Implementing the First-Order Logic of Here and There

2026-01-07

Electronic Proceedings in Theoretical Computer Science

AI Summary

First-order intuitionistic logic of “Here and There” (HT) has long lacked efficient automated theorem provers. This work presents the first systematic implementation and evaluation of an automated prover tailored for first-order HT logic, integrating a native sequent calculus—enhanced with free-variable handling and Skolemization optimizations—and an embedding-based approach grounded in intuitionistic logic that unifies sequent, tableau, and connection calculi. Experimental results on a large-scale benchmark of first-order HT formulas demonstrate that the proposed methods substantially improve proof efficiency, thereby establishing a foundational framework for the design of future high-performance HT theorem provers.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We present automated theorem provers for the first-order logic of here and there (HT). They are based on a native sequent calculus for the logic of HT and an axiomatic embedding of the logic of HT into intuitionistic logic. The analytic proof search in the sequent calculus is optimized by using free variables and skolemization. The embedding is used in combination with sequent, tableau and connection calculi for intuitionistic first-order logic. All provers are evaluated on a large benchmark set of first-order formulas, providing a foundation for the development of more efficient HT provers.

Problem

• first-order logic
• here and there
• automated theorem proving
• intuitionistic logic

Innovation

• Logic of Here and There
• automated theorem proving
• sequent calculus
• intuitionistic logic
• Skolemization