Hyper Tableaux and Disjunctive Logic Programming

For disjunctive logic programs (DLPs) there are several proposals for deening interpreters, There have also been diierent approaches to assign least xpoints to DLPs. This paper proves that there exist an eecient proof procedure, namely hyper tableaux, which can be understood as a direct implementation of some of the well known xpoint iteration techniques. We show how a hyper tableaux refutation can be transformed into a restart model elimination refutation. This result links the bottom-up to a top-down semantics for DLPs, and thus generalizes the standard result in Lloyd, 1987] saying that any nite iteration of the T-operator for deenite programs can be simulated top-town in a SLD-refutation. A diierent approach to obtain a top-down calculus is to replace all literals in the input clause set by their complements. We demonstrate that in this setting hyper tableaux generalize Rajasekar's SLO-Resolution. In the next section we give the proof theoretical part of this paper, which is based on the hyper tableaux calculus from Baumgartner et al., 1996]. In the following two sections we compare this calculus to xpoint iteration techniques: there is one seminal paper by Minker and Rajasekar Minker and Rajasekar, 1990] which introduces a consequence operator to deene a semantics for positive disjunctive logic programs by xpoint iteration over states. We will relate hyper tableaux to this iteration. Another approach by Fernandez and Minker (Fernandez and Minker, 91]), gives a bottom up evaluation of hierarchical disjunctive databases. We will demonstrate, that this approach is a special case of hyper tableaux. In Section 5 we discuss the relation of hyper tableaux to SLO-resolution, and in Section 6 we relate hyper tableaux to restart model elimination. 1 Preliminaries In what follows, we assume that the reader is familiar with the basic concepts of rst-order logic. A clause is a multiset of literals, usually written as the disjunction A 1 _ _ A m _ :B 1 _ _ :B n or the implication A 1 the variables occurring in clauses are considered implicitly as being universally quantiied, a clause is considered logically as a disjunction of literals, and a ((nite) clause set is taken as a conjunction of clauses. A ground clause is a clause containing no variables. Literal K is an instance of literal L, written as K L or L K, ii K = LL for some substitution. Let L denote the complement of a literal L. Two literals …