An Algebraic Extension of Intuitionistic Linear Logic: The $L_!^S$-Calculus and Its Categorical Model

2025-04-16

AI Summary

This paper addresses the challenge of syntactically capturing linearity in Intuitionistic Linear Logic (ILL). It introduces a novel proof calculus, $L_!^S$, the first to integrate the $!$ modality with algebraic operations—specifically scalar multiplication and term addition—within a linear $lambda$-calculus, thereby rendering linearity explicitly expressible at the syntactic level. Unlike traditional ILL systems, $L_!^S$ supports logical modeling of measurement-free quantum programming and admits direct encoding of vectors and matrices. The authors establish its metatheory, proving confluence, strong normalization, and type preservation. Furthermore, they construct a sound and complete denotational semantics based on linear categories equipped with biproducts. This work provides a foundational framework unifying linear logic with algebraic quantum computation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We introduce the $L_!^S$-calculus, a linear lambda-calculus extended with scalar multiplication and term addition, that acts as a proof language for intuitionistic linear logic (ILL). These algebraic operations enable the direct expression of linearity at the syntactic level, a property not typically available in standard proof-term calculi. Building upon previous work, we develop the $L_!^S$-calculus as an extension of the $L^S$-calculus with the $!$ modality. We prove key meta-theoretical properties--subject reduction, confluence, strong normalisation, and an introduction property--as well as preserve the expressiveness of the original $L^S$-calculus, including the encoding of vectors and matrices, and the correspondence between proof-terms and linear functions. A denotational semantics is provided in the framework of linear categories with biproducts, ensuring a sound and adequate interpretation of the calculus. This work is part of a broader programme aiming to build a measurement-free quantum programming language grounded in linear logic.

Problem

• Extends intuitionistic linear logic with algebraic operations
• Develops $L_!^S$-calculus for syntactic linearity expression
• Provides denotational semantics for quantum programming language

Innovation

• Extends linear lambda-calculus with algebraic operations
• Develops $L_!^S$-calculus with $!$ modality
• Provides denotational semantics in linear categories