Function-free Programs Research Articles

Transformation-based bottom-up computation of the well-founded model

We present a framework for expressing bottom-up algorithms to compute the well-founded model of non-disjunctive logic programs. Our method is based on the notion of conditional facts and elementary program transformations studied by BRASS and DIX (Brass and Dix, 1994; Brass and Dix, 1999) for disjunctive programs. However, even if we restrict their framework to nondisjunctive programs, their ‘residual program’ can grow to exponential size, whereas for function-free programs our ‘program remainder’ is always polynomial in the size of the extensional database (EDB). We show that particular orderings of our transformations (we call them strategies) correspond to well-known computational methods like the alternating fixpoint approach (Van Gelder, 1989; Van Gelder, 1993), the well-founded magic sets method (Kemp et al., 1995) and the magic alternating fixpoint procedure (Morishita, 1996). However, due to the confluence of our calculi (first noted in Brass and Dix, 1998), we come up with computations of the well-founded model that are provably better than these methods. In contrast to other approaches, our transformation method treats magic set transformed programs correctly, i.e. it always computes a relevant part of the well-founded model of the original program. These results show that our approach is a valuable tool to analyze, compare, and optimize existing evaluation methods or to create new strategies that are automatically proven to be correct if they can be described by a sequence of transformations in our framework. We have also developed a prototypical implementation. Experiments illustrate that the theoretical results carry over to the implemented prototype and may be used to optimize real life systems.

Loop checking in SLD-derivations by well-quasi-ordering of goals

Loop checking mechanisms are used to detect and prune infinite SLD derivations, through run time checks which are introduced in logic program interpreters. Simple loop checks, i.e. checks which do not depend on the specific logic program, have been widely studied in literature. Since no sound and complete loop check exists even in the case of function-free programs, several subclasses have been characterized for which sound and complete loop checks can be determined. In this paper a theoretical framework for analysing properties of loop check mechanisms for logic programs is proposed, which exploits general mathematical results about well-quasi-ordered ( wqo) sets. In a way, the method can be viewed as a counterpart of well known techniques based on well-founded partial-ordering, used in termination proofs for rewriting systems and for logic programs. The main results are obtained on the basis of a combinatorial analysis of properties of wqo sets of goals. As shown in the paper, subclasses of programs, for which sound and complete simple loop checks exist, can be easily framed in the wqo approach. Reasons for the different behaviours of subsumption loop checks based on list and multiset goal comparisons are also plainly highlighted.

Constructive negation under the well-founded semantics

Constructive negation derives constraint answers for non-ground negative literals. Its incorporation into query evaluation under the well-founded semantics introduces two problems. One is the detection of repeated subgoals and the elimination of redundant answers, which is required in order to guarantee termination. The other is the interaction between constraint answers of non-ground negative literals and recursion through negation. This paper presents SLG CN for effective query evaluation with constructive negation under the well-founded semantics. It has two unique features. First, it supports reduction of constraint answers and redundant answer elimination and provides the first termination result for goal-oriented query evaluation with constructive negation for function-free programs. Second, it avoids repeated computation in a subgoal. Even if a non-ground negative literal depends upon some ground negative literals whose truth values are not completely determined when they are selected, the constraints and bindings for variables in the non-ground negative literal can still be propagated once and for all.

Disjunctive Stable Models: Unfounded Sets, Fixpoint Semantics, and Computation

Disjunctive logic programs have become a powerful tool in knowledge representation and commonsense reasoning. This paper focuses on stable model semantics, currently the most widely acknowledged semantics for disjunctive logic programs. After presenting a new notion of unfounded sets for disjunctive logic programs, we provide two declarative characterizations of stable models in terms of unfounded sets. One shows that the set of stable models coincides with the family of unfounded-free models (i.e., a model is stable iff it contains no unfounded atoms). The other proves that stable models can be defined equivalently by a property of their false literals, as a model is stable iff the set of its false literals coincides with its greatest unfounded set. We then generalize the well-founded WPoperator to disjunctive logic programs, give a fixpoint semantics for disjunctive stable models and present an algorithm for computing the stable models of function-free programs. The algorithm's soundness and completeness are proved and some complexity issues are discussed.

An Extension of Van Gelder's Alternating Fixpoint to Magic Programs

We study the application of the magic-sets transformation technique to Datalog⌝(function-free programs with negation) that may not have two-valued well-founded models. In this general setting, the well-founded model of the original program does not always agree with the well-founded model of the magic program derived by commonly used left-to-right sideways information-passing strategies on the query. In order to correct this disagreement we present a novel method that is obtained by slightly and naturally tailoring Van Gelder's alternating fixpoint technique to any magic program.

Tabled evaluation with delaying for general logic programs

SLD resolution with negation as finite failure (SLDNF) reflects the procedural interpretation of predicate calculus as a programming language and forms the computational basis for Prolog systems. Despite its advantages for stack-based memory management, SLDNF is often not appropriate for query evaluation for three reasons: (a) it may not terminate due to infinite positive recursion; (b) it may be terminate due to infinite recursion through negation; and (c) it may repeatedly evaluate the same literal in a rule body, leading to unacceptable performance. We address all three problems for goal-oriented query evaluation of general logic programs by presenting tabled evaluation with delaying, called SLG resolution. It has three distinctive features: (i) SLG resolution is a partial deduction procedure, consisting of seven fundamental transformations. A query is transformed step by step into a set of answers. The use of transformations separates logical issues of query evaluation from procedural ones. SLG allows an arbitrary computation rule for selecting a literal from a rule body and an arbitrary control strategy for selecting transformations to apply. (ii) SLG resolution is sound and search space complete with respect to the well-founded partial model for all non-floundering queries, and preserves all three-valued stable models. To evaluate a query under differenc three-valued stable models, SLG resolution can be enhanced by further processing of the answers of subgoals relevant to a query. (iii) SLG resolution avoids both positive and negative loops and always terminates for programs with the bounded-term-size property. It has a polynomial time data complexity for well-founded negation of function-free programs. Through a delaying mechanism for handling ground negative literals involved in loops, SLG resolution avoids the repetition of any of its derivation steps. Restricted forms of SLG resolution are identified for definite, locally stratified, and modularly stratified programs, shedding light on the role each transformation plays.

Query evaluation in deductive databases with alternating fixpoint semantics

First-order formulas allow natural descriptions of queries and rules. Van Gelder's alternating fixpoint semantics extends the well-founded semantics of normal logic programs to general logic programs with arbitrary first-order formulas in rule bodies. However, an implementation of general logic programs through the standard translation into normal logic programs does not preserve the alternating fixpoint semantics. This paper presents a direct method for goal-oriented query evaluation of general logic programs. Every general logic program is first transformed into a normal form where the body of each rule is either an existential conjunction of literals or a universal disjunction of literals. Techniques of memoing and loop checking are incorporated so that termination and polynomial-time data complexity are guaranteed for deductive databases (or function-free programs). Results of the soundness and search space completeness are established.

Redundancy Elimination and Loop Checks for Logic Programs

A simple analysis of the arguments developed by Bol et al. (Theoret. Comput. Sci.86, 35-79 (1991)) shows that an actual reason for the nonexistence of a complete sound simple check for all function-free programs is the presence in the resolvents of potentially unlimited sequences of atoms chained by common variables. This hints that a limitation of the number of variables generating this kind of chain could guarantee the applicability of complete simple loop checks. This line is followed in the paper, and quite general classes of logic programs are characterized, without any direct imposition on the structures of the rules. This objective is accomplished by exploiting a variant of SLD-resolution, which is able to perform a systematic elimination of redundant atoms from resolvents. As a notable result, it turns out that the equality loop check is complete for our class of logic programs. This seems to suggest that the necessity of using subsumption loop checks instead of equality checks is essentially due to the presence of redundant atoms in resolvents.

A procedural semantics for well-founded negation in logic programs

We introduce global SLS-resolution, a procedural semantics for well-founded negation as defined by Van Gelder, Ross, and Schlipf. Global SLS-resolution extends Przymusinski's SLS-resolution and may be applied to all programs, whether locally stratified or not. Global SLS-resolution is defined in terms of global trees, a new data structure representing the dependence of goals on derived negative subgoals. We prove that global SLS-resolution is sound with respect to the well-founded semantics and complete for nonfloundering queries. Although not effective in general, global SLS-resolution is effective for classes of “acrylic” programs and can be augmented with a memoing device to be effective for all function-free programs.