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Non-flat ABA is an Instance of Bipolar Argumentation

Markus Ulbricht, Nico Potyka, Anna Rapberger, Francesca Toni

TL;DR

The paper addresses the challenge of representing non-flat assumption-based argumentation (ABA) within a bipolar abstract framework (BAF). It introduces instantiated BAFs (F_D) that encode ABA arguments, attacks, and a closure-based support relation, and then enhances this with premise-augmented BAFs (pBAFs) to faithfully capture admissible and complete ABA semantics. The authors establish mappings between ABA extensions and BAF/pBAF extensions under complete-based semantics, provide a detailed complexity analysis showing that (p)BAFs typically lie one level lower in the polynomial hierarchy than general non-flat ABA, and propose dispute trees to support computation and explainability. This work yields practical instantiation-based reasoning tools for non-flat ABA and opens avenues for richer explanations in bipolar argumentation. The results demonstrate a principled bridge between structured ABA and bipolar semantics, with explicit semantics, complexity results, and a pathway toward scalable solvers and explanations.

Abstract

Assumption-based Argumentation (ABA) is a well-known structured argumentation formalism, whereby arguments and attacks between them are drawn from rules, defeasible assumptions and their contraries. A common restriction imposed on ABA frameworks (ABAFs) is that they are flat, i.e., each of the defeasible assumptions can only be assumed, but not derived. While it is known that flat ABAFs can be translated into abstract argumentation frameworks (AFs) as proposed by Dung, no translation exists from general, possibly non-flat ABAFs into any kind of abstract argumentation formalism. In this paper, we close this gap and show that bipolar AFs (BAFs) can instantiate general ABAFs. To this end we develop suitable, novel BAF semantics which borrow from the notion of deductive support. We investigate basic properties of our BAFs, including computational complexity, and prove the desired relation to ABAFs under several semantics. Finally, in order to support computation and explainability, we propose the notion of dispute trees for our BAF semantics.

Non-flat ABA is an Instance of Bipolar Argumentation

TL;DR

The paper addresses the challenge of representing non-flat assumption-based argumentation (ABA) within a bipolar abstract framework (BAF). It introduces instantiated BAFs (F_D) that encode ABA arguments, attacks, and a closure-based support relation, and then enhances this with premise-augmented BAFs (pBAFs) to faithfully capture admissible and complete ABA semantics. The authors establish mappings between ABA extensions and BAF/pBAF extensions under complete-based semantics, provide a detailed complexity analysis showing that (p)BAFs typically lie one level lower in the polynomial hierarchy than general non-flat ABA, and propose dispute trees to support computation and explainability. This work yields practical instantiation-based reasoning tools for non-flat ABA and opens avenues for richer explanations in bipolar argumentation. The results demonstrate a principled bridge between structured ABA and bipolar semantics, with explicit semantics, complexity results, and a pathway toward scalable solvers and explanations.

Abstract

Assumption-based Argumentation (ABA) is a well-known structured argumentation formalism, whereby arguments and attacks between them are drawn from rules, defeasible assumptions and their contraries. A common restriction imposed on ABA frameworks (ABAFs) is that they are flat, i.e., each of the defeasible assumptions can only be assumed, but not derived. While it is known that flat ABAFs can be translated into abstract argumentation frameworks (AFs) as proposed by Dung, no translation exists from general, possibly non-flat ABAFs into any kind of abstract argumentation formalism. In this paper, we close this gap and show that bipolar AFs (BAFs) can instantiate general ABAFs. To this end we develop suitable, novel BAF semantics which borrow from the notion of deductive support. We investigate basic properties of our BAFs, including computational complexity, and prove the desired relation to ABAFs under several semantics. Finally, in order to support computation and explainability, we propose the notion of dispute trees for our BAF semantics.
Paper Structure (15 sections, 36 theorems, 11 equations, 3 figures, 1 table)

This paper contains 15 sections, 36 theorems, 11 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Abstract Argumentation.
  4. Assumption-based Argumentation.
  5. Closed Extensions for Bipolar AFs
  6. Instantiated BAFs
  7. From ABA to BAF
  8. From BAF to ABA
  9. BAFs and Admissible Semantics
  10. Computational Complexity
  11. Discussion and Related Work
  12. Conclusion
  13. Proof Details of Section \ref{['sec:ABA vs BAF']}
  14. Proof Details of Section \ref{['sec:pbafs']}
  15. Proof Details of Section \ref{['sec:computational complexity']}

Key Result

Lemma 3.4

Let $\mathcal{F} = (A, \mathrm{Att}, \mathrm{Sup})$ be a BAF and let $E\subseteq A$ and $a\in A$. Then $E$ defends $a$ iff for each attacker $b$ of a it holds that $E$ attacks $\mathit{cl}(\{b\})$.

Figures (3)

  • Figure 1: Construction \ref{['constr:co non-empty reduction']} applied to the formula $\Phi = \{x_1,x_2\}, \{\neg x_1, x_3\}, \{\neg x_1, \neg x_3\}$
  • Figure 2: Construction \ref{['constr:co and unsat forumla']} applied to the formula $\Psi = \{x_1,x_2\}, \{\neg x_1, x_3\}, \{\neg x_1, \neg x_3\}$
  • Figure 3: Construction \ref{['constr:skep adm pbaf']} applied to the formula $\Psi = \{x_1,x_2\}, \{\neg x_1, x_3\}, \{\neg x_1, \neg x_3\}$

Theorems & Definitions (82)

  • Example 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Lemma 3.4
  • Definition 3.5
  • ...and 72 more