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17th International Conference on Logic Programming and Non-monotonic Reasoning

作者： Pollaci, Samuele Kostopoulos, Babis Denecker, Marc Bogaerts, Bart Vrije Univ Brussel Brussels Belgium Katholieke Univ Leuven Leuven Belgium Harokopio Univ Athens Athens Greece

ISBN: (纸本)9783031742088;9783031742095

approximation fixpoint theory (AFT) is an algebraic framework designed to study the semantics of non-monotonic logics. Despite its success, AFT is not readily applicable to higher-order definitions. To solve such an issue, we devise a formal mathematical framework employing concepts drawn from Category theory. In particular, we make use of the notion of Cartesian closed category to inductively construct higher-order approximation spaces while preserving the structures necessary for the correct application of AFT. We show that this novel theoretical approach extends standard AFT to a higher-order environment, and generalizes the AFT setting of [7].

关键词： approximation fixpoint theory Higher-order definitions Category theory

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Analyzing Semantics of Aggregate Answer Set Programming Using approximation fixpoint theory

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theory AND PRACTICE OF LOGIC PROGRAMMING 2022年第4期22卷 523-537页

作者： Vanbesien, Linde Bruynooghe, Maurice Denecker, Marc Katholieke Univ Leuven Dept Comp Sci Leuven Belgium

Aggregates provide a concise way to express complex knowledge. The problem of selecting an appropriate formalization of aggregates for answer set programming (ASP) remains unsettled. This paper revisits it from the viewpoint of approximation fixpoint theory (AFT). We introduce an AFT formalization equivalent with the Gelfond-Lifschitz reduct for basic ASP programs and we extend it to handle aggregates. We analyze how existing approaches relate to our framework. We hope this work sheds some new light on the issue of a proper formalization of aggregates.

关键词： aggregates approximation fixpoint theory Answer set Programming

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Categorical approximation fixpoint theory 18th

Categorical Approximation Fixpoint Theory

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18th European Conference on Logics in Artificial Intelligence (JELIA)

作者： Charalambidis, Angelos Rondogiannis, Panos Natl & Kapodistrian Univ Athens Dept Informat & Telecommun Athens Greece Harokopio Univ Dept Informat & Telemat Athens Greece

ISBN: (纸本)9783031436185;9783031436192

approximation fixpoint theory (AFT) is a powerful framework that has been widely used for defining the semantics of nonmonotonic formalisms in artificial intelligence and logic programming. In particular, AFT is used to derive the fixed points of (potentially non-monotonic) operators over complete lattices. However, in certain application domains, there arise operators defined over structures that are not necessarily complete lattices. Therefore, the quest for a more general version of AFT has been lingering as an interesting research direction. We develop an extension of AFT, namely Categorical AFT, that allows us to study the fixed points of (potentially non-functorial) operators defined over categories. Since categories are more general structures than complete lattices, we argue that our approach provides a more general and unified framework for the study of non-monotonicity. The versatility of category theory creates the potential of new insights and applications.

关键词： approximation fixpoint theory Category theory Non-Monotonicity Domain theory

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Non-deterministic approximation fixpoint theory and its application in disjunctive logic programming

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ARTIFICIAL INTELLIGENCE 2024年 331卷

作者： Heyninck, Jesse Arieli, Ofer Bogaerts, Bart Open Univ Heerlen Netherlands Univ Cape Town Cape Town South Africa Tel Aviv Acad Coll Sch Comp Sci Tel Aviv Israel Vrije Univ Brussel Brussels Belgium

approximation fixpoint theory (AFT) is 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 dealing with non -deterministic constructs that allow to handle indefinite information, represented e.g. by disjunctive formulas. This is done by generalizing the main constructions and corresponding results of AFT 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.

关键词： Logic Programming Answer Set Programming approximation fixpoint theory Knowledge Representation

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Argumentative Reflections of approximation fixpoint theory 8

Argumentative Reflections of Approximation Fixpoint Theory

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Biennial International Conference on Computational Models of Argument (COMMA)

作者： Heyninck, Jesse Arieli, Ofer TU Dortmund Dept Comp Sci Dortmund Germany Acad Coll Tel Aviv Sch Comp Sci Tel Aviv Israel

ISBN: (纸本)9781643681078;9781643681061

In this paper we show that non-monotonic formalisms that are represented by approximation fixpoint theory can also be represented by formal argumentation frameworks. By this, we are able not only to recapture and generalize many forms of non-monotonic reasoning in the context of argumentation theory, but also introduce new argumentative representations that have not been considered so far.

关键词： Non-monotonic reasoning approximation fixpoint theory assumption-based argumentation

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approximation fixpoint theory and the Well-Founded Semantics of Higher-Order Logic Programs

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theory AND PRACTICE OF LOGIC PROGRAMMING 2018年第3-4期18卷 421-437页

作者： Charalambidis, Angelos Rondogiannis, Panos Symeonidou, Ioanna Univ Athens Dept Informat & Telecommun Athens Greece

We define a novel, extensional, three-valued semantics for higher-order logic programs with negation. The new semantics is based on interpreting the types of the source language as three-valued Fitting-monotonic functions at all levels of the type hierarchy. We prove that there exists a bijection between such Fitting-monotonic functions and pairs of two-valued-result functions where the first member of the pair is monotone-antimonotone and the second member is antimonotone-monotone. By deriving an extension of consistent approximation fixpoint theory (Denecker et al. 2004) and utilizing the above bijection, we define an iterative procedure that produces for any given higher-order logic program a distinguished extensional model. We demonstrate that this model is actually a minimal one. Moreover, we prove that our construction generalizes the familiar well-founded semantics for classical logic programs, making in this way our proposal an appealing formulation for capturing the well-founded semantics for higher-order logic programs.

关键词： Higher-Order Logic Programming Negation in Logic Programming approximation fixpoint theory

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approximation fixpoint theory and the Well-Founded Semantics of Higher-Order Logic Programs

Approximation Fixpoint Theory and the Well-Founded Semantics...

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34th International Conference on Logic Programming (ICLP)

作者： Charalambidis, Angelos Rondogiannis, Panos Symeonidou, Ioanna Univ Athens Dept Informat & Telecommun Athens Greece

关键词： Higher-Order Logic Programming Negation in Logic Programming approximation fixpoint theory

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Analyzing the computational complexity of abstract dialectical frameworks via approximation fixpoint theory

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ARTIFICIAL INTELLIGENCE 2015年 226卷 34-74页

作者： Strass, Hannes Wallner, Johannes Peter Univ Leipzig Inst Comp Sci D-04109 Leipzig Germany Vienna Univ Technol Inst Informat Syst A-1040 Vienna Austria

The dialectical frameworks (ADFs) have recently been proposed as a versatile generalization of Dung's abstract argumentation frameworks (AFs). In this paper, we present a comprehensive analysis of the computational complexity of ADFs. Our results show that while ADFs are one level up in the polynomial hierarchy compared to AFs, there is a useful subclass of ADFs which is as complex as AFs while arguably offering more modeling capacities. As a technical vehicle, we employ the approximation fixpoint theory of Denecker, Marek and Truszczynski, thus showing that it is also a useful tool for complexity analysis of operator-based semantics. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Abstract dialectical frameworks Computational complexity approximation fixpoint theory

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Embedding justification theory in approximation fixpoint

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ARTIFICIAL INTELLIGENCE 2024年 331卷

作者： Marynissen, Simon Bogaerts, Bart Denecker, Marc Katholieke Univ Leuven Celestijnenlaan 200A B-3001 Leuven Belgium Vrije Univ Brussel Pleinlaan 9 B-1050 Brussels Belgium

approximation fixpoint theory (AFT) and Justification theory (JT) are two frameworks to unify logical formalisms. AFT studies semantics in terms of fixpoints of lattice operators, and JT in terms of so-called justifications, which are explanations of why certain facts do or do not hold in a model. While the approaches differ, the frameworks were designed with similar goals in mind, namely to study the different semantics that arise in (mainly) non -monotonic logics. The first contribution of our current paper is to provide a formal link between the two frameworks. To be precise, we show that every justification frame induces an approximator and that this mapping from JT to AFT preserves all major semantics. The second contribution exploits this correspondence to extend JT with a novel class of semantics, namely ultimate semantics : we formally show that ultimate semantics can be obtained in JT by a syntactic transformation on the justification frame, essentially performing a sort of resolution on the rules.

关键词： Justification theory approximation fixpoint theory Knowledge representation

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Non-deterministic approximation Operators: Ultimate Operators, Semi-equilibrium Semantics, and Aggregates

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theory AND PRACTICE OF LOGIC PROGRAMMING 2023年第4期23卷 632-647页

作者： Heyninck, Jesse Bogaerts, Bart Open Univ Heerlen Netherlands Vrij Univ Brussels Brussels Belgium

approximation fixpoint theory (AFT) is an abstract and general algebraic framework for studying the semantics of non-monotonic logics. In recent work, AFT was generalized to non-deterministic operators, that is, operators whose range are sets of elements rather than single elements. In this paper, we make three further contributions to non-deterministic AFT: (1) we define and study ultimate approximations of non-deterministic operators, (2) we give an algebraic formulation of the semi-equilibrium semantics by Amendola et al., and (3) we generalize the characterizations of disjunctive logic programs to disjunctive logic programs with aggregates.

关键词： approximation fixpoint theory disjunctive logic programming semi-equilibrium semantics

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