Through the Looking Glass, and what Horn Clause Programs Found There

TL;DR

This paper investigates Dual Horn clauses, revealing their symmetries with Horn clauses, and demonstrates their practical applications in constructive negation, explainability in AI, and efficient reasoning with polynomial complexity.

Contribution

It introduces the concept of Dual Horn clauses as a constructive negation tool, provides a compilation scheme to integrate them with Horn clauses, and applies them to enhance AI reasoning and explainability.

Findings

Dual Horn clauses enable constructive falsification explanations.

They have polynomial complexity similar to Horn clauses.

A compilation scheme allows seamless integration with Horn clause programs.

Abstract

Dual Horn clauses mirror key properties of Horn clauses. This paper explores the ``other side of the looking glass'' to reveal some expected and unexpected symmetries and their practical uses. We revisit Dual Horn clauses as enablers of a form of constructive negation that supports goal-driven forward reasoning and is valid both intuitionistically and classically. In particular, we explore the ability to falsify a counterfactual hypothesis in the context of a background theory expressed as a Dual Horn clause program. With Dual Horn clause programs, by contrast to negation as failure, the variable bindings in their computed answers provide explanations for the reasons why a statement is successfully falsified. Moreover, in the propositional case, by contrast to negation as failure as implemented with stable models semantics in ASP systems, and similarly to Horn clause programs, Dual Horn clause programs have polynomial complexity. After specifying their execution model with a metainterpreter, we devise a compilation scheme from Dual Horn clause programs to Horn clause programs, ensuring their execution with no performance penalty and we design the embedded SymLP language to support combined Horn clause and Dual Horn clause programs. As a (motivating) application, we cast LLM reasoning chains into propositional Horn and Dual Horn clauses that work together to constructively prove and disprove goals and enhance Generative AI with explainability of reasoning chains.