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Simple Contrapositive Assumption-Based Frameworks

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Logic Programming and Nonmonotonic Reasoning (LPNMR 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11481))

Abstract

Assumption-based argumentation is one of the most prominent formalisms for logical (or structured) argumentation. It has been shown useful for representing defeasible reasoning and has tight links to logic programming. In this paper we study the Dung semantics for extended forms of assumption-based argumentation frameworks (ABFs), based on any contrapositive propositional logic, and whose defeasible rules are expressed by arbitrary formulas in that logic. In particular, new results on the well-founded semantics for such ABFs are reported, the redundancy of the closure condition is shown, and the use of disjunctive attacks is investigated. Finally, some useful properties of the generalized frameworks are considered.

This work is supported by the Israel Science Foundation (grant number 817/15). The first author is also supported by the Sofja Kovalevskaja award of the Alexander von Humboldt-Foundation, funded by the German Ministry for Education and Research.

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Notes

  1. 1.

    While both this paper and [13] refer to Dung semantics for simple contrapositive ABFs, the topics that each paper addresses are different, thus the papers are complementary.

  2. 2.

    An extended abstract of this paper appears in the proceedings of AAMAS’2019.

  3. 3.

    That is, \(\psi ,\lnot \psi \vdash \mathsf{F}\). In explosive logics every formula follows from inconsistent assertions.

  4. 4.

    In the sequel, some proofs will be sketched or omitted altogether due to space restrictions.

  5. 5.

    In particular, the emptyset does not defend q from the attack \(p,\lnot p \vdash \lnot q\).

  6. 6.

    This is exactly the reason why the restriction to closed sets is imposed when standard attacks are used, while for disjunctive attacks this is not necessary.

  7. 7.

    Note that this writing is somewhat ambiguous, since, e.g. when \(\varGamma ,Ab,\psi \) are the premises, \(\psi \) may be either a strict or a defeasible assumption. This will not cause problems in what follows.

  8. 8.

    We refer to [13] for an example that shows that is not preferential even for ABFs with standard (non-disjunctive) attacks.

  9. 9.

    Note that by Theorem 2, , and so is not only cumulative, but also preferential.

  10. 10.

    We note that works such as [12] use similar terminology when referring to attacks among arguments, but the nature of the attacks (disjunctive formulas vs. conjunctive formulas), as well as the context of those works (other structured frameworks), are different.

  11. 11.

    The fact that a redundant closure condition reduces the computational complexity has been exploited in [10], for the analysis of flat ABFs (i.e., for ABFs in which no assumptions are derivable from other assumptions), in which case the closure assumption is indeed redundant. Our results now establish that for a wide class of non-flat ABFs, the closure condition can be safely dropped.

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Author information

Authors and Affiliations

  1. Institute of Philosophy II, Ruhr University Bochum, Bochum, Germany

    Jesse Heyninck

  2. School of Computer Science, The Academic College of Tel-Aviv, Tel Aviv, Israel

    Ofer Arieli

Authors
  1. Jesse Heyninck
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  2. Ofer Arieli
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Correspondence to Jesse Heyninck .

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Editors and Affiliations

  1. Saint Joseph's University, Philadelphia, PA, USA

    Marcello Balduccini

  2. University of Nebraska at Omaha, Omaha, USA

    Yuliya Lierler

  3. TU Wien, Vienna, Austria

    Stefan Woltran

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Heyninck, J., Arieli, O. (2019). Simple Contrapositive Assumption-Based Frameworks. In: Balduccini, M., Lierler, Y., Woltran, S. (eds) Logic Programming and Nonmonotonic Reasoning. LPNMR 2019. Lecture Notes in Computer Science(), vol 11481. Springer, Cham. https://doi.org/10.1007/978-3-030-20528-7_7

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  • DOI: https://doi.org/10.1007/978-3-030-20528-7_7

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  • Online ISBN: 978-3-030-20528-7

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