Interpolable Formulas in Equilibrium Logic and Answer Set Programming
Interpolation is an important property of classical and many non-classical logics that has been shown to have interesting applications in computer science and AI. Here we study the Interpolation Property for the the non-monotonic system of equilibrium logic, establishing weaker or stronger forms of interpolation depending on the precise interpretation of the inference relation. These results also yield a form of interpolation for ground logic programs under the answer sets semantics. For disjunctive logic programs we also study the property of uniform interpolation that is closely related to the concept of variable forgetting. The first-order version of equilibrium logic has analogous Interpolation properties whenever the collection of equilibrium models is (first-order) definable. Since this is the case for so-called safe programs and theories, it applies to the usual situations that arise in practical answer set programming.
💡 Research Summary
The paper investigates the interpolation property within equilibrium logic, a non‑monotonic logical framework that underlies answer set programming (ASP). Interpolation, a well‑studied feature of classical logics, states that for any two formulas A and B such that A entails B, there exists an intermediate formula I that uses only the symbols common to A and B and satisfies A ⊢ I and I ⊢ B. Extending this notion to equilibrium logic is non‑trivial because the semantics involve equilibrium models—classical models that are also minimal with respect to a specific ordering.
The authors first distinguish two notions of entailment in equilibrium logic. The “strong” entailment (⊨ₑ) requires that every equilibrium model of A is also a model of B, while the “weak” entailment (⊢ₑ) adds a classical model condition before imposing minimality. For each entailment they prove a corresponding interpolation theorem. In the strong case, given A ⊨ₑ B, they construct an intermediate formula I that contains only the shared variables and preserves equilibrium minimality; the construction adapts the classical Craig interpolation proof by strengthening the interpolant with additional minimality constraints. In the weak case, a similar result holds under more restrictive assumptions, effectively providing a bridge between classical and equilibrium reasoning.
Having established logical interpolation, the paper translates these results to the realm of ASP. For a (possibly disjunctive) logic program P and a query Q, if P entails Q under the answer‑set semantics, there exists a “interpolating program” I that uses only the atoms common to P and Q and such that P entails I and I entails Q. The authors further explore uniform interpolation for disjunctive programs, which is tightly linked to the operation of variable forgetting. Variable forgetting removes selected atoms while preserving the set of answer sets projected onto the remaining atoms; the paper shows that this operation can be used to compute the uniform interpolant, and it provides an algorithmic sketch together with complexity considerations.
The investigation is then lifted to first‑order equilibrium logic. Here the authors note that interpolation holds whenever the collection of equilibrium models is first‑order definable. This condition is satisfied by “safe” programs and theories, where variables are appropriately restricted, ensuring that the equilibrium models can be captured by a first‑order theory. Consequently, all the interpolation results for the propositional case extend to these practical first‑order ASP settings.
Beyond the technical theorems, the authors discuss several applications. Interpolation enables modular reasoning: large knowledge bases can be decomposed into components linked by interpolants, facilitating verification of interfaces and reuse of proofs. Variable forgetting, as a concrete realization of uniform interpolation, supports knowledge hiding and abstraction without losing answer‑set semantics. The paper thus bridges a gap between abstract logical properties and concrete ASP engineering tasks, offering both theoretical insight and practical tools for knowledge representation, modular program design, and efficient reasoning in non‑monotonic settings.