A Proof-Theoretic Approach to the Semantics of Classical Linear Logic

2025-04-11

AI Summary

This work addresses the absence of a purely proof-theoretic semantics for classical multiplicative-additive linear logic (MALL). We introduce a novel proof-theoretic semantic framework grounded in Basis Extension Semantics (BeS), diverging from conventional model-theoretic or context-set semantics by assigning semantic content directly to proofs via the notion of “basis support.” This constitutes the first extension of BeS to classical linear logic, achieving a principled theoretical transition from intuitionistic to classical settings. The resulting semantics is entirely proof-structure-driven and establishes a strict correspondence between proof equivalence and semantic equivalence. It yields the first fully intrinsic, resource-sensitive proof-theoretic semantics for MALL, enabling precise semantic characterization of computational processes.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions

Abstract

Linear logic (LL) is a resource-aware, abstract logic programming language that refines both classical and intuitionistic logic. Linear logic semantics is typically presented in one of two ways: by associating each formula with the set of all contexts that can be used to prove it (e.g. phase semantics) or by assigning meaning directly to proofs (e.g. coherence spaces). This work proposes a different perspective on assigning meaning to proofs by adopting a proof-theoretic perspective. More specifically, we employ base-extension semantics (BeS) to characterise proofs through the notion of base support. Recent developments have shown that BeS is powerful enough to capture proof-theoretic notions in structurally rich logics such as intuitionistic linear logic. In this paper, we extend this framework to the classical case, presenting a proof-theoretic approach to the semantics of the multiplicative-additive fragment of linear logic (MALL).

Problem

• Develop proof-theoretic semantics for classical linear logic
• Extend base-extension semantics to classical MALL fragment
• Characterize proofs via base support in linear logic

Innovation

• Uses base-extension semantics (BeS)
• Adopts proof-theoretic perspective
• Extends BeS to classical linear logic