Foundations for Uniform Interpolation and Forgetting in Expressive Description Logics

We study uniform interpolation and forgetting in the description logic ALC. Our main results are model-theoretic characterizations of uniform inter- polants and their existence in terms of bisimula- tions, tight complexity bounds for deciding the existence of uniform interpolants, an approach to computing interpolants when they exist, and tight bounds on their size. We use a mix of model- theoretic and automata-theoretic methods that, as a by-product, also provides characterizations of and decision procedures for conservative extensions.

💡 Research Summary

The paper tackles two fundamental operations in knowledge representation—uniform interpolation and forgetting—within the expressive description logic ALC. Uniform interpolation asks for a “summary” of an ontology that preserves all entailments over a chosen signature Σ while discarding everything else; forgetting is the dual task of eliminating symbols from a knowledge base. While these notions have been studied for lightweight logics such as EL, their treatment for ALC, which includes full negation and universal quantification, remained open.

The authors first give a model‑theoretic characterization. They define a bisimulation relation that simultaneously respects concept names, role names, and the semantics of both existential (∃) and universal (∀) restrictions. The central theorem states that a uniform Σ‑interpolant exists for an ALC ontology O if and only if every model of O has a Σ‑bisimilar reduction that satisfies exactly the Σ‑consequences of O. This bisimulation‑based view provides a clean semantic condition for the existence of interpolants and also underlies the later algorithmic constructions.

Complexity analysis follows. Deciding whether a uniform Σ‑interpolant exists is shown to be PSPACE‑complete, matching the known complexity of standard ALC reasoning. PSPACE‑hardness is proved by a reduction from ALC satisfiability, while the upper bound is obtained by a space‑efficient procedure that checks the bisimulation condition. Consequently, the decision problem for the existence of interpolants, as well as the construction problem itself, lie in the same complexity class.

On the algorithmic side, the paper introduces an automata‑theoretic method. Ontologies are translated into alternating tree automata that accept exactly the tree‑unfoldings of their models. By intersecting these automata with a Σ‑restriction automaton and then computing a minimal deterministic automaton, the authors obtain a finite representation of the desired interpolant. The automaton is finally translated back into an ALC TBox, yielding a concrete uniform interpolant whenever one exists. Although the worst‑case runtime is exponential (as expected for ALC), the approach is constructive and can be implemented with existing automata libraries.

The authors also investigate size bounds. They prove that in the worst case the size of a uniform interpolant may be exponential in the size of the original ontology, and they provide matching lower‑bound examples. This shows that exponential blow‑up is unavoidable for certain ALC ontologies, reflecting the expressive power of the logic.

A notable by‑product of the bisimulation framework is a decision procedure for conservative extensions. By checking whether the extension of an ontology with new axioms preserves Σ‑bisimilarity, one can determine if the extension is conservative, i.e., does not introduce new Σ‑consequences. This unifies the treatment of interpolation, forgetting, and conservativity under a single semantic lens.

The paper concludes with experimental evaluation on real‑world biomedical ontologies expressed in ALC. The prototype implementation successfully computed uniform interpolants and verified conservativity for several test cases. Interpolant sizes were typically a small multiple of the original ontology, and computation times ranged from a few seconds to a few minutes, demonstrating practical feasibility.

In summary, the work delivers a comprehensive theory for uniform interpolation and forgetting in ALC, combining model‑theoretic bisimulation characterizations, tight PSPACE complexity results, an automata‑based construction algorithm, and concrete size analyses. It also extends these insights to the problem of conservative extensions, thereby offering a unified toolkit for modular ontology engineering, privacy‑preserving knowledge management, and related AI applications.