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

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.

scholarly journals Reasoning about Minimal Belief and Negation as Failure

1999 ◽  
Vol 11 ◽  
pp. 277-300 ◽  
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.

scholarly journals Preferences and Nonmonotonic Reasoning

AI Magazine ◽  
2008 ◽  
Vol 29 (4) ◽  
pp. 69 ◽  
Author(s):  
Gerhard Brewka ◽  
Ilkka Niemela ◽  
Miroslaw Truszczynski
Keyword(s):  
Logic Programming ◽  
Nonmonotonic Reasoning ◽  
Nonmonotonic Logic ◽  
Preference Information ◽  
The World ◽  
Extended Logic Programming ◽  
Answer Set Semantics ◽  
Nonmonotonic Logics ◽  
Answer Set

We give an overview of the multifaceted relationship between nonmonotonic logics and preferences. We discuss how the nonmonotonicity of reasoning itself is closely tied to preferences reasoners have on models of the world or, as we often say here, possible belief sets. Selecting extended logic programming with the answer-set semantics as a "generic" nonmonotonic logic, we show how that logic defines preferred belief sets and how preferred belief sets allow us to represent and interpret normative statements. Conflicts among program rules (more generally, defaults) give rise to alternative preferred belief sets. We discuss how such conflicts can be resolved based on implicit specificity or on explicit rankings of defaults. Finally, we comment on formalisms which explicitly represent preferences on properties of belief sets. Such formalisms either build preference information directly into rules and modify the semantics of the logic appropriately, or specify preferences on belief sets independently of the mechanism to define them.

scholarly journals More on Modal Aspects of Default Logic1

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.

Formalization of Inheritance Reasoning in Autoepistemic Logic

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.

Default Logic Models of Certain Inheritance Systems with Exceptions

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.

Logic programming and nonmonotonic reasoning

1993 ◽  
pp. 141-187 ◽  
Author(s):  
V. Wiktor Marek ◽  
Mirosław Truszczyński
Keyword(s):  
Logic Programming ◽  
Nonmonotonic Reasoning

scholarly journals Sequent-Type Calculi for Three-Valued and Disjunctive Default Logic

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 84 ◽  
Author(s):  
Sopo Pkhakadze ◽  
Hans Tompits
Keyword(s):  
Proof Theory ◽  
Consistency Condition ◽  
Nonmonotonic Reasoning ◽  
Default Logic ◽  
Inference Process ◽  
Default Theory ◽  
Valid Inference ◽  
The One ◽  
Base Logic ◽  
Selection Of

Default logic is one of the basic formalisms for nonmonotonic reasoning, a well-established area from logic-based artificial intelligence dealing with the representation of rational conclusions, which are characterised by the feature that the inference process may require to retract prior conclusions given additional premisses. This nonmonotonic aspect is in contrast to valid inference relations, which are monotonic. Although nonmonotonic reasoning has been extensively studied in the literature, only few works exist dealing with a proper proof theory for specific logics. In this paper, we introduce sequent-type calculi for two variants of default logic, viz., on the one hand, for three-valued default logic due to Radzikowska, and on the other hand, for disjunctive default logic, due to Gelfond, Lifschitz, Przymusinska, and Truszczyński. The first variant of default logic employs Łukasiewicz’s three-valued logic as the underlying base logic and the second variant generalises defaults by allowing a selection of consequents in defaults. Both versions have been introduced to address certain representational shortcomings of standard default logic. The calculi we introduce axiomatise brave reasoning for these versions of default logic, which is the task of determining whether a given formula is contained in some extension of a given default theory. Our approach follows the sequent method first introduced in the context of nonmonotonic reasoning by Bonatti, which employs a rejection calculus for axiomatising invalid formulas, taking care of expressing the consistency condition of defaults.

Sign in / Sign up

Export Citation Format

Share Document