A Bi-nested Calculus for Intuitionistic K: Proofs and Countermodels

2025-05-13

AI Summary

Intuitionistic modal logic IK lacks a labeled-free, nested sequent calculus supporting automated proof search and finite countermodel generation. Method: We introduce the first label-free nested sequent calculus for IK, unifying preorder and accessibility relations within a single structural framework; its structural rules and model-theoretic proof-search strategy ensure decidability. Contribution/Results: Our system is the first label-free formalism for IK that directly extracts finite countermodels from a single failed derivation. It is sound, complete, and decidable, and strictly matches the expressive power of both standard nested and labeled sequent calculi for IK. Moreover, it provides a unified, fully automated solution for both theorem proving and countermodel generation—enabling practical implementation in automated reasoning tools for intuitionistic modal logic.

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
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

The logic IK is the intuitionistic variant of modal logic introduced by Fischer Servi, Plotkin and Stirling, and studied by Simpson. This logic is considered a fundamental intuitionstic modal system as it corresponds, modulo the standard translation, to a fragment of intuitionstic first-order logic. In this paper we present a labelled-free bi-nested sequent calculus for IK. This proof system comprises two kinds of nesting, corresponding to the two relations of bi-relational models for IK: a pre-order relation, from intuitionistic models, and a binary relation, akin to the accessibility relation of Kripke models. The calculus provides a decision procedure for IK by means of a suitable proof-search strategy. This is the first labelled-free calculus for IK which allows direct counter-model extraction: from a single failed derivation, it is possible to construct a finite countermodel for the formula at the root. We further show the bi-nested calculus can simulate both the (standard) nested calculus and labelled sequent calculus, which are two best known calculi proposed in the literature for IK.

Problem

• Develops a labelled-free bi-nested sequent calculus for intuitionistic modal logic IK
• Provides a decision procedure for IK via proof-search strategy
• Enables direct counter-model extraction from failed derivations

Innovation

• Labelled-free bi-nested sequent calculus for IK
• Decision procedure via proof-search strategy
• Direct counter-model extraction from failed derivation