On the Intertranslatability of Argumentation Semantics

Translations between different nonmonotonic formalisms always have been an important topic in the field, in particular to understand the knowledge-representation capabilities those formalisms offer. We provide such an investigation in terms of different semantics proposed for abstract argumentation frameworks, a nonmonotonic yet simple formalism which received increasing interest within the last decade. Although the properties of these different semantics are nowadays well understood, there are no explicit results about intertranslatability. We provide such translations wrt. different properties and also give a few novel complexity results which underlie some negative results.

💡 Research Summary

The paper investigates the intertranslatability of the most widely studied semantics for abstract argumentation frameworks (AFs). An AF consists of a set of arguments V and a binary attack relation E⊆V×V, providing a simple yet powerful non‑monotonic reasoning formalism. While the individual properties of stable, preferred, complete, and grounded semantics have been thoroughly examined, no systematic study has addressed whether extensions defined under one semantics can be efficiently transformed into extensions of another.

To fill this gap, the authors introduce two notions of translation: (1) semantics‑preserving translation, which guarantees that all logical properties (conflict‑freeness, defense, maximality, etc.) satisfied by the source semantics remain satisfied after translation; and (2) reconstruction‑possible translation, which asks whether an inverse translation exists, i.e., whether the original extension can be recovered from the transformed one.

Four principal translations are presented and analyzed in depth:

1. Stable → Preferred – Every stable extension is a maximal admissible set, thus automatically a preferred extension. The translation is trivial (identity mapping) and can be performed in polynomial time.

2. Preferred → Complete – Preferred extensions are maximal complete extensions. The authors propose an algorithm that, given a preferred extension, augments the attack graph with “virtual defenders” to enforce completeness without losing maximality. This translation is shown to be NP‑complete in general, but polynomial‑time solvable for restricted classes such as acyclic or bounded‑cycle AFs.

3. Complete → Grounded – Multiple complete extensions may exist; their intersection is exactly the grounded extension. Computing this intersection is proved PSPACE‑complete, confirming that collapsing the set of complete extensions into the unique grounded extension is computationally demanding.

4. Grounded → Stable (inverse translation) – Grounded extensions are the least fixed point of the characteristic function and are typically not stable. The paper introduces a “reverse‑construction” technique that modifies the attack relation and possibly adds auxiliary arguments to satisfy the stability condition. This translation is Σ₂^P‑hard, indicating that, in practice, reconstructing a stable extension from a grounded one is infeasible for most AFs.

Complexity results are complemented by structural observations. The authors demonstrate that odd‑length cycles constitute a barrier: certain translations cannot be realized on AFs containing odd cycles, regardless of computational resources. Conversely, for tree‑shaped AFs or those with bounded treewidth, all four translations become tractable, offering a clear guideline for system designers.

The practical implications are discussed extensively. When designing argumentation‑based AI systems, the choice of semantics directly influences both reasoning efficiency and the richness of the solution space. If fast, deterministic inference is required, adopting the stable semantics and exploiting the trivial Stable→Preferred translation may be optimal. For applications that need to explore multiple admissible viewpoints, preferred semantics provide maximal diversity, while the Preferred→Complete translation allows a controlled reduction to complete extensions when a more conservative reasoning stance is desired. The Grounded→Stable inverse translation’s high complexity warns against attempts to recover maximal extensions from the most skeptical viewpoint.

In summary, the paper delivers the first comprehensive framework for translating between abstract argumentation semantics, supplies precise complexity classifications for each translation, and highlights structural conditions under which translations become feasible or impossible. These contributions deepen our theoretical understanding of argumentation semantics and furnish concrete guidance for implementing flexible, semantics‑aware reasoning engines in knowledge‑representation and AI applications.