# Logic of Differentiable Logics: Towards a Uniform Semantics of DL

arXiv: https://arxiv.org/abs/2303.10650 · HTML: https://mlsanity.com/paper/2303.10650/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 49 references that are in the map.
- First posted on arXiv 19 Mar 2023.

## Builds on (most cited first, 20 of 20)

- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [A Semantic Loss Function for Deep Learning with Symbolic Knowledge](https://mlsanity.com/paper/1711.11157/) (2017)
- [Beta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Complete and Incomplete Neural Network Robustness Verification](https://mlsanity.com/paper/2103.06624/) (2021)
- [Logic Tensor Networks](https://mlsanity.com/paper/2012.13635/) (2020)
- [Efficient Formal Safety Analysis of Neural Networks](https://mlsanity.com/paper/1809.08098/) (2018)
- [Neural-Symbolic Computing: An Effective Methodology for Principled Integration of Machine Learning and Reasoning](https://mlsanity.com/paper/1905.06088/) (2019)
- [Harnessing Deep Neural Networks with Logic Rules](https://mlsanity.com/paper/1603.06318/) (2016)
- [Analyzing Differentiable Fuzzy Logic Operators](https://mlsanity.com/paper/2002.06100/) (2020)
- [The Second International Verification of Neural Networks Competition (VNN-COMP 2021): Summary and Results](https://mlsanity.com/paper/2109.00498/) (2021)
- [MultiplexNet: Towards Fully Satisfied Logical Constraints in Neural Networks](https://mlsanity.com/paper/2111.01564/) (2021)
- [Deep Learning with Logical Constraints](https://mlsanity.com/paper/2205.00523/) (2022)
- [Neural Network Robustness as a Verification Property: A Principled Case Study](https://mlsanity.com/paper/2104.01396/) (2021)
- [On Robustness Metrics for Learning STL Tasks](https://mlsanity.com/paper/2003.06041/) (2020)
- [Cross-Entropy Loss and Low-Rank Features Have Responsibility for Adversarial Examples](https://mlsanity.com/paper/1901.08360/) (2019)
- [Analyzing Differentiable Fuzzy Implications](https://mlsanity.com/paper/2006.03472/) (2020)
- [Neuro-Symbolic Constraint Programming for Structured Prediction](https://mlsanity.com/paper/2103.17232/) (2021)
- [Smoothness Analysis for Probabilistic Programs with Application to Optimised Variational Inference](https://mlsanity.com/paper/2208.10530/) (2022)
- [Propositional Logics for the Lawvere Quantale](https://mlsanity.com/paper/2302.01224/) (2023)

## Built on by (most cited first, 15 of 15)

- [NLP Verification: Towards a General Methodology for Certifying Robustness](https://mlsanity.com/paper/2403.10144/) (2024)
- [Vehicle: Bridging the Embedding Gap in the Verification of Neuro-Symbolic Programs](https://mlsanity.com/paper/2401.06379/) (2024)
- [Neural Network Verification is a Programming Language Challenge](https://mlsanity.com/paper/2501.05867/) (2025)
- [ULLER: A Unified Language for Learning and Reasoning](https://mlsanity.com/paper/2405.00532/) (2024)
- [Defining neurosymbolic AI](https://mlsanity.com/paper/2507.11127/) (2025)
- [ANTONIO: Towards a Systematic Method of Generating NLP Benchmarks for Verification](https://mlsanity.com/paper/2305.04003/) (2023)
- [Understanding the Logic of Direct Preference Alignment through Logic](https://mlsanity.com/paper/2412.17696/) (2024)
- [To Neuro-Symbolic Classification and Beyond by Compiling Description Logic Ontologies to Probabilistic Circuits](https://mlsanity.com/paper/2601.14894/) (2026)
- [Gradient-Based Optimization on G"odel Logic as Discrete Local Search](https://mlsanity.com/paper/2503.01817/) (2025)
- [Comparing Differentiable Logics for Learning Systems: A Research Preview](https://mlsanity.com/paper/2311.09809/) (2023)
- [Proof-Carrying Neuro-Symbolic Code](https://mlsanity.com/paper/2504.12031/) (2025)
- [A General Framework for Property-Driven Machine Learning](https://mlsanity.com/paper/2505.00466/) (2025)
- [A Neurosymbolic Framework for Bias Correction in Convolutional Neural Networks](https://mlsanity.com/paper/2405.15886/) (2024)
- [Creating a Formally Verified Neural Network for Autonomous Navigation: An Experience Report](https://mlsanity.com/paper/2411.14163/) (2024)
- [NeSyCat Torch: A Differentiable Tensor Implementation of Categorical Semantics for Neurosymbolic Learning](https://mlsanity.com/paper/2606.19279/) (2026)

## Abstract (arXiv)

Differentiable logics (DL) have recently been proposed as a method of training neural networks to satisfy logical specifications. A DL consists of a syntax in which specifications are stated and an interpretation function that translates expressions in the syntax into loss functions. These loss functions can then be used during training with standard gradient descent algorithms. The variety of existing DLs and the differing levels of formality with which they are treated makes a systematic comparative study of their properties and implementations difficult. This paper remedies this problem by suggesting a meta-language for defining DLs that we call the Logic of Differentiable Logics, or LDL. Syntactically, it generalises the syntax of existing DLs to FOL, and for the first time introduces the formalism for reasoning about vectors and learners. Semantically, it introduces a general interpretation function that can be instantiated to define loss functions arising from different existing DLs. We use LDL to establish several theoretical properties of existing DLs, and to conduct their empirical study in neural network verification.