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On the Complexity of the Grounded Semantics for Infinite Argumentation Frameworks

Uri Andrews, Luca San Mauro

TL;DR

This work analyzes the grounded extension in infinite Dung-style argumentation frameworks using computability and set theory. It introduces the grounding ordinal to measure the transfinite length of the iterative defense process and proves a sharp upper bound: for computable AFs, grounding ordinals are bounded by $ω_1^{CK}$, while the grounded acceptance problem is $Π^1_1$-complete. It constructs computable AFs realizing every computable ordinal and even $ω_1^{CK}$, and develops a tree-rank analysis linking entry into the grounded extension to ordinal stages. The paper also shows that KP-set theory suffices to define grounded extensions and investigates the Turing-degree landscape, showing that not all $Π^1_1$ degrees arise as grounded extensions, and raises questions about a complete degree characterization.

Abstract

Argumentation frameworks, consisting of arguments and an attack relation representing conflicts, are fundamental for formally studying reasoning under conflicting information. We use methods from mathematical logic, specifically computability and set theory, to analyze the grounded extension, a widely-used model of maximally skeptical reasoning, defined as the least fixed-point of a natural defense operator. Without additional constraints, finding this fixed-point requires transfinite iterations. We identify the exact ordinal number corresponding to the length of this iterative process and determine the complexity of deciding grounded acceptance, showing it to be maximally complex. This shows a marked distinction from the finite case where the grounded extension is polynomial-time computable, thus simpler than other reasoning problems explored in formal argumentation.

On the Complexity of the Grounded Semantics for Infinite Argumentation Frameworks

TL;DR

This work analyzes the grounded extension in infinite Dung-style argumentation frameworks using computability and set theory. It introduces the grounding ordinal to measure the transfinite length of the iterative defense process and proves a sharp upper bound: for computable AFs, grounding ordinals are bounded by , while the grounded acceptance problem is -complete. It constructs computable AFs realizing every computable ordinal and even , and develops a tree-rank analysis linking entry into the grounded extension to ordinal stages. The paper also shows that KP-set theory suffices to define grounded extensions and investigates the Turing-degree landscape, showing that not all degrees arise as grounded extensions, and raises questions about a complete degree characterization.

Abstract

Argumentation frameworks, consisting of arguments and an attack relation representing conflicts, are fundamental for formally studying reasoning under conflicting information. We use methods from mathematical logic, specifically computability and set theory, to analyze the grounded extension, a widely-used model of maximally skeptical reasoning, defined as the least fixed-point of a natural defense operator. Without additional constraints, finding this fixed-point requires transfinite iterations. We identify the exact ordinal number corresponding to the length of this iterative process and determine the complexity of deciding grounded acceptance, showing it to be maximally complex. This shows a marked distinction from the finite case where the grounded extension is polynomial-time computable, thus simpler than other reasoning problems explored in formal argumentation.

Paper Structure

This paper contains 14 sections, 37 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Argumentation theory background
  4. Computability Background
  5. AFs with maximally large grounding ordinals and $\Pi^1_1$-completeness
  6. Upper bounds on grounding ordinals
  7. Rank analysis for groundedness
  8. Concluding Upper Bounds
  9. Conclusions beyond computable AFs
  10. The Turing degrees of grounded extensions
  11. Additional details for Section \ref{['sec:maximially long and hard']}
  12. Additional details for Section \ref{['sec:upper bounds']}
  13. Additional details for Section \ref{['sec:conclusions for non-computable AFs']}
  14. Additional details for Section \ref{['sec:Turing degrees missing']}

Key Result

Theorem 1

For computable AFs, the decision problem of determining membership in the grounded extension is $\Pi^1_1$-complete.This result establishes the complexity of the skeptical acceptance of arguments with respect to the complete semantics for infinite AFs, providing a proof of a theorem announced but lef

Figures (1)

  • Figure 1: Example from Baumann-Spanring BaumannSpanring of an AF with grounding ordinal $\omega\cdot2$.

Theorems & Definitions (72)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Theorem 4
  • Definition 5
  • Definition 6
  • proof
  • Definition 8
  • Definition 9
  • Definition 11
  • ...and 62 more