First-order answer set programming as constructive proof search

TL;DR

This paper establishes a formal connection between first-order answer set programming and intuitionistic proof theory, showing their equivalence in certain logical fragments and revealing deep links between constructive provability and stable entailment.

Contribution

It introduces polynomial translations between FOASP and the Sigma_1 fragment of first-order intuitionistic logic, demonstrating their logical equivalence and providing new insights into answer set construction.

Findings

Sigma_1 formulas with fixed arity are as strong as FOASP

Constructive provability aligns with stable entailment

Answer set construction parallels intuitionistic refutation

Abstract

We propose an interpretation of the first-order answer set programming (FOASP) in terms of intuitionistic proof theory. It is obtained by two polynomial translations between FOASP and the bounded-arity fragment of the Sigma_1 level of the Mints hierarchy in first-order intuitionistic logic. It follows that Sigma_1 formulas using predicates of fixed arity (in particular unary) is of the same strength as FOASP. Our construction reveals a close similarity between constructive provability and stable entailment, or equivalently, between the construction of an answer set and an intuitionistic refutation. This paper is under consideration for publication in Theory and Practice of Logic Programming