Trichotomy and Dichotomy Results on the Complexity of Reasoning with Disjunctive Logic Programs

We present trichotomy results characterizing the complexity of reasoning with disjunctive logic programs. To this end, we introduce a certain definition schema for classes of programs based on a set of allowed arities of rules. We show that each such class of programs has a finite representation, and for each of the classes definable in the schema we characterize the complexity of the existence of an answer set problem. Next, we derive similar characterizations of the complexity of skeptical and credulous reasoning with disjunctive logic programs. Such results are of potential interest. On the one hand, they reveal some reasons responsible for the hardness of computing answer sets. On the other hand, they identify classes of problem instances, for which the problem is “easy” (in P) or “easier than in general” (in NP). We obtain similar results for the complexity of reasoning with disjunctive programs under the supported-model semantics. To appear in Theory and Practice of Logic Programming (TPLP)

💡 Research Summary

The paper investigates the computational complexity of reasoning tasks for disjunctive logic programs (DLPs) by introducing a systematic classification based on the arities of rules. An “arity schema” is defined as a set Δ ⊆ ℕ×ℕ of allowed pairs (h, b), where h denotes the number of atoms in the head of a rule and b the number of atoms in its body. For any Δ, the class P(Δ) consists of all DLPs whose every rule respects (h, b) ∈ Δ. A key technical result shows that even when Δ is infinite, it can be finitely represented by a regular expression (e.g., (≤k, ≤ℓ)), which makes the analysis tractable.

The authors first focus on the existence‑of‑an‑answer‑set (EAS) problem for P(Δ). They prove a trichotomy theorem: depending on the shape of Δ, the EAS problem for P(Δ) is either (i) P‑complete, (ii) NP‑complete, or (iii) Σ₂^P‑complete. Roughly, if the head arity is bounded by 1 and the body arity is bounded by a constant, the problem lies in P; if the head arity may be 2 or more while the body remains bounded, the problem jumps to NP; and if both head and body arities are unrestricted (or only loosely bounded), the problem reaches the second level of the polynomial hierarchy (Σ₂^P). The proof relies on reductions from canonical complete problems and on constructing appropriate normal forms for DLPs.

Next, the paper extends the analysis to skeptical and credulous reasoning. Skeptical reasoning asks whether a given literal holds in all answer sets, while credulous reasoning asks whether it holds in some answer set. Using the same arity classification, the authors derive a dichotomy for each reasoning mode: for Δ in the P‑complete region, both skeptical and credulous reasoning are P‑complete; for Δ in the NP‑complete region, skeptical reasoning becomes coNP‑complete and credulous reasoning remains NP‑complete; for Δ in the Σ₂^P‑complete region, skeptical reasoning is Π₂^P‑complete and credulous reasoning is Σ₂^P‑complete. These results precisely locate the increase in difficulty caused by allowing more expressive rule heads or larger bodies.

The study also covers the supported‑model semantics, which is weaker than the answer‑set semantics but still widely used. The authors show that the same trichotomy applies: the existence of a supported model for P(Δ) is P‑complete, NP‑complete, or Σ₂^P‑complete according to Δ, and the corresponding skeptical/credulous reasoning complexities follow the same pattern as for answer sets.

Beyond the formal theorems, the paper discusses practical implications. By selecting an arity schema Δ that falls into the “easy” (P) region, a system designer can guarantee polynomial‑time reasoning, which is valuable for applications requiring fast, predictable performance. Conversely, if a problem inherently needs disjunctions with large heads or unrestricted bodies, the analysis warns that the reasoning task will likely be intractable (NP‑ or Σ₂^P‑hard), suggesting the need for approximation, heuristics, or restricted modeling. The finite representation of infinite Δ also enables automated tools to check whether a given program belongs to a tractable class.

In summary, the work provides a clean, arity‑based taxonomy of DLP reasoning complexities, delivering both a theoretical foundation (trichotomy and dichotomy theorems) and actionable guidance for developers of logic‑programming systems. It clarifies exactly which syntactic features drive computational hardness and identifies large families of programs where reasoning remains efficiently solvable.