Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rejection in Abstract Argumentation: Harder Than Acceptance?

Johannes K. Fichte, Markus Hecher, Yasir Mahmood, Arne Meier

TL;DR

This work introduces rejection augmented AFs (RCFs), extending abstract argumentation with per-argument rejection conditions (RCs) that are modeled as CNF formulas or ASP programs. By attaching RCs to arguments, RCs enable flexible, non-monotonic constraints on extensions, and the authors show RCs can express more than traditional CAFs or ADFs, sometimes increasing computational hardness by an additional level in the polynomial hierarchy. They provide a comprehensive complexity analysis across RC fragments (simple, propositional, tight, normal, disjunctive) and semantics, incorporating treewidth to obtain tight, ETH-aligned upper and lower bounds. The results demonstrate that rejection can be strictly harder than acceptance in many cases, and they establish simulation results showing RCFs can emulate CAFs and certain extensions-between semantics. The work also outlines practical modeling via ASP and points to future work in richer RC languages, non-ground RCs, and empirical evaluation with ASP solvers.

Abstract

Abstract argumentation is a popular toolkit for modeling, evaluating, and comparing arguments. Relationships between arguments are specified in argumentation frameworks (AFs), and conditions are placed on sets (extensions) of arguments that allow AFs to be evaluated. For more expressiveness, AFs are augmented with \emph{acceptance conditions} on directly interacting arguments or a constraint on the admissible sets of arguments, resulting in dialectic frameworks or constrained argumentation frameworks. In this paper, we consider flexible conditions for \emph{rejecting} an argument from an extension, which we call rejection conditions (RCs). On the technical level, we associate each argument with a specific logic program. We analyze the resulting complexity, including the structural parameter treewidth. Rejection AFs are highly expressive, giving rise to natural problems on higher levels of the polynomial hierarchy.

Rejection in Abstract Argumentation: Harder Than Acceptance?

TL;DR

This work introduces rejection augmented AFs (RCFs), extending abstract argumentation with per-argument rejection conditions (RCs) that are modeled as CNF formulas or ASP programs. By attaching RCs to arguments, RCs enable flexible, non-monotonic constraints on extensions, and the authors show RCs can express more than traditional CAFs or ADFs, sometimes increasing computational hardness by an additional level in the polynomial hierarchy. They provide a comprehensive complexity analysis across RC fragments (simple, propositional, tight, normal, disjunctive) and semantics, incorporating treewidth to obtain tight, ETH-aligned upper and lower bounds. The results demonstrate that rejection can be strictly harder than acceptance in many cases, and they establish simulation results showing RCFs can emulate CAFs and certain extensions-between semantics. The work also outlines practical modeling via ASP and points to future work in richer RC languages, non-ground RCs, and empirical evaluation with ASP solvers.

Abstract

Abstract argumentation is a popular toolkit for modeling, evaluating, and comparing arguments. Relationships between arguments are specified in argumentation frameworks (AFs), and conditions are placed on sets (extensions) of arguments that allow AFs to be evaluated. For more expressiveness, AFs are augmented with \emph{acceptance conditions} on directly interacting arguments or a constraint on the admissible sets of arguments, resulting in dialectic frameworks or constrained argumentation frameworks. In this paper, we consider flexible conditions for \emph{rejecting} an argument from an extension, which we call rejection conditions (RCs). On the technical level, we associate each argument with a specific logic program. We analyze the resulting complexity, including the structural parameter treewidth. Rejection AFs are highly expressive, giving rise to natural problems on higher levels of the polynomial hierarchy.
Paper Structure (24 sections, 15 theorems, 7 equations, 5 figures, 1 table)

This paper contains 24 sections, 15 theorems, 7 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related Works.
  4. Preliminaries
  5. Abstract Argumentation.
  6. Answer Set Programs (ASP).
  7. QBFs.
  8. Tree Decompositions and Treewidth.
  9. TDs for AFs.
  10. Rejection Augmented AFs (RCFs)
  11. Semantics.
  12. Using ASP Programs.
  13. Simulating Frameworks
  14. Comparison to CAFs.
  15. RCFs for Extensions In-Between Semantics.
  16. ...and 9 more sections

Key Result

Proposition 2

For any arbitrary QBF $\phi$ of quantifier rank $\ell>0$, the problem $\ell\hbox{-} \textsc{QSat}\xspace$ can be solved in time $\mathop{\mathrm{\mathsf{tower}}}\nolimits(\ell, \mathcal{O}(\mathsf{tw}(\mathcal{G}_{\varphi})))\cdot\mathop{\mathrm{{poly}}}\nolimits(\left|\mathrm{var}(\phi)\right|)$.

Figures (5)

  • Figure 1: Argumentation framework for Example \ref{['ex:AF']}.
  • Figure 2: RCF modeling an excerpt of every day research.
  • Figure 3: ASP enables non-monotonicity for RCs.
  • Figure 4: "Sub-framework" of an AF with stable extension.
  • Figure 5: Primal graph and a TD of Figure \ref{['fig:CCF-complex']}.

Theorems & Definitions (39)

  • Example 1
  • Proposition 2: Chen04a
  • Proposition 3: FichteHecherPfandler20
  • Definition 4
  • Definition 5
  • Example 6
  • Example 7
  • Definition 8
  • Lemma 11
  • proof
  • ...and 29 more