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Home / Proceedings / Proceedings of the AAAI Conference on Artificial Intelligence, 29 / No.1: The Twenty-Ninth Conference on Artificial Intelligence

Grounded Fixpoints

March 8, 2023

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Authors

Bart Bogaerts

KU Leuven


Joost Vennekens

KU Leuven


Marc Denecker

KU Leuven


DOI:

10.1609/aaai.v29i1.9388


Abstract:

Algebraical fixpoint theory is an invaluable instrument for studying semantics of logics. For example, all major semantics of logic programming, autoepistemic logic, default logic and more recently, abstract argumentation have been shown to be induced by the different types of fixpoints defined in approximation fixpoint theory (AFT). In this paper, we add a new type of fixpoint to AFT: a grounded fixpoint of lattice operator O : L → L is defined as a lattice element x ∈ L such that O(x) = x and for all v ∈ L such that O(v ∧ x) ≤ v, it holds that x ≤ v. On the algebraical level, we show that all grounded fixpoints are minimal fixpoints approximated by the well-founded fixpoint and that all stable fixpoints are grounded. On the logical level, grounded fixpoints provide a new mathematically simple and compact type of semantics for any logic with a (possibly non-monotone) semantic operator. We explain the intuition underlying this semantics in the context of logic programming by pointing out that grounded fixpoints of the immediate consequence operator are interpretations that have no non-trivial unfounded sets. We also analyse the complexity of the induced semantics. Summarised, grounded fixpoint semantics is a new, probably the simplest and most compact, element in the family of semantics that capture basic intuitions and principles of various non-monotonic logics. 

Topics: AAAI

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HOW TO CITE:

Bart Bogaerts|| Joost Vennekens|| Marc Denecker Grounded Fixpoints Proceedings of the AAAI Conference on Artificial Intelligence, 29 (2015) .

Bart Bogaerts|| Joost Vennekens|| Marc Denecker Grounded Fixpoints AAAI 2015, .

Bart Bogaerts|| Joost Vennekens|| Marc Denecker (2015). Grounded Fixpoints. Proceedings of the AAAI Conference on Artificial Intelligence, 29, .

Bart Bogaerts|| Joost Vennekens|| Marc Denecker. Grounded Fixpoints. Proceedings of the AAAI Conference on Artificial Intelligence, 29 2015 p..

Bart Bogaerts|| Joost Vennekens|| Marc Denecker. 2015. Grounded Fixpoints. "Proceedings of the AAAI Conference on Artificial Intelligence, 29". .

Bart Bogaerts|| Joost Vennekens|| Marc Denecker. (2015) "Grounded Fixpoints", Proceedings of the AAAI Conference on Artificial Intelligence, 29, p.

Bart Bogaerts|| Joost Vennekens|| Marc Denecker, "Grounded Fixpoints", AAAI, p., 2015.

Bart Bogaerts|| Joost Vennekens|| Marc Denecker. "Grounded Fixpoints". Proceedings of the AAAI Conference on Artificial Intelligence, 29, 2015, p..

Bart Bogaerts|| Joost Vennekens|| Marc Denecker. "Grounded Fixpoints". Proceedings of the AAAI Conference on Artificial Intelligence, 29, (2015): .

Bart Bogaerts|| Joost Vennekens|| Marc Denecker. Grounded Fixpoints. AAAI[Internet]. 2015[cited 2023]; .


ISSN: 2374-3468


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