Proving Correctness and Completeness of Normal Programs - a Declarative Approach

We advocate a declarative approach to proving properties of logic programs. Total correctness can be separated into correctness, completeness and clean termination; the latter includes non-floundering. Only clean termination depends on the operational semantics, in particular on the selection rule. We show how to deal with correctness and completeness in a declarative way, treating programs only from the logical point of view. Specifications used in this approach are interpretations (or theories). We point out that specifications for correctness may differ from those for completeness, as usually there are answers which are neither considered erroneous nor required to be computed. We present proof methods for correctness and completeness for definite programs and generalize them to normal programs. For normal programs we use the 3-valued completion semantics; this is a standard semantics corresponding to negation as finite failure. The proof methods employ solely the classical 2-valued logic. We use a 2-valued characterization of the 3-valued completion semantics which may be of separate interest. The presented methods are compared with an approach based on operational semantics. We also employ the ideas of this work to generalize a known method of proving termination of normal programs.Comment: To appear in Theory and Practice of Logic Programming (TPLP). 44 page

The Constructive method for query containment checking (extended version)

We present a new method that checks Query Containment for queries with negated derived atoms and/or integrity constraints. Existing methods for Query Containment checking that deal with these cases do not check actually containment but another related property called uniform containment, which is a sufficient but not necessary condition for containment. Our method can be seen as an extension of the canonical databases approach beyond the class of conjunctive queries.Postprint (published version

SLT-Resolution for the Well-Founded Semantics

Global SLS-resolution and SLG-resolution are two representative mechanisms for top-down evaluation of the well-founded semantics of general logic programs. Global SLS-resolution is linear for query evaluation but suffers from infinite loops and redundant computations. In contrast, SLG-resolution resolves infinite loops and redundant computations by means of tabling, but it is not linear. The principal disadvantage of a non-linear approach is that it cannot be implemented using a simple, efficient stack-based memory structure nor can it be easily extended to handle some strictly sequential operators such as cuts in Prolog. In this paper, we present a linear tabling method, called SLT-resolution, for top-down evaluation of the well-founded semantics. SLT-resolution is a substantial extension of SLDNF-resolution with tabling. Its main features include: (1) It resolves infinite loops and redundant computations while preserving the linearity. (2) It is terminating, and sound and complete w.r.t. the well-founded semantics for programs with the bounded-term-size property with non-floundering queries. Its time complexity is comparable with SLG-resolution and polynomial for function-free logic programs. (3) Because of its linearity for query evaluation, SLT-resolution bridges the gap between the well-founded semantics and standard Prolog implementation techniques. It can be implemented by an extension to any existing Prolog abstract machines such as WAM or ATOAM.Comment: Slight modificatio

Transforming acyclic programs

An unfold/fold transformation system is a source-to-source rewriting methodology devised to improve the efficiency of a program. Any such transformation should preserve the main properties of the initial program: among them, termination. In the field of logic programming, the class of acyclic programs plays an important role in this respect, since it is closely related to the one of terminating programs. The two classes coincide when negation is not allowed in the bodies of the clauses. We prove that the Unfold/Fold transformation system defined by Tamaki and Sato preserves the acyclicity of the initial program. From this result, it follows that when the transformation is applied to an acyclic program, then the finite failure set for definite programs is preserved; in the case of normal programs, all major declarative and operational semantics are preserved as well. These results cannot be extended to the class of left-terminating programs without modifying the definition of the transformation

A Complete Axiomatization of the Three valued Completion of Logic Programs

We prove the completeness of extended SLDNF-resolution for the new class of e-programs with respect to the three-valued completion of a logic program. Not only the class of allowed programs but also the class of definite programs are contained in the class of ε-programs. To understand better the three-valued completion of a logic program we introduce a formal system for three-valued logic in which one can derive exactly the three-valued consequences of the completion of a logic program. The system is proof theoretically interesting, since it is a fragment of Gentzen's sequent calculus L