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Abstract Argumentation with Subargument Relations

Beishui Liao

TL;DR

This paper addresses the limitation of Dung-style abstract argumentation, where internal argument structure is hidden behind a purely attack-based representation. It introduces Subargument Argumentation Frameworks (SAF) that treat subargument relations as a primitive, orthogonal to direct conflict represented by Att, and develops structure-sensitive defence, admissibility, and fixpoint semantics that preserve the core Dung architecture while making structural dependencies explicit. A key result is that forgetting subargument structure preserves extension-based semantics but loses important structural information, establishing that SAFs are strictly more expressive than attack-only abstractions; this expressiveness is captured via a core-based lifting approach separating conflict handling from status propagation. The framework supports principled explanations through local, subargument-grounded justifications, maintains a conservative extension of Dung semantics, and clarifies how structural information can drive explainability and dynamic updates in argumentation systems.

Abstract

Dung's abstract argumentation framework characterises argument acceptability solely via an attack relation, deliberately abstracting from the internal structure of arguments. While this level of abstraction has enabled a rich body of results, it limits the ability to represent structural dependencies that are central in many structured argumentation formalisms, in particular subargument relations. Existing extensions, including bipolar argumentation frameworks, introduce support relations, but these do not capture the asymmetric and constitutive nature of subarguments or their interaction with attacks. In this paper, we study abstract argumentation frameworks enriched with an explicit subargument relation, treated alongside attack as a basic relation. We analyse how subargument relations interact with attacks and examine their impact on fundamental semantic properties. This framework provides a principled abstraction of structural information and clarifies the role of subarguments in abstract acceptability reasoning.

Abstract Argumentation with Subargument Relations

TL;DR

This paper addresses the limitation of Dung-style abstract argumentation, where internal argument structure is hidden behind a purely attack-based representation. It introduces Subargument Argumentation Frameworks (SAF) that treat subargument relations as a primitive, orthogonal to direct conflict represented by Att, and develops structure-sensitive defence, admissibility, and fixpoint semantics that preserve the core Dung architecture while making structural dependencies explicit. A key result is that forgetting subargument structure preserves extension-based semantics but loses important structural information, establishing that SAFs are strictly more expressive than attack-only abstractions; this expressiveness is captured via a core-based lifting approach separating conflict handling from status propagation. The framework supports principled explanations through local, subargument-grounded justifications, maintains a conservative extension of Dung semantics, and clarifies how structural information can drive explainability and dynamic updates in argumentation systems.

Abstract

Dung's abstract argumentation framework characterises argument acceptability solely via an attack relation, deliberately abstracting from the internal structure of arguments. While this level of abstraction has enabled a rich body of results, it limits the ability to represent structural dependencies that are central in many structured argumentation formalisms, in particular subargument relations. Existing extensions, including bipolar argumentation frameworks, introduce support relations, but these do not capture the asymmetric and constitutive nature of subarguments or their interaction with attacks. In this paper, we study abstract argumentation frameworks enriched with an explicit subargument relation, treated alongside attack as a basic relation. We analyse how subargument relations interact with attacks and examine their impact on fundamental semantic properties. This framework provides a principled abstraction of structural information and clarifies the role of subarguments in abstract acceptability reasoning.
Paper Structure (29 sections, 17 theorems, 33 equations)

This paper contains 29 sections, 17 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dung argumentation frameworks
  4. Characteristic function and fixpoints
  5. Graph reachability
  6. A Motivating Example
  7. The setting.
  8. Abstraction by attacks alone.
  9. Structural grounding of defence.
  10. Propagation beyond direct conflict.
  11. Implications.
  12. Subargument Frameworks and Structure-sensitive Semantics
  13. Argumentation Frameworks with Subargument Relations
  14. Structured-sensitive Defence
  15. Admissibility and Fixpoints
  16. ...and 14 more sections

Key Result

Proposition 1

Let $\mathcal{F}=(A,\mathit{Att},\mathit{Sub})$ be an SAF and $E\subseteq A$. Then $\mathsf{F}_{\mathcal{F}}(E)=\mathsf{Def}_{\mathcal{F}}(E)$ is subargument-closed, i.e.,

Theorems & Definitions (45)

  • Definition 1: Subargument Argumentation Framework
  • Definition 2: Structure-sensitive defence
  • Definition 3: Structure-aware conflict-freeness
  • Definition 4: Admissible set
  • Definition 5: Characteristic function
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Definition 6: Complete extension
  • ...and 35 more