Structural Resolution: a Framework for Coinductive Proof Search and Proof Construction in Horn Clause Logic

Logic programming (LP) is a programming language based on first-order Horn clause logic that uses SLD-resolution as a semi-decision procedure. Finite SLD-computations are inductively sound and complete with respect to least Herbrand models of logic programs. Dually, the corecursive approach to SLD-resolution views infinite SLD-computations as successively approximating infinite terms contained in programs’ greatest complete Herbrand models. State-of-the-art algorithms implementing corecursion in LP are based on loop detection. However, such algorithms support inference of logical entailment only for rational terms, and they do not account for the important property of productivity in infinite SLD-computations. Loop detection thus lags behind coinductive methods in interactive theorem proving (ITP) and term-rewriting systems (TRS). Structural resolution is a newly proposed alternative to SLD-resolution that makes it possible to define and semi-decide a notion of productivity approrpriate to LP. In this paper we show that productivity supports the development of a new coinductive proof principle for LP that semi-decides logical entailment by observing finite fragments of resolution computations for productive programs. This severs the dependence of coinductive proof on term rationality, and puts coinductive methods in LP on par with productivity-based observational approaches to coinduction in ITP and TRS. We prove soundness of structural resolution relative to Herbrand model semantics for productive inductive, coinductive, and mixed inductive-coinductive logic programs.