Clopen Knowledge Bases: Combining Description Logics and Answer Set Programming

Answer Set Programming (ASP) and ontology languages like Description Logics (DLs) play leading roles in Knowledge Representation and Reasoning (KR&R). ASP and DLs have largely orthogonal features because they make very different assumptions regarding the completeness of information, and thus reasoning techniques and algorithms that are deployed in ASP are significantly different from the ones used in DLs. Combining ASP, which makes the closed-world assumption (CWA), with DLs, which make the open-world assumption (OWA), into expressive hybrid languages that would enjoy the positive features of both has received significant attention in the last decade (see, e.g., [23, 24, 7, 20, 19]). However, progress on understanding the relationship between different hybrid languages, and their relationship with more standard languages like plain ASP, has been limited, as has the development of efficient reasoning algorithms and implementations. These and related problems are investigated in this paper for a new hybrid language called Clopen Knowledge Bases (CKBs), which generalizes and improves the prominent r-hybrid language [23] and DL+LOG [24]. Each CKB is a triple H = (P , φ,Σ), where P is a disjunctive Datalog program with “not” literals in rule bodies, φ is theory (e.g., in first-order logic), and Σ is a set of predicate symbols. Intuitively, Σ specifies the predicates that should be interpreted under the OWA; the remaining predicates should be interpreted under the CWA (see Section 3 for details). The contributions of this paper can be summarized as follows: ◦ We introduce CKBs, and define for them a stable model semantics, inspired the semantics given by Rosati to r-hybrid and DL+LOG KBs. In a nutshell, the major difference between the latter formalisms and CKBs is that CKBs allow to use CWA predicates in the theory. This allows for more convenient knowledge representation, but also causes technical challenges. ◦ We study automated reasoning in CKBs. To this end, in Section 4 we provide a general decidability result for checking entailment of ground atoms and consistency testing in CKBs H = (P , φ,Σ), where φ is expressed in the guarded negation fragment of FO (GNFO) [4]. This is a very expressive fragment that subsumes the more prominent guarded fragment of FO, as well as many expressive DLs. We give a NEXPTIME2EXPTIME upper bound for inference from GNFO-based CKBs (we note that satisfiability of GNFO formulas is 2EXPTIME-hard). ◦ In Section 5 we study reasoning in CKBs H = (P , φ,Σ), where φ is expressed in the very expressive DL ALCHIO. We show that the (combined) complexity of