Reasoning with Forest Logic Programs and f-hybrid Knowledge Bases

Open Answer Set Programming (OASP) is an undecidable framework for integrating ontologies and rules. Although several decidable fragments of OASP have been identified, few reasoning procedures exist. In this article, we provide a sound, complete, and terminating algorithm for satisfiability checking w.r.t. Forest Logic Programs (FoLPs), a fragment of OASP where rules have a tree shape and allow for inequality atoms and constants. The algorithm establishes a decidability result for FoLPs. Although believed to be decidable, so far only the decidability for two small subsets of FoLPs, local FoLPs and acyclic FoLPs, has been shown. We further introduce f-hybrid knowledge bases, a hybrid framework where \SHOQ{} knowledge bases and forest logic programs co-exist, and we show that reasoning with such knowledge bases can be reduced to reasoning with forest logic programs only. We note that f-hybrid knowledge bases do not require the usual (weakly) DL-safety of the rule component, providing thus a genuine alternative approach to current integration approaches of ontologies and rules.

💡 Research Summary

The paper tackles the undecidability of Open Answer Set Programming (OASP) by introducing Forest Logic Programs (FoLPs), a syntactically restricted fragment where rules have a tree shape, may contain constants and inequality atoms, and enjoy the “forest model property”. This property guarantees that if a unary predicate is satisfiable, there exists a labeled forest model—each constant serves as a root of a tree, and additional anonymous trees may be attached. FoLPs thus extend Conceptual Logic Programs (CoLPs) by allowing constants, enabling the simulation of SHOQ nominals.

A tableau‑based algorithm is presented for checking satisfiability with respect to FoLPs. The algorithm builds a completion structure that incrementally expands a partial model. Because OASP’s minimal‑model semantics differs from the closed‑world assumption of ordinary ASP, a simple subset‑blocking condition is insufficient. The authors therefore devise a more sophisticated blocking mechanism that combines depth limits with a double‑exponential bound on the number of individuals required. The procedure is proven sound, complete, and terminating, with worst‑case double‑exponential time complexity (one exponential level higher than reasoning with simple CoLPs). Importantly, FoLPs, as well as their subclasses (local, acyclic, and simple FoLPs), possess the bounded finite model property, which underlies the termination guarantee.

Building on this foundation, the paper defines f‑hybrid knowledge bases (f‑KBs), which integrate a SHOQ TBox/ABox with a FoLP rule component. Unlike existing hybrid approaches that enforce (weak) DL‑safety on the rule part, f‑KBs impose no such restriction; instead, the entire hybrid knowledge base is translated into an equivalent FoLP. Consequently, reasoning over f‑KBs reduces to the FoLP satisfiability problem already solved by the tableau algorithm. This yields a genuine non‑monotonic integration of ontologies and rules, supporting minimal‑model semantics for the rule component.

The authors also introduce simple FoLPs, a subclass that allows constants and inequalities while restricting predicate recursion. This restriction simplifies the blocking condition and lowers the complexity by one exponential level, making implementation more feasible.

Overall, the work delivers (1) a decidability result for the full FoLP fragment via a sound, complete, terminating algorithm; (2) an expressive hybrid formalism (f‑KBs) that combines SHOQ with FoLPs without DL‑safety constraints; and (3) complexity analyses showing double‑exponential worst‑case performance, with opportunities for lower complexity in restricted subclasses. The contributions advance the state of the art in integrating description logics and rule‑based reasoning under open answer set semantics.