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Defeasibility in Answer Set Programs with Defaults and Argumentation Rules

A unifying framework for defeasible reasoning in disjunctive logic programs under answer set semantics using argumentation rules.
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Defeasibility in Answer Set Programs with Defaults and Argumentation Rules

Defeasible reasoning allows logical axioms to be true “by default” while permitting other conflicting axioms to undercut or negate them. This paper introduces ASPDA (Answer Set Programs via Argumentation Rules), a unifying framework for defeasible reasoning in disjunctive logic programs under answer set semantics.

The Challenge of Defeasible Reasoning

Current approaches to defeasible reasoning suffer from fragmentation. Multiple dissimilar and incompatible theories exist, making the field difficult to navigate and different techniques hard to compare. Most existing work focuses on either well-founded semantics or stable model semantics, but not both.

ASPDA addresses this fragmentation by extending the authors’ earlier LPDA framework (Logic Programming with Defaults and Argumentation Theories) from well-founded semantics to answer set programming. Together, LPDA and ASPDA provide a unified approach to most existing defeasible reasoning theories in logic programming.

Core Framework

Tagged Rules and Argumentation

ASPDA uses tagged rules where each rule carries an identifier (@tag). Rules can be either strict (definite statements) or defeasible (default assumptions that can be defeated).

@r1 loaded(?Gun,?Time+1) :- loaded(?Gun,?Time).  // Defeasible frame axiom
@r2 neg loaded(?Gun,?Time+1) :- shoot(?Gun,?Time). // Strict rule

Argumentation rulesets define when rule instances become defeated through the predicate $defeated. These rulesets capture different theories of defeasible reasoning while maintaining uniform syntax.

Answer Set Semantics Extension

The framework extends traditional answer set semantics by modifying truth valuation for defeasible rules. A defeasible rule is satisfied if either:

  • The rule is defeated (all head literals have $defeated true)
  • The rule body is false
  • The rule head is true

This creates a feedback loop where the truth of $defeated predicates influences which rules contribute to the answer set.

Key Technical Results

Reduction to Standard ASP

Theorem: Every ASPDA program can be reduced to ordinary answer set programming. For each defeasible rule with head disjunction L₁ ∨ ... ∨ Lₙ, the reduction creates multiple rules of the form:

∨ᵢ∈ₖ Lᵢ :- Body ∧ ⋀ᵢ∈ₖ naf $defeated(r,Lᵢ) ∧ ⋀ⱼ∈ₙ₋ₖ $defeated(r,Lⱼ)

This reduction enables implementation using existing ASP solvers like DLV.

Head-Cycle-Free Programs

For head-cycle-free ASPDA programs, a shifting transformation reduces disjunctive programs to non-disjunctive ones. This mirrors classical results for regular ASP but requires more complex transformations due to the defeasibility mechanism.

Argumentation Ruleset Examples

A-GCLP (Generalized Courteous Logic Programming)

This ruleset implements priority-based defeat:

$defeated(?T,?L) :- $defeats(?T',?L',?T,?L).
$defeats(?T1,?L1,?T2,?L2) :- 
    $refutes(?T1,?L1,?T2,?L2) ∧ 
    naf $defeated(?T1,?L1) ∧ 
    naf #strict(?T2,?L2).

Rules are defeated when higher-priority rules produce conflicting conclusions.

Defeasible Logic Adaptation

An adaptation of Antoniou et al.’s Defeasible Logic uses different defeat conditions:

$defeated(?T,?L) :- 
    $conflict(?T,?T') ∧ 
    head(?T',?L') ∧ 
    $definitely(?L').

This creates different behavior in cases with priority cycles.

Practical Applications

The Turkey Shoot example demonstrates ASPDA’s capabilities. Given uncertain initial conditions (one gun loaded, unknown which), the framework correctly infers that firing guns in sequence will kill the turkey regardless of initial uncertainty.

The framework handles:

  • Frame axioms (persistence rules)
  • Action effects (state changes)
  • Priority conflicts (higher-priority rules override defaults)
  • Disjunctive uncertainty (multiple possible initial states)

Implementation Advantages

ASPDA provides several benefits over existing approaches:

  1. Unified syntax: Different defeasible theories use the same rule format
  2. Modular argumentation: Theories are separated from domain knowledge
  3. Standard reduction: Existing ASP solvers can execute ASPDA programs
  4. Compositional semantics: Multiple argumentation theories can coexist

Unlike meta-interpretation approaches or custom transformations for each preference strategy, ASPDA provides a single model-theoretic semantics. The variable component is the argumentation ruleset (typically 20-30 rules) rather than complex program transformations requiring hundreds of lines of code.

Conclusion

ASPDA successfully unifies defeasible reasoning approaches under answer set semantics. The framework’s reduction theorems prove its implementability, while the head-cycle-free results show that many disjunctive defeasible programs can be handled efficiently. By abstracting defeasibility mechanisms into argumentation rulesets, ASPDA enables practitioners to choose appropriate reasoning strategies while maintaining uniform syntax and semantics.

The work demonstrates that complex defeasible reasoning can be elegantly captured through the interaction between domain rules and meta-level argumentation principles, providing a foundation for practical defeasible reasoning systems.