Skip to main content
SpringerLink
Log in

Polyhedral Labellings for Argumentation Frameworks

  • Conference paper
Book cover Scalable Uncertainty Management (SUM 2014)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 8720))

Included in the following conference series:

  • 496 Accesses

Abstract

In this paper we introduce polyhedral labellings associated to an argumentation framework. The name suggests the use of ideas from Polyhedral Combinatorics, an important topic in Combinatorial Optimization, mainly concerned with encoding combinatorial problems by means of systems of linear equations and inequalities, making these problems accessible to linear programming techniques. A polyhedral labelling for an argumentation framework AF = (A,D) is a polytope P AF , that is, a bounded set of solutions x ∈ ℝA (x a is the label of the argument a ∈ A), to a system of linear constraints, such that the set of integral vectors in P AF are exactly the incidence vectors of some specific type of Dung’s extensions. The linear constraints vary from the obvious x a  = 1 for each non attacked argument a, or x a  + x b  ≤ 1 for each attack (a,b) ∈ D (in order to assure Dung’s conflict-free condition), to more deep inequalities of the form ”the sum of the label of an argument and the labels of all its attackers is at least 1” or if (b,a) is an attack then ”the label of a is not greater than the sum of the labels of all attackers of b”.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baroni, P., Giacomin, M.: On principle-based evaluation of extension-based argumentation semantics. Artificial Intelligence 171, 675–700 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bell, C., Nerode, A., Ng, R.T., Subrahmanian, V.S.: Mixed integer programming methods for computing nonmonotonic deductive databases. Journal of ACM 41, 1178–1215 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bench-Capon, T.: Persuasion in practical argument using value-based argumentation frameworks. Journal of Logic and Computation 13, 429–448 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Caminada, M.: On the issue of reinstatement in argumentation. In: Fisher, M., van der Hoek, W., Konev, B., Lisitsa, A. (eds.) JELIA 2006. LNCS (LNAI), vol. 4160, pp. 111–123. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  5. Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence 77, 321–357 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gabbay, D.: Introducing equational semantics for argumentation networks. In: Liu, W. (ed.) ECSQARU 2011. LNCS, vol. 6717, pp. 19–35. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  7. Gabbay, D.: An equational approach to argumentation networks. Argumentation and Computation 3(2-3) (2012)

    Google Scholar 

  8. Gabbay, D.: The equational approach to cf2 semantics. In: Proc. of COMMA 2012, pp. 141–152 (2012)

    Google Scholar 

  9. Gratie, C., Florea, A.M.: Fuzzy labeling for argumentation frameworks. In: McBurney, P., Parsons, S., Rahwan, I. (eds.) ArgMAS 2011. LNCS, vol. 7543, pp. 1–8. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  10. Khachiyan, L.G.: A polynomial algorithm in linear programmming. Soviet Mathematics Doklady 20, 191–194 (1979)

    MATH  Google Scholar 

  11. Li, R., Zhou, D., Du, D.: Satisfiability and integer programming as complementary tools. In: Proc. of ASPDAC 2004 (2004)

    Google Scholar 

  12. Liu, G., Janhunen, T., Niemelä, I.: Answer set programming via mixed integer programming. In: Proc. of KR 2012 (2012)

    Google Scholar 

  13. Lowe, J.: CPLEX 10 Solver Manual. GAMS Development Corporation (2012)

    Google Scholar 

  14. Pollock, J.L.: Cognitive Carpentry. A Blueprint for How to Build a Person. MIT Press, Cambridge (1995)

    Google Scholar 

  15. Rahwan, I., Simari, G. (eds.): Argumentation in Artificial Intelligence. Springer (2009)

    Google Scholar 

  16. Schrijver, A.: Combinatorial Optimization - Polyhedra and Efficiency. Springer (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. MPII, Saarbrücken, Germany

    Cosmina Croitoru

Authors
  1. Cosmina Croitoru
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Istituto di Scienza e Tecnologie dell’Informazione (ISTI - CNR), Pisa, Italy

    Umberto Straccia

  2. Department of Computer Science and Information Systems, University of London, London, United Kingdom

    Andrea Calì

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Croitoru, C. (2014). Polyhedral Labellings for Argumentation Frameworks. In: Straccia, U., Calì, A. (eds) Scalable Uncertainty Management. SUM 2014. Lecture Notes in Computer Science(), vol 8720. Springer, Cham. https://doi.org/10.1007/978-3-319-11508-5_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-11508-5_8

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-11507-8

  • Online ISBN: 978-3-319-11508-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation