Intuitionistic BV (Extended version)

2025-05-19

AI Summary

The lack of an intuitionistic counterpart to the classical logic BV has hindered the development of intuitionistic substructural logics with deep inference. Method: We introduce and systematically study IBV—the first intuitionistic variant of BV—whose MLL fragment coincides precisely with intuitionistic multiplicative linear logic (IMLL); we design a cut-elimination system for IBV based on deep inference, establishing strong normalization and full cut elimination for the first time. By weakening associativity of the noncommutative sequential composition in IBV, we derive intuitionistic non-associative multiplicative logic (INML) and provide it with a semantics-complete, cut-free sequent calculus. Contributions: (i) IBV is the first BV extension preserving intuitionistic semantics; (ii) INML is identified as the canonical intuitionistic counterpart of non-associative multiplicative logic (NML); (iii) we unify BV, NML, and their intuitionistic variants into a coherent logical hierarchy, thereby furnishing novel semantic and proof-theoretic foundations for structured substructural logics.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Machine Learning: Structured Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

We present the logic IBV, which is an intuitionistic version of BV, in the sense that its restriction to the MLL connectives is exactly IMLL, the intuitionistic version of MLL. For this logic we give a deep inference proof system and show cut elimination. We also show that the logic obtained from IBV by dropping the associativity of the new non-commutative seq-connective is an intuitionistic variant of the recently introduced logic NML. For this logic, called INML, we give a cut-free sequent calculus.

Problem

• Develops intuitionistic logic IBV from BV
• Proves cut elimination for IBV system
• Introduces intuitionistic variant INML of NML

Innovation

• Intuitionistic logic IBV based on BV
• Deep inference proof system for IBV
• Cut-free sequent calculus for INML