# Evaluating Relaxations of Logic for Neural Networks: A Comprehensive Study

arXiv: https://arxiv.org/abs/2107.13646 · HTML: https://mlsanity.com/paper/2107.13646/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 33 references that are in the map.
- First posted on arXiv 28 Jul 2021.

## Builds on (most cited first, 16 of 16)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [A Semantic Loss Function for Deep Learning with Symbolic Knowledge](https://mlsanity.com/paper/1711.11157/) (2017)
- [Learning Explanatory Rules from Noisy Data](https://mlsanity.com/paper/1711.04574/) (2017)
- [Logic Tensor Networks for Semantic Image Interpretation](https://mlsanity.com/paper/1705.08968/) (2017)
- [Logic-Guided Data Augmentation and Regularization for Consistent Question Answering](https://mlsanity.com/paper/2004.10157/) (2020)
- [A Logic-Driven Framework for Consistency of Neural Models](https://mlsanity.com/paper/1909.00126/) (2019)
- [Hinge-Loss Markov Random Fields and Probabilistic Soft Logic](https://mlsanity.com/paper/1505.04406/) (2015)
- [Joint Constrained Learning for Event-Event Relation Extraction](https://mlsanity.com/paper/2010.06727/) (2020)
- [Augmenting Neural Networks with First-order Logic](https://mlsanity.com/paper/1906.06298/) (2019)
- [Adversarial Sets for Regularising Neural Link Predictors](https://mlsanity.com/paper/1707.07596/) (2017)
- [Analyzing Differentiable Fuzzy Implications](https://mlsanity.com/paper/2006.03472/) (2020)
- [Towards a Formal Distributional Semantics: Simulating Logical Calculi with Tensors](https://mlsanity.com/paper/1304.5823/) (2013)
- [Structured Tuning for Semantic Role Labeling](https://mlsanity.com/paper/2005.00496/) (2020)
- [On the relation between Loss Functions and T-Norms](https://mlsanity.com/paper/1907.07904/) (2019)
- [Deep Adaptive Semantic Logic (DASL): Compiling Declarative Knowledge into Deep Neural Networks](https://mlsanity.com/paper/2003.07344/) (2020)

## Built on by (most cited first, 8 of 8)

- [A-NeSI: A Scalable Approximate Method for Probabilistic Neurosymbolic Inference](https://mlsanity.com/paper/2212.12393/) (2022)
- [A Pseudo-Semantic Loss for Autoregressive Models with Logical Constraints](https://mlsanity.com/paper/2312.03905/) (2023)
- [Semantic Strengthening of Neuro-Symbolic Learning](https://mlsanity.com/paper/2302.14207/) (2023)
- [EXPLAIN, AGREE, LEARN: Scaling Learning for Neural Probabilistic Logic](https://mlsanity.com/paper/2408.08133/) (2024)
- [Understanding the Logic of Direct Preference Alignment through Logic](https://mlsanity.com/paper/2412.17696/) (2024)
- [The Integer Linear Programming Inference Cookbook](https://mlsanity.com/paper/2307.00171/) (2023)
- [Gradient-Based Optimization on G"odel Logic as Discrete Local Search](https://mlsanity.com/paper/2503.01817/) (2025)
- [A neurosymbolic Approach with Epistemic Deep Learning for Hierarchical Image Classification](https://mlsanity.com/paper/2605.16383/) (2026)

## Abstract (arXiv)

Symbolic knowledge can provide crucial inductive bias for training neural models, especially in low data regimes. A successful strategy for incorporating such knowledge involves relaxing logical statements into sub-differentiable losses for optimization. In this paper, we study the question of how best to relax logical expressions that represent labeled examples and knowledge about a problem; we focus on sub-differentiable t-norm relaxations of logic. We present theoretical and empirical criteria for characterizing which relaxation would perform best in various scenarios. In our theoretical study driven by the goal of preserving tautologies, the Lukasiewicz t-norm performs best. However, in our empirical analysis on the text chunking and digit recognition tasks, the product t-norm achieves best predictive performance. We analyze this apparent discrepancy, and conclude with a list of best practices for defining loss functions via logic.