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On the Equivalence between Logic Programming and SETAF

João Alcântara, Renan Cordeiro, Samy Sá

TL;DR

This paper establishes a semantic and structural bridge between Normal Logic Programs ($NLP$) and Sets of Attacking Arguments Frameworks ($SETAF$). It develops translations $NLP\rightarrow SETAF$ and $SETAF\rightarrow NLP$, proving that key semantics align across the two formalisms, including an unexpected equivalence between $L$-stable models and semi-stable labellings. By introducing Redundancy-Free Atomic Logic Programs (RFALP) and a confluence-guaranteed transformation sequence $\mapsto_{\textit{UTPM}}$, the authors show RFALPs are as expressive as NLPs and that the translations are inverse for RFALP-SETaF pairs. Collectively, the work argues that NLPs and SETAFs are essentially the same formalism, with RFALP providing a robust, structure-preserving bridge that enables cross-pollination of techniques between logic programming and argumentation. The results advance a unified view of nonmonotonic reasoning and open avenues for applying SETAF-based reasoning methods to NLPs and vice versa.

Abstract

A framework with sets of attacking arguments (SETAF) is an extension of the well-known Dung's Abstract Argumentation Frameworks (AAFs) that allows joint attacks on arguments. In this paper, we provide a translation from Normal Logic Programs (NLPs) to SETAFs and vice versa, from SETAFs to NLPs. We show that there is pairwise equivalence between their semantics, including the equivalence between L-stable and semi-stable semantics. Furthermore, for a class of NLPs called Redundancy-Free Atomic Logic Programs (RFALPs), there is also a structural equivalence as these back-and-forth translations are each other's inverse. Then, we show that RFALPs are as expressive as NLPs by transforming any NLP into an equivalent RFALP through a series of program transformations already known in the literature. We also show that these program transformations are confluent, meaning that every NLP will be transformed into a unique RFALP. The results presented in this paper enhance our understanding that NLPs and SETAFs are essentially the same formalism. Under consideration in Theory and Practice of Logic Programming (TPLP).

On the Equivalence between Logic Programming and SETAF

TL;DR

This paper establishes a semantic and structural bridge between Normal Logic Programs () and Sets of Attacking Arguments Frameworks (). It develops translations and , proving that key semantics align across the two formalisms, including an unexpected equivalence between -stable models and semi-stable labellings. By introducing Redundancy-Free Atomic Logic Programs (RFALP) and a confluence-guaranteed transformation sequence , the authors show RFALPs are as expressive as NLPs and that the translations are inverse for RFALP-SETaF pairs. Collectively, the work argues that NLPs and SETAFs are essentially the same formalism, with RFALP providing a robust, structure-preserving bridge that enables cross-pollination of techniques between logic programming and argumentation. The results advance a unified view of nonmonotonic reasoning and open avenues for applying SETAF-based reasoning methods to NLPs and vice versa.

Abstract

A framework with sets of attacking arguments (SETAF) is an extension of the well-known Dung's Abstract Argumentation Frameworks (AAFs) that allows joint attacks on arguments. In this paper, we provide a translation from Normal Logic Programs (NLPs) to SETAFs and vice versa, from SETAFs to NLPs. We show that there is pairwise equivalence between their semantics, including the equivalence between L-stable and semi-stable semantics. Furthermore, for a class of NLPs called Redundancy-Free Atomic Logic Programs (RFALPs), there is also a structural equivalence as these back-and-forth translations are each other's inverse. Then, we show that RFALPs are as expressive as NLPs by transforming any NLP into an equivalent RFALP through a series of program transformations already known in the literature. We also show that these program transformations are confluent, meaning that every NLP will be transformed into a unique RFALP. The results presented in this paper enhance our understanding that NLPs and SETAFs are essentially the same formalism. Under consideration in Theory and Practice of Logic Programming (TPLP).
Paper Structure (16 sections, 65 theorems, 20 equations, 5 figures, 1 table)

This paper contains 16 sections, 65 theorems, 20 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. SETAF
  4. Logic Programs and Semantics
  5. From $\textit{NLP}$ to $\textit{SETAF}$
  6. $\textit{SETAF}$ Construction
  7. Equivalence Results
  8. From $\textit{SETAF}$ to $\textit{NLP}$
  9. On the relation between $\textit{RFALP}$s and $\textit{SETAF}$s
  10. On the Expressiveness of $\textit{RFALP}$s
  11. Conclusion and Future Works
  12. Proofs of Theorems
  13. Theorems and Proofs from Section \ref{['s:nlp-setaf']}
  14. Theorems and Proofs from Section \ref{['s:setaflp']}
  15. Theorems and Proofs from Section \ref{['s:slp-setaf']}
  16. ...and 1 more sections

Key Result

Lemma 0

Let $P$ be an $\textit{NLP}$, $\mathcal{I} = \left< T, F \right>$ an interpretation and $\Omega_P(\mathcal{I}) = \left< T', F' \right>$ the least 3-valued model of $\frac{P}{\mathcal{I}}$. It holds

Figures (5)

  • Figure 1: A $\textit{SETAF}$$\mathfrak{A}$
  • Figure 2: Statements constructed from $P$
  • Figure 3: A $\textit{SETAF}$$\mathfrak{A}_P = (\mathcal{A}_P,\mathit{Att}_P)$
  • Figure 4: $\textit{NLP}$$P$ and its associated $\textit{SETAF}$$\mathfrak{A}_P$
  • Figure 5: Possible $\textit{SETAF}$s associated with $P$

Theorems & Definitions (130)

  • Definition 1: $\textit{SETAF}$ nielsen2006generalization
  • Definition 2: Labellings flouris2019comprehensive
  • Definition 3: Semantics flouris2019comprehensive
  • Example 1
  • Definition 4: caminada15equivalence
  • Definition 5: 3-Valued Herbrand Interpretation przymusinski90well-founded
  • Definition 6: $\Psi$ Operator przymusinski90well-founded
  • Definition 7
  • Example 2
  • Definition 8: Statements and Arguments
  • ...and 120 more