A Type-Theoretic Approach to Structural Resolution

TL;DR

This paper introduces a type-theoretic framework for analyzing structural resolution in logic programming, highlighting its advantages in handling corecursion and providing a formal basis for understanding its operational properties.

Contribution

It develops a type system to analyze S-resolution and SLD-resolution, and introduces realizability transformation to ensure productivity and non-overlapping properties in logic programs.

Findings

S-resolution allows systematic separation of derivation steps.

Realizability transformation makes programs productive and non-overlapping.

S-resolution and SLD-resolution are equivalent for programs with these properties.

Abstract

Structural resolution (or S-resolution) is a newly proposed alternative to SLD-resolution that allows a systematic separation of derivations into term-matching and unification steps. Productive logic programs are those for which term-matching reduction on any query must terminate. For productive programs with coinductive meaning, finite term-rewriting reductions can be seen as measures of observation in an infinite derivation. Ability of handling corecursion in a productive way is an attractive computational feature of S-resolution. In this paper, we make first steps towards a better conceptual understanding of operational properties of S-resolution as compared to SLD-resolution. To this aim, we propose a type system for the analysis of both SLD-resolution and S-resolution. We formulate S-resolution and SLD-resolution as reduction systems, and show their soundness relative to the type system. One of the central methods of this paper is realizability transformation, which makes logic programs productive and non-overlapping. We show that S-resolution and SLD-resolution are only equivalent for programs with these two properties.