Complexity Classifications for Logic-Based Argumentation

We consider logic-based argumentation in which an argument is a pair (Φ, α), where the support Φ is a minimal consistent set of formulae taken from a given knowledge base (usually denoted by Δ) that entails the claim α (a formula). We study the complexity of three central problems in argumentation: the existence of a support Φ⊆Δ, the verification of a support, and the relevance problem (given ψ, is there a support Φ such that ψ ∈ Φ?). When arguments are given in the full language of propositional logic, these problems are computationally costly tasks: the verification problem is DP-complete; the others are Σ<sup>p</sup><sub>2</sub>-complete. We study these problems in Schaefer's famous framework where the considered propositional formulae are in generalized conjunctive normal form. This means that formulae are conjunctions of constraints built upon a fixed finite set of Boolean relations Γ (the constraint language). We show that according to the properties of this language Γ, deciding whether there exists a support for a claim in a given knowledge base is either polynomial, NP-complete, coNP-complete, or Σ<sup>p</sup><sub>2</sub>-complete. We present a dichotomous classification, P or DP-complete, for the verification problem and a trichotomous classification for the relevance problem into either polynomial, NP-complete, or Σ<sup>p</sup><sub>2</sub>-complete. These last two classifications are obtained by means of algebraic tools.

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