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Relative Expressiveness of Defeasible Logics II

Michael J. Maher

TL;DR

The paper advances the study of relative expressiveness in the DL family of defeasible logics by formalizing a simulation-based framework and proving that all logics in DL are equally expressive when considering additions of facts, and that individual defeat is as expressive as team defeat when considering additions of rules. It achieves this via two key transformation strategies: ABsimAP (blocked-ambiguity simulates propagated-ambiguity) and APsimAB (propagated-ambiguity simulates blocked-ambiguity), each carefully encoding inference, support, and defeat mechanics with fresh literals and auxiliary rules. To address additions of rules, extended transformations TD are developed to align team- and individual-defeat semantics, yielding mutual expressiveness results for the corresponding logics. Collectively, the work shows that the notable differences among DL logics stem solely from ambiguity handling, thereby completing the expressiveness landscape for DL and suggesting avenues for linking to related defeasible frameworks. The formal apparatus relies on modular theory additions, language-separation constraints, and monotone fixpoint arguments to establish the simulation-based equivalences. The conclusions underscore the robustness of DL’s expressiveness landscape and point toward further cross-framework comparisons (e.g., with MG99 WFDL and other defeasible systems).

Abstract

(Maher 2012) introduced an approach for relative expressiveness of defeasible logics, and two notions of relative expressiveness were investigated. Using the first of these definitions of relative expressiveness, we show that all the defeasible logics in the DL framework are equally expressive under this formulation of relative expressiveness. The second formulation of relative expressiveness is stronger than the first. However, we show that logics incorporating individual defeat are equally expressive as the corresponding logics with team defeat. Thus the only differences in expressiveness of logics in DL arise from differences in how ambiguity is handled. This completes the study of relative expressiveness in DL begun in \cite{Maher12}.

Relative Expressiveness of Defeasible Logics II

TL;DR

The paper advances the study of relative expressiveness in the DL family of defeasible logics by formalizing a simulation-based framework and proving that all logics in DL are equally expressive when considering additions of facts, and that individual defeat is as expressive as team defeat when considering additions of rules. It achieves this via two key transformation strategies: ABsimAP (blocked-ambiguity simulates propagated-ambiguity) and APsimAB (propagated-ambiguity simulates blocked-ambiguity), each carefully encoding inference, support, and defeat mechanics with fresh literals and auxiliary rules. To address additions of rules, extended transformations TD are developed to align team- and individual-defeat semantics, yielding mutual expressiveness results for the corresponding logics. Collectively, the work shows that the notable differences among DL logics stem solely from ambiguity handling, thereby completing the expressiveness landscape for DL and suggesting avenues for linking to related defeasible frameworks. The formal apparatus relies on modular theory additions, language-separation constraints, and monotone fixpoint arguments to establish the simulation-based equivalences. The conclusions underscore the robustness of DL’s expressiveness landscape and point toward further cross-framework comparisons (e.g., with MG99 WFDL and other defeasible systems).

Abstract

(Maher 2012) introduced an approach for relative expressiveness of defeasible logics, and two notions of relative expressiveness were investigated. Using the first of these definitions of relative expressiveness, we show that all the defeasible logics in the DL framework are equally expressive under this formulation of relative expressiveness. The second formulation of relative expressiveness is stronger than the first. However, we show that logics incorporating individual defeat are equally expressive as the corresponding logics with team defeat. Thus the only differences in expressiveness of logics in DL arise from differences in how ambiguity is handled. This completes the study of relative expressiveness in DL begun in \cite{Maher12}.

Paper Structure

This paper contains 15 sections, 19 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Defeasible Logic
  3. Simulating Defeasible Logics
  4. Blocked Ambiguity Simulates Propagated Ambiguity
  5. Propagated Ambiguity Simulates Blocked Ambiguity
  6. Simulation of Individual Defeat wrt Addition of Rules
  7. Simulation of Team Defeat wrt Addition of Rules
  8. Conclusions
  9. Appendix
  10. Inference Rules for ${\bf DL}$
  11. Proofs of results
  12. Blocked Ambiguity Simulates Propagated Ambiguity
  13. Propagated Ambiguity Simulates Blocked Ambiguity
  14. Simulation of Individual Defeat wrt Addition of Rules
  15. Simulation of Team Defeat wrt Addition of Rules

Key Result

Theorem 1

Let $D$ be a defeasible theory.

Figures (1)

  • Figure 1: Relative expressiveness of logics in ${\bf DL}$ using simulation wrt addition of rules

Theorems & Definitions (41)

  • Theorem 1: Inclusion Theorem TOCL10
  • Definition 2
  • Definition 3
  • Example 4
  • Theorem 5
  • Definition 6
  • Example 7
  • Theorem 8
  • Definition 9
  • Example 10
  • ...and 31 more