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On the Correspondence of Non-flat Assumption-based Argumentation and Logic Programming with Negation as Failure in the Head

Anna Rapberger, Markus Ulbricht, Francesca Toni

TL;DR

This work shows a correspondence between non-flat ABA and LPs with negation as failure in their head and defines set-stable ABA semantics, originally defined for the fragment of non-flat ABA called bipolar ABA.

Abstract

The relation between (a fragment of) assumption-based argumentation (ABA) and logic programs (LPs) under stable model semantics is well-studied. However, for obtaining this relation, the ABA framework needs to be restricted to being flat, i.e., a fragment where the (defeasible) assumptions can never be entailed, only assumed to be true or false. Here, we remove this restriction and show a correspondence between non-flat ABA and LPs with negation as failure in their head. We then extend this result to so-called set-stable ABA semantics, originally defined for the fragment of non-flat ABA called bipolar ABA. We showcase how to define set-stable semantics for LPs with negation as failure in their head and show the correspondence to set-stable ABA semantics.

On the Correspondence of Non-flat Assumption-based Argumentation and Logic Programming with Negation as Failure in the Head

TL;DR

This work shows a correspondence between non-flat ABA and LPs with negation as failure in their head and defines set-stable ABA semantics, originally defined for the fragment of non-flat ABA called bipolar ABA.

Abstract

The relation between (a fragment of) assumption-based argumentation (ABA) and logic programs (LPs) under stable model semantics is well-studied. However, for obtaining this relation, the ABA framework needs to be restricted to being flat, i.e., a fragment where the (defeasible) assumptions can never be entailed, only assumed to be true or false. Here, we remove this restriction and show a correspondence between non-flat ABA and LPs with negation as failure in their head. We then extend this result to so-called set-stable ABA semantics, originally defined for the fragment of non-flat ABA called bipolar ABA. We showcase how to define set-stable semantics for LPs with negation as failure in their head and show the correspondence to set-stable ABA semantics.
Paper Structure (16 sections, 10 theorems, 33 equations)

This paper contains 16 sections, 10 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Logic programs with negation as failure in head
  4. Assumption-based Argumentation
  5. Stable Semantics Correspondence
  6. From LPs to ABAFs
  7. From ABAFs to LPs
  8. Relating LP and LP-ABA
  9. From ABA to LP-ABA
  10. Denial Integrity Constraints in ABA
  11. Set-Stable Model Semantics
  12. Bipolar LPs and Set-Stable Semantics
  13. Set-stable Semantics in general (non-bipolar) LPs
  14. Relating ABA and LP under set-stable semantics
  15. Set-stable Semantics for General (non-bipolar) ABAFs
  16. ...and 1 more sections

Key Result

Theorem 3.4

Let $P$ be an LP and $D_P$ the ABAF corresponding to $P$. Then $I$ is a stable model of $P$ iff $\Delta(I)\in{\mathit{stb}}(D_P)$.

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • Definition 3.1
  • Example 3.2
  • ...and 34 more