Sharing and Linear Logic with Restricted Access (Extended Version)

2025-01-27

AI Summary

This paper addresses the modeling difficulty of evaluation strategies in linear logic arising from the conflation of “sharing” and “access” semantics. We propose Modal-Separated Classical Linear Logic (MSCLL), which decomposes the traditional exponential modality ! into two distinct modalities: ! (permitting copying and erasure but not access) and • (enabling further access), thereby formally distinguishing “copyability” from “accessibility” at the logical level for the first time. Based on MSCLL, we define the λ!•-calculus, establishing its type safety, confluence, and strong normalization. We further develop a Girard-style embedding into Multiplicative-Exponential Linear Logic (MELL) and prove cut elimination, showing that MSCLL is a conservative extension of MELL. The framework uniformly models call-by-name, call-by-value, and call-by-need–like evaluation strategies, providing a novel foundation for sharing-sensitive program semantics and type systems.

Technology Category

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

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

The two Girard translations provide two different means of obtaining embeddings of Intuitionistic Logic into Linear Logic, corresponding to different lambda-calculus calling mechanisms. The translations, mapping A ->B respectively to !A -o B and !(A -o B), have been shown to correspond respectively to call-by-name and call-by-value. In this work, we split the of-course modality of linear logic into two modalities, written"!"and"$ullet$". Intuitively, the modality"!"specifies a subproof that can be duplicated and erased, but may not necessarily be"accessed", i.e. interacted with, while the combined modality"$!ullet$"specifies a subproof that can moreover be accessed. The resulting system, called MSCLL, enjoys cut-elimination and is conservative over MELL. We study how restricting access to subproofs provides ways to control sharing in evaluation strategies. For this, we introduce a term-assignment for an intuitionistic fragment of MSCLL, called the $lambda!ullet$-calculus, which we show to enjoy subject reduction, confluence, and strong normalization of the simply typed fragment. We propose three sound and complete translations that respectively simulate call-by-name, call-by-value, and a variant of call-by-name that shares the evaluation of its arguments (similarly as in call-by-need). The translations are extended to simulate the Bang-calculus, as well as weak reduction strategies.

Problem

• Linear Logic Optimization
• Data Sharing Control
• Programming Efficiency

Innovation

• Linear Logic Decomposition
• MSCLL System
• Lambda Calculus Enhancement