Evaluation of Queries under Closed-World Assumption

Abstract

The minimal entailment ⊢Min has been characterized elsewhere by

$P \vdash _{{\text{Min}}} \varphi {\text{iff }}Cn(P \cup \{ \varphi \} ) \cap Pos \subseteq Cn(P),$

where Cn is the first-order consequence operation, P is a set of clauses (indefinite deductive data base; in short: a data base), ϕ is a clause (a query), and Pos is the set of positive (that is, bodiless) ground clauses. In this paper, we address the problem of the computational feasibility of criterion (1). Our objective is to find a query evaluation algorithm that decides P ⊢Min ϕ by what we call indefinite modeling, without actually computing all ground positive consequences of P and P ∪ {ϕ}. For this purpose, we introduce the concept of minimal indefinite Herbrand model MP of P, which is defined as the set of subsumption-minimal ground positive clauses provable from P. The algorithm first computes MP by finding the least fixed-point of an indefinite consequence operator μ TIP. Next, the algorithm verifies whether every ground positive clause derivable from MP ∪ {ϕ} by one application of the parallel positive resolution rule (in short: the PPR rule) is subsumed by an element of MP. We prove that the PPR rule, which can derive only positive clauses, is positively complete, that is, every positive clause provable from a data base P is derivable from P by means of subsumption and finitely many applications of PPR. From this we conclude that the presented algorithm is partially correct and that it eventually halts if both P and MP are finite. Moreover, we indicate how the algorithm can be modified to handle data bases with infinite indefinite Herbrand models. This modification leads to a concept of universal model that allows for nonground clauses in its Herbrand base and appears to be a good candidate for representation of indefinite deductive data bases.