Deductive Inference for the Interiors and Exteriors of Horn Theories

In this paper, we investigate the deductive inference for the interiors and exteriors of Horn knowledge bases, where the interiors and exteriors were introduced by Makino and Ibaraki to study stability properties of knowledge bases. We present a linear time algorithm for the deduction for the interiors and show that it is co-NP-complete for the deduction for the exteriors. Under model-based representation, we show that the deduction problem for interiors is NP-complete while the one for exteriors is co-NP-complete. As for Horn envelopes of the exteriors, we show that it is linearly solvable under model-based representation, while it is co-NP-complete under formula-based representation. We also discuss the polynomially solvable cases for all the intractable problems.

💡 Research Summary

The paper investigates deductive inference for the interiors and exteriors of Horn knowledge bases, concepts originally introduced by Makino and Ibaraki to capture stability properties of logical theories. An α‑interior of a Horn theory Σ consists of all models v such that every assignment within Hamming distance ≤α from v also satisfies Σ; conversely, an α‑exterior contains those models for which at least one α‑neighbor satisfies Σ. These notions allow one to ask, for a given clause φ, whether φ is entailed by the interior (Σ ⊨intα φ) or by the exterior (Σ ⊨extα φ).

The authors first treat the problem under the usual clause‑based representation. For interiors, they exploit the defining property of Horn clauses—at most one positive literal—to show that it suffices to examine the minimal models of Σ. By constructing the minimal model in linear time and checking the α‑neighborhood condition, they obtain a deterministic O(|Σ|+|φ|) algorithm. Thus interior deduction is tractable (linear time).

In contrast, exterior deduction is shown to be co‑NP‑complete. The hardness proof reduces the complement of SAT to the exterior entailment problem: given any CNF ψ, one builds a Horn theory Σ and a clause φ such that ψ is unsatisfiable iff Σ ⊨extα φ. The reduction works even when α is a fixed constant, demonstrating that the existence of a single α‑neighbor satisfying Σ introduces an inherent NP‑hard existential component; the complement of this problem lands in co‑NP, establishing co‑NP‑completeness.

The paper then switches to a model‑based representation, where a Horn theory is given explicitly as a set M of its models. Under this view, interior entailment asks whether every model in M has all its α‑neighbors also in M. This decision problem is NP‑complete: a reduction from 3‑SAT shows that checking the universal neighbor condition captures the difficulty of SAT. Exterior entailment becomes the complementary problem—does there exist a model in M with an α‑neighbor in M?—and is therefore co‑NP‑complete.

Next, the authors study the Horn envelope of the exterior, i.e., the smallest Horn theory that contains the α‑exterior. They prove that, when the input is given as a model set, the envelope can be computed in linear time by generating Horn clauses that capture each model’s minimal positive support. However, when the input is a clause set, computing the envelope is co‑NP‑complete, because one must verify that no non‑Horn clause is required to represent the exterior, a task equivalent to checking universal properties over exponentially many candidates.

Finally, the paper identifies several polynomial‑time solvable subclasses. If α is a constant (e.g., α = 1) or the Horn theory consists of a single clause, interior deduction remains linear. Similarly, when the clause size is bounded or the theory is already in a certain normal form, both interior and exterior problems admit efficient algorithms. These tractable cases are relevant for practical knowledge‑base systems where the underlying Horn theories often have limited structure.

In summary, the work delineates a clear computational boundary: interior deduction for Horn theories is efficiently solvable under clause‑based representation, while exterior deduction is inherently hard (co‑NP‑complete) in both clause‑ and model‑based settings. The Horn envelope of the exterior is easy to obtain from models but hard from clauses. The results provide a nuanced understanding of stability versus flexibility in Horn knowledge bases and guide the choice of representation for applications that require either robust inference (interiors) or controlled adaptability (exteriors).