A Foundation for Differentiable Logics using Dependent Type Theory

2026-02-27

AI Summary

This work addresses the lack of a systematic algebraic and proof-theoretic foundation for differentiable logics, as well as the insufficient integration of differentiability into fuzzy logics—despite their well-developed algebraic structures—and the absence of a unified framework for comparing these paradigms. To bridge this gap, we present the first unified dependently typed framework formalized in the Coq proof assistant using the MathComp library, encompassing the syntax, residuated lattice semantics, analytical properties, and proof systems of prominent fuzzy logics (Yager, Łukasiewicz, Gödel, Product) and differentiable logics (DL2, STL). Our key contributions include novel sequent calculi for DL2 and STL∞ with verified soundness, the first formalization and integration of L’Hôpital’s rule into MathComp, and proofs of the existence of positive derivatives for critical connectives, thereby establishing a formal foundation for differentiable reasoning across multiple logical systems.

Technology Category

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

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Differentiable logics are a family of quantitative logics originated in the machine learning literature. Because of their origin, differentiable logics often come equipped with analytic properties that guarantee that they are differentiable. However, they usually lack an accompanying theory that describes their algebraic and proof-theoretic properties. Meanwhile, fuzzy logics, seen as substructural logics, have been studied algebraically and proof-theoretically, and some fuzzy logics with desirable analytic properties have also been used in machine learning. Our aim is to systematically compare analytic, algebraic and proof-theoretical properties of both fuzzy and differentiable logics. To this end, we formalize differentiable and fuzzy logics in a unified framework, encoded using the Mathcomp library in the Rocq proof assistant. We propose a single language specification to encompass multiple logics, using intrinsic typing to only allow valid and well-typed formulas for each of the logics that we encode: Yager, {\L}ukasiewicz, G\"{o}del and product fuzzy logics, as well as the differentiable logics DL2 and STL. Algebraically, we show how these logics can be interpreted using residuated lattices, which are prevalent in the theory of substructural logics. Analytically, we formalise the existence of a positive derivative for certain logical connectives, and to this end we formalise L'H\^opital's, contributing it to the Mathcomp library. Proof-theoretically, we formalise established sequent calculi for fuzzy logics, and we propose new sequent calculi for DL2 and STL$_{\infty}$, and formalise their soundness in our framework.

Problem

• differentiable logics
• fuzzy logics
• residuated lattices
• proof theory
• substructural logics

Innovation

• Differentiable Logics
• Fuzzy Logics
• Dependent Type Theory
• Residuated Lattices
• Formal Verification