Embedding Non-Ground Logic Programs into Autoepistemic Logic for Knowledge Base Combination

In the context of the Semantic Web, several approaches to the combination of ontologies, given in terms of theories of classical first-order logic and rule bases, have been proposed. They either cast rules into classical logic or limit the interaction between rules and ontologies. Autoepistemic logic (AEL) is an attractive formalism which allows to overcome these limitations, by serving as a uniform host language to embed ontologies and nonmonotonic logic programs into it. For the latter, so far only the propositional setting has been considered. In this paper, we present three embeddings of normal and three embeddings of disjunctive non-ground logic programs under the stable model semantics into first-order AEL. While the embeddings all correspond with respect to objective ground atoms, differences arise when considering non-atomic formulas and combinations with first-order theories. We compare the embeddings with respect to stable expansions and autoepistemic consequences, considering the embeddings by themselves, as well as combinations with classical theories. Our results reveal differences and correspondences of the embeddings and provide useful guidance in the choice of a particular embedding for knowledge combination.

💡 Research Summary

The paper addresses the problem of integrating ontologies, expressed as first‑order (FO) theories, with rule bases, expressed as logic programs, in the context of the Semantic Web. Existing approaches either translate rules into classical logic or restrict the interaction between the two components, which limits the expressive power of non‑monotonic reasoning. Autoepistemic Logic (AEL), which extends classical logic with a single modal belief operator L, offers a uniform host language that can naturally capture both monotonic ontological knowledge and non‑monotonic rule‑based knowledge.

The authors extend previous work that considered only propositional AEL embeddings to the non‑ground case, i.e., to logic programs that contain variables and function symbols. They define six embeddings into first‑order AEL (FO‑AEL): three for normal (non‑disjunctive) programs—τ_HP, τ_EB, and τ_EH—and three for disjunctive programs—τ_∨HP, τ_∨EB, and τ_∨EH. Each embedding rewrites a rule by inserting the modal operator L to represent “known” literals and by using ¬L to represent negation‑as‑failure. For example, a normal rule a ← b, not c becomes under τ_HP the formula L b → a ∧ ¬L c → a. The three variants differ in how they treat the head, the body, and the interaction with the modal operator.

A central technical result (Theorem 5.3) shows that all six embeddings are faithful with respect to objective ground atoms: the set of stable models of the original program coincides one‑to‑one with the set of objective ground atoms that appear in the stable expansions of any of the embeddings. Thus, at the level of ground atoms the embeddings preserve semantics. However, when one looks at non‑atomic formulas (e.g., formulas containing L, disjunctions, or nested negation) the embeddings diverge. This divergence becomes crucial when the embedded program is combined with an FO theory, because the additional modal structure can affect the theory’s deductive closure.

The paper conducts a systematic comparative study of the embeddings along two dimensions. First, it classifies programs into three syntactic categories—ground, safe, and arbitrary—and establishes inclusion and equivalence relationships among the sets of consequences produced by the different embeddings (Propositions 5.5‑5.9, Theorems 5.14 and 5.15). For safe programs, τ_HP and τ_EB yield the same non‑ground consequences, while τ_EH is strictly stronger for arbitrary programs. Second, it examines combinations of an FO theory Φ with a program P, considering three important families of FO theories: Horn, universal, and generalized Horn (the latter encompassing RDF‑Schema, OWL‑RL, OWL‑QL, etc.). The main combination theorem (Theorem 6.2) gives a complete picture:

• With a Horn theory and a safe program, τ_HP and τ_EB are interchangeable, but τ_EH may produce additional consequences.
• With a generalized Horn theory and an arbitrary program, only τ_EH guarantees full correspondence; the other embeddings may miss some consequences.
• With a universal theory and a ground program, all embeddings coincide.

These results provide precise conditions under which one can replace one embedding by another without altering the overall knowledge base semantics.

To illustrate practical relevance, the authors apply their framework to a running Semantic Web example involving RDF triples, an OWL ontology, and a non‑monotonic rule that defines “IndieMovie”. They show how different embeddings affect query answering over the combined knowledge base. If the ontology’s negative facts must be taken into account by the rule, the τ_EH embedding (which treats the modal operator more aggressively) is appropriate. If the goal is to keep the ontology and the rule base as independent as possible, τ_HP offers a more conservative integration.

The paper also discusses how the proposed FO‑AEL embeddings relate to other combination formalisms such as MKNF and quantified equilibrium logic (QEL). Unlike MKNF, which requires a separate translation layer, FO‑AEL can directly host both ontological axioms and rule clauses, allowing a more seamless integration.

In conclusion, the study delivers a thorough theoretical foundation for embedding non‑ground logic programs into FO‑AEL, establishes the exact logical relationships among six natural embeddings, and delineates when each embedding is suitable for knowledge‑base combination. The findings are directly applicable to Semantic Web technologies and suggest extensions to other modeling environments (e.g., UML + OCL).