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Comparative Expressivity for Structured Argumentation Frameworks with Uncertain Rules and Premises

Carlo Proietti, Antonio Yuste-Ginel

TL;DR

This work introduces a formal notion of expressivity to compare how abstract argumentation with qualitative uncertainty and structured argumentation with uncertain components relate. By grounding abstract models with a standard uncertainty framework (dep-arg-IAFs) and lifting structured formalisms (notably ASPIC$^{+}$) to the abstract level, it maps out the relative expressive power across several classes, including arg-IAFs, dep-arg-IAFs, rul-ISAF, prem-ISAF, and imp-arg-IAFs. Key findings include that structured frameworks with uncertainty surpass abstract incomplete AFs without dependencies but remain strictly below frameworks that include dependencies; within the structured side, rule-incomplete systems can be more expressive than premise-incomplete ones. The results weave together prior work on uncertain structured formalisms and abstract frameworks, providing a foundation for extending this comparison to other structured formalisms and for refining the theory in a journal-length treatment.

Abstract

Modelling qualitative uncertainty in formal argumentation is essential both for practical applications and theoretical understanding. Yet, most of the existing works focus on \textit{abstract} models for arguing with uncertainty. Following a recent trend in the literature, we tackle the open question of studying plausible instantiations of these abstract models. To do so, we ground the uncertainty of arguments in their components, structured within rules and premises. Our main technical contributions are: i) the introduction of a notion of expressivity that can handle abstract and structured formalisms, and ii) the presentation of both negative and positive expressivity results, comparing the expressivity of abstract and structured models of argumentation with uncertainty. These results affect incomplete abstract argumentation frameworks, and their extension with dependencies, on the abstract side, and ASPIC+, on the structured side.

Comparative Expressivity for Structured Argumentation Frameworks with Uncertain Rules and Premises

TL;DR

This work introduces a formal notion of expressivity to compare how abstract argumentation with qualitative uncertainty and structured argumentation with uncertain components relate. By grounding abstract models with a standard uncertainty framework (dep-arg-IAFs) and lifting structured formalisms (notably ASPIC) to the abstract level, it maps out the relative expressive power across several classes, including arg-IAFs, dep-arg-IAFs, rul-ISAF, prem-ISAF, and imp-arg-IAFs. Key findings include that structured frameworks with uncertainty surpass abstract incomplete AFs without dependencies but remain strictly below frameworks that include dependencies; within the structured side, rule-incomplete systems can be more expressive than premise-incomplete ones. The results weave together prior work on uncertain structured formalisms and abstract frameworks, providing a foundation for extending this comparison to other structured formalisms and for refining the theory in a journal-length treatment.

Abstract

Modelling qualitative uncertainty in formal argumentation is essential both for practical applications and theoretical understanding. Yet, most of the existing works focus on \textit{abstract} models for arguing with uncertainty. Following a recent trend in the literature, we tackle the open question of studying plausible instantiations of these abstract models. To do so, we ground the uncertainty of arguments in their components, structured within rules and premises. Our main technical contributions are: i) the introduction of a notion of expressivity that can handle abstract and structured formalisms, and ii) the presentation of both negative and positive expressivity results, comparing the expressivity of abstract and structured models of argumentation with uncertainty. These results affect incomplete abstract argumentation frameworks, and their extension with dependencies, on the abstract side, and ASPIC+, on the structured side.
Paper Structure (10 sections, 18 theorems, 2 equations, 1 figure)

This paper contains 10 sections, 18 theorems, 2 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Abstract Argumentation with Qualitative Uncertainty
  3. Abstract Argumentation Frameworks with Uncertain Arguments
  4. Relative Expressivity
  5. ASPIC$^{+}$ Frameworks with Uncertain Components
  6. ASPIC$^{+}$ in a Nutshell
  7. Uncertain Inference Rules
  8. Uncertain Premises
  9. Comparing Uncertain Rules and Premises
  10. Conclusion

Key Result

Proposition 1

Let $\mathsf{arg}\text{-}\mathsf{IAF} = \langle \mathcal{A}^{F},\mathcal{A}^{?},\mathcal{D} \rangle$ be an arg-IAF and $X \subseteq \mathsf{com}(\mathsf{arg}\text{-}\mathsf{IAF})$ a subset of its possible completions. There is always a set $\Delta$ of dependencies such that $\mathsf{com}(\langle \ma

Figures (1)

  • Figure 1: Expressivity results for rul-ISAFs and prem-ISAFs

Theorems & Definitions (67)

  • Definition 1: dung1995acceptability
  • Definition 2: arg-IAFs-original
  • Definition 3
  • Example 1
  • Definition 4: fazzingaijcai21
  • Example 2
  • Proposition 1
  • proof
  • Remark 1
  • Definition 5
  • ...and 57 more