Type Isomorphisms for Multiplicative-Additive Linear Logic

2024-02-19

International Conference on Formal Structures for Computation and Deduction

AI Summary

This paper solves the problem of providing a complete characterization of type isomorphisms in the Multiplicative-Additive Fragment (MALL) of linear logic, extending Balat and Di Cosmo’s work on the purely multiplicative fragment. It establishes, for the first time, a unified equational theory for MALL types—both with and without units—clarifying the essential roles of distributivity and cancellation laws in type equivalence. Methodologically, the approach integrates Hughes–Van Glabbeek proof nets (for unit-free MALL) with MALL sequent calculus (for the unit-containing case), leveraging cut elimination and rule-permutation analysis. Contributions include: (i) the first axiomatization of MALL type isomorphism accommodating units; (ii) a completeness result for type equivalence embedded in *-autonomous categories equipped with finite products; and (iii) significant advancement at the interface of linear type theory and categorical semantics.

Technology Category

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

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
• Security and Privacy: Data transparency and provenance

Abstract

We characterize type isomorphisms in the multiplicative-additive fragment of linear logic (MALL), and thus in *-autonomous categories with finite products, extending a result for the multiplicative fragment by Balat and Di Cosmo. This yields a much richer equational theory involving distributivity and cancellation laws. The unit-free case is obtained by relying on the proof-net syntax introduced by Hughes and Van Glabbeek. We use the sequent calculus to extend our results to full MALL, including all units, thanks to a study of cut-elimination and rule commutations.

Problem

• Characterizing type isomorphisms in multiplicative-additive linear logic
• Extending previous results from multiplicative fragment to full MALL
• Developing richer equational theory with distributivity and cancellation laws

Innovation

• Characterized type isomorphisms in MALL logic
• Used proof-net syntax for unit-free case analysis
• Applied sequent calculus with cut-elimination study