Quantified Equilibrium Logic and Hybrid Rules

Abstract

Abstract. In the ongoing discussion about combining rules and Ontologies on the Semantic Web a recurring issue is how to combine first-order classical logic with nonmonotonic rule languages. Whereas several modular approaches to define a combined semantics for such hybrid knowledge bases focus mainly on decidability issues, we tackle the matter from a more general point of view. In this paper we show how Quantified Equilibrium Logic (QEL) can function as a unified framework which embraces classical logic as well as disjunctive logic programs under the (open) answer set semantics. In the proposed variant of QEL we relax the unique names assumption, which was present in earlier versions of QEL. Moreover, we show that this framework elegantly captures the existing modular approaches for hybrid knowledge bases in a unified way.

Integrating Rules and Ontologies in the First-Order Stable Model Semantics (Preliminary Report)

“…By Choice(p) we denote the conjunction of "choice formulas" ∀x(p(x) ∨ ¬p(x)) for all predicate constants p in p where x is a list of distinct object variables whose length is the same as the arity of p. According to the QELbased integration approach in (de Bruijn et al 2007b), an HT-interpretation I of signature C, P T ∪ P P is a model of the hybrid knowledge K = (T , P) iff it is an equilibrium model of FO(T ) ∧ FO(P) ∧ Choice(P T ). Formula FO(T ) ∧ FO(P) ∧ Choice(P T ) is called the stable closure of K. The following proposition shows the relationship between the QEL-based approach and our approach.…”

“…Finally, in the tight integration under a unifying logic approach, T and P are treated uniformly by translating them into a uniform logic, and there is no principled separation between Σ T and Σ P . Examples are Hybrid MKNF KB (Motik & Rosati 2010), the first-order autoepistemic logic based integration (de Bruijn et al 2007a), and the QEL-based integration (de Bruijn et al 2007b). This approach is attractive since it provides a seamless integration of DLs and logic programs, and inforCopyright c 2011, American Association for Artificial Intelligence (www.aaai.org).…”

“…Then we present a reformulation of DL + log in the first-order stable model semantics, and show how the small predicate property, that is ensured by semi-safety, can be used to strengthen some complexity results related to DL + log. Then we relate our approach to the approach in (de Bruijn et al 2007b) that uses Quantified Equilibrium Logic as a unifying logic and, as a corollary, show a reformulation of g-hybrid knowledge bases. Next we relate our approach to nonmonotonic dlprograms.…”

“…Proposition 7 For any g-hybrid knowledge base K = (T , P), an interpretation I of signature C, P T ∪ P P is a model of K in the sense of (de Bruijn et al 2007b) …”

A Revised Concept of Safety for General Answer Set Programs

“…Two variations of this semantics, the open [8] and generalised open answer set [9] semantics, consider non-ground programs and open domains, thereby relaxing the PNA. For the present version of QEL the correspondence to answer sets can be summarised as follows (see [17,18,3]). If ϕ is a universal sentence in L = C, P (see §3 below), a total QHT model U, T, T of ϕ is an equilibrium model of ϕ iff T, T is a propositional equilibrium model of the grounding of ϕ with respect to the universe U .…”