

Vol 11 ◽

pp. 277-300 ◽

Cited By ~ 4

Author(s):

R. Rosati

Keyword(s):

Logic Programming ◽

Nonmonotonic Reasoning ◽

Default Logic ◽

Polynomial Hierarchy ◽

Autoepistemic Logic ◽

The Third ◽

Negation As Failure ◽

Negative Introspection

We investigate the problem of reasoning in the propositional fragment of MBNF, the logic of minimal belief and negation as failure introduced by Lifschitz, which can be considered as a unifying framework for several nonmonotonic formalisms, including default logic, autoepistemic logic, circumscription, epistemic queries, and logic programming. We characterize the complexity and provide algorithms for reasoning in propositional MBNF. In particular, we show that entailment in propositional MBNF lies at the third level of the polynomial hierarchy, hence it is harder than reasoning in all the above mentioned propositional formalisms for nonmonotonic reasoning. We also prove the exact correspondence between negation as failure in MBNF and negative introspection in Moore's autoepistemic logic.

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Approximation Fixpoint Theory for Non-Deterministic Operators and Its Application in Disjunctive Logic Programming

10.24963/kr.2021/32 ◽

2021 ◽

Author(s):

Jesse Heyninck ◽

Ofer Arieli

Keyword(s):

Logic Programming ◽

Nonmonotonic Reasoning ◽

Default Logic ◽

Autoepistemic Logic ◽

Algebraic Framework ◽

Nonmonotonic Logics

Approximation fixpoint theory (AFT) constitutes an abstract and general algebraic framework for studying the semantics of nonmonotonic logics. It provides a unifying study of the semantics of different formalisms for nonmonotonic reasoning, such as logic programming, default logic and autoepistemic logic. In this paper, we extend AFT to non-deterministic constructs such as disjunctive information. This is done by generalizing the main constructions and corresponding results to non-deterministic operators, whose ranges are sets of elements rather than single elements. The applicability and usefulness of this generalization is illustrated in the context of disjunctive logic programming.

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Nonmonotonic reasoning and modal logic, from negation as failure to default logic

Uncertainty in Knowledge Bases - Lecture Notes in Computer Science ◽

10.1007/bfb0028107 ◽

2005 ◽

pp. 223-231

Author(s):

Philippe Balbiani

Keyword(s):

Modal Logic ◽

Nonmonotonic Reasoning ◽

Default Logic ◽

Negation As Failure

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On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games: 25 years later

Argument & Computation ◽

10.3233/aac-200901 ◽

2020 ◽

Vol 11 (1-2) ◽

pp. 1-14

Author(s):

Pietro Baroni ◽

Francesca Toni ◽

Bart Verheij

Keyword(s):

Logic Programming ◽

Nonmonotonic Reasoning

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More on Modal Aspects of Default Logic1

Fundamenta Informaticae ◽

10.3233/fi-1992-171-207 ◽

1992 ◽

Vol 17 (1-2) ◽

pp. 99-116

Author(s):

V. Wiktor Marek ◽

Miroslaw Truszczynski

Keyword(s):

Modal Logic ◽

Default Logic ◽

Nonmonotonic Logic ◽

Default Reasoning ◽

Default Theory ◽

Autoepistemic Logic ◽

The Relationship ◽

Natural Classes ◽

Weak Extension

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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Formalization of Inheritance Reasoning in Autoepistemic Logic

Fundamenta Informaticae ◽

10.3233/fi-1990-13403 ◽

1990 ◽

Vol 13 (4) ◽

pp. 403-443

Author(s):

Michael Gelfond ◽

Halina Przymusinska

Keyword(s):

General Framework ◽

Nonmonotonic Reasoning ◽

Commonsense Reasoning ◽

Autoepistemic Logic

Current research in the area of nonmonotonic reasoning suggests that autoepistemic logic provides a general framework for formalizing commonsense reasoning in various domains of discourse. The goal of this paper is to investigate the suitability of autoepistemic logic for formalization of some forms of inheritance reasoning. To this end we propose a new semantics for inheritance networks with exceptions based on autoepistemic logic.

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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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