Quantitative Linear Logic

2026-05-13

AI Summary

This work proposes a differentiable “soft” additive connective semantics to seamlessly integrate logical specifications into probabilistic and quantified systems. By reformulating sequent calculus and incorporating additive and multiplicative analyses over the real numbers, the authors develop a parameterized Quantified Linear Logic system, pQLL, which unifies hypersequents from fuzzy logic with deep inference. The framework introduces a hardness parameter \( p \) that enables a continuous transition from soft to classical logic, and establishes a soft lattice semantics under which cut-elimination and completeness with respect to enriched residuated lattices are proven. Notably, as \( p \to \infty \), pQLL converges to Multiplicative Additive Linear Logic (MALL), thereby bridging differentiable reasoning with classical logical systems.

Abstract

Real-valued logics have seen a renewed interest in verification for probabilistic and quantitative systems, in particular machine learning models, where they can be used to directly integrate specifications in the training objective. To do so effectively one has to strike a balance between the logical properties of the connectives and their semantics. A major hurdle in this sense is to give ``soft'' (i.e. differentiable) semantics to additive connectives -- in linear and fuzzy logics, additives are necessarily ``hard'' lattice operations. In this paper, we solve this problem by combining an accurate analysis of the properties of sum and product on the reals with a significant revision of sequent calculus. We introduce `quantitative sequent calculi', which simultaneously generalize hypersequent calculi of fuzzy logics and deep inference, and in which validity of a proof and provability of a sequent are real-valued quantities. We present a family of calculi, pQLL, indexed by a hardness degree $p$, prove cut-elimination theorem for them, and show completeness for enriched residuated `soft' lattices. For $p = \infty$, pQLL reduces to MALL, with provability in pQLL converging to provability in MALL when $p \to \infty$.

Problem

• Quantitative Logic
• Additive Connectives
• Soft Semantics
• Differentiable Logic
• Real-valued Logics

Innovation

• Quantitative Linear Logic
• Soft Connectives
• Differentiable Semantics
• Sequent Calculus
• Cut-elimination