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Multi-valued GRAPPA

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Book cover Logics in Artificial Intelligence (JELIA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11468))

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Abstract

Abstract dialectical frameworks (ADFs) are generalizations of Dung’s argumentation frameworks which allow arbitrary relationships among arguments to be expressed. In particular, arguments can not only attack each other, they also may provide support for other arguments and interact in various complex ways. The ADF approach has recently been extended in two different ways. On the one hand, GRAPPA is a framework that applies the key notions underlying ADFs – in particular their operator-based semantics – directly to arbitrary labelled graphs. This allows users to represent argumentation scenarios in their favourite graphical representations without giving up the firm ground of well-defined semantics. On the other hand, ADFs have been further generalized to the multi-valued case to enable fine-grained acceptance values. In this paper we unify these approaches and develop a multi-valued version of GRAPPA combining the advantages of both extensions.

This research has been supported by DFG (Research Unit 1513 and project BR 1817/7-2) and FWF (project I2854).

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Notes

  1. 1.

    The approach in [6] differs from [10] and [9] in the treatment of non-flat information orderings. In this paper we only consider flat orderings where \(\mathbf {u} \) is the only value considered less informative than others.

  2. 2.

    We call a parent \(n'\) of a node n l-parent if the link \((n',n)\) is labelled with l.

  3. 3.

    We will often leave \(\varSigma \) implicit in definitions from now on.

  4. 4.

    The values \(val^m(max_{W,\preccurlyeq }(l))\) and \(val^m(min_{W,\preccurlyeq }(l))\) are only defined when \(\hat{\preccurlyeq }\) is an order that has a maximal, respectively, minimal, element for every subset of V. The expressions \(max_{W,\preccurlyeq }(l)\) and \(min_{W,\preccurlyeq }(l)\) may only be used when this is the case.

  5. 5.

    The use of \(min_{W}\) and \(max_{W}\) is restricted to settings where the label domain L is numeric.

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Authors and Affiliations

  1. Computer Science Department, Leipzig University, Leipzig, Germany

    Gerhard Brewka

  2. Institute of Logic and Computation, Vienna University of Technology, Vienna, Austria

    Jörg Pührer & Stefan Woltran

Authors
  1. Gerhard Brewka
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  2. Jörg Pührer
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  3. Stefan Woltran
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Correspondence to Gerhard Brewka .

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Editors and Affiliations

  1. University of Calabria, Rende, Italy

    Francesco Calimeri

  2. University of Calabria, Rende, Italy

    Nicola Leone

  3. Mathematics and Computer Science, University of Calabria, Rende, Italy

    Marco Manna

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Brewka, G., Pührer, J., Woltran, S. (2019). Multi-valued GRAPPA. In: Calimeri, F., Leone, N., Manna, M. (eds) Logics in Artificial Intelligence. JELIA 2019. Lecture Notes in Computer Science(), vol 11468. Springer, Cham. https://doi.org/10.1007/978-3-030-19570-0_6

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  • DOI: https://doi.org/10.1007/978-3-030-19570-0_6

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