# Softened Symbol Grounding for Neuro-symbolic Systems

arXiv: https://arxiv.org/abs/2403.00323 · HTML: https://mlsanity.com/paper/2403.00323/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 78 references that are in the map.
- First posted on arXiv 1 Mar 2024.

## Builds on (most cited first, 30 of 32)

- [Vision-and-Language Navigation: Interpreting visually-grounded navigation instructions in real environments](https://mlsanity.com/paper/1711.07280/) (2017)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [Speaker-Follower Models for Vision-and-Language Navigation](https://mlsanity.com/paper/1806.02724/) (2018)
- [Neural-Symbolic VQA: Disentangling Reasoning from Vision and Language Understanding](https://mlsanity.com/paper/1810.02338/) (2018)
- [A Semantic Loss Function for Deep Learning with Symbolic Knowledge](https://mlsanity.com/paper/1711.11157/) (2017)
- [Neural-Symbolic Learning and Reasoning: A Survey and Interpretation](https://mlsanity.com/paper/1711.03902/) (2017)
- [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019)
- [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)
- [Stochastic Optimization of Sorting Networks via Continuous Relaxations](https://mlsanity.com/paper/1903.08850/) (2019)
- [Neural Symbolic Machines: Learning Semantic Parsers on Freebase with Weak Supervision](https://mlsanity.com/paper/1611.00020/) (2016)
- [NeurASP: Embracing Neural Networks into Answer Set Programming](https://mlsanity.com/paper/2307.07700/) (2023)
- [On the Convergence of SGD with Biased Gradients](https://mlsanity.com/paper/2008.00051/) (2020)
- [From Statistical Relational to Neurosymbolic Artificial Intelligence: a Survey](https://mlsanity.com/paper/2108.11451/) (2021)
- [Neural Arithmetic Logic Units](https://mlsanity.com/paper/1808.00508/) (2018)
- [Closed Loop Neural-Symbolic Learning via Integrating Neural Perception, Grammar Parsing, and Symbolic Reasoning](https://mlsanity.com/paper/2006.06649/) (2020)
- [DeepStochLog: Neural Stochastic Logic Programming](https://mlsanity.com/paper/2106.12574/) (2021)
- [Memory Augmented Policy Optimization for Program Synthesis and Semantic Parsing](https://mlsanity.com/paper/1807.02322/) (2018)
- [MultiplexNet: Towards Fully Satisfied Logical Constraints in Neural Networks](https://mlsanity.com/paper/2111.01564/) (2021)
- [Probabilistic Neural-symbolic Models for Interpretable Visual Question Answering](https://mlsanity.com/paper/1902.07864/) (2019)
- [Neural-Symbolic Solver for Math Word Problems with Auxiliary Tasks](https://mlsanity.com/paper/2107.01431/) (2021)
- [NeuPSL: Neural Probabilistic Soft Logic](https://mlsanity.com/paper/2205.14268/) (2022)
- [Enforcing Policy Feasibility Constraints through Differentiable Projection for Energy Optimization](https://mlsanity.com/paper/2105.08881/) (2021)
- [Learning by Fixing: Solving Math Word Problems with Weak Supervision](https://mlsanity.com/paper/2012.10582/) (2020)
- [Techniques for Symbol Grounding with SATNet](https://mlsanity.com/paper/2106.11072/) (2021)
- [A Scalable Approximate Model Counter](https://mlsanity.com/paper/1306.5726/) (2013)
- [On Stackelberg Mixed Strategies](https://mlsanity.com/paper/1705.07476/) (2017)
- [Unbiased Markov chain Monte Carlo with couplings](https://mlsanity.com/paper/1708.03625/) (2017)
- [Constrained Sampling and Counting: Universal Hashing Meets SAT Solving](https://mlsanity.com/paper/1512.06633/) (2015)
- [Sampling Constraint Satisfaction Solutions in the Local Lemma Regime](https://mlsanity.com/paper/2011.03915/) (2020)

## Built on by (most cited first, 16 of 16)

- [On the Independence Assumption in Neurosymbolic Learning](https://mlsanity.com/paper/2404.08458/) (2024)
- [Large language models for artificial general intelligence (AGI): A survey of foundational principles and approaches](https://mlsanity.com/paper/2501.03151/) (2025)
- [Neuro-symbolic Learning Yielding Logical Constraints](https://mlsanity.com/paper/2410.20957/) (2024)
- [Neuro-Symbolic AI: Explainability, Challenges, and Future Trends](https://mlsanity.com/paper/2411.04383/) (2024)
- [On the Hardness of Probabilistic Neurosymbolic Learning](https://mlsanity.com/paper/2406.04472/) (2024)
- [A Study on Neuro-Symbolic Artificial Intelligence: Healthcare Perspectives](https://mlsanity.com/paper/2503.18213/) (2025)
- [Imbalances in Neurosymbolic Learning: Characterization and Mitigating Strategies](https://mlsanity.com/paper/2407.10000/) (2024)
- [Neuro-Symbolic AI for Cybersecurity: State of the Art, Challenges, and Opportunities](https://mlsanity.com/paper/2509.06921/) (2025)
- [Efficiently Learning Probabilistic Logical Models by Cheaply Ranking Mined Rules](https://mlsanity.com/paper/2409.16238/) (2024)
- [Process-of-Thought Reasoning for Videos](https://mlsanity.com/paper/2602.07689/) (2026)
- [On Improving Neurosymbolic Learning by Exploiting the Representation Space](https://mlsanity.com/paper/2602.07973/) (2026)
- [Bridging the Gap: Representation Spaces in Neuro-Symbolic AI](https://mlsanity.com/paper/2411.04393/) (2024)
- [Neuro-symbolic Weak Supervision: Theory and Semantics](https://mlsanity.com/paper/2503.18509/) (2025)
- [Visual Perceptual to Conceptual First-Order Rule Learning Networks](https://mlsanity.com/paper/2604.07897/) (2026)
- [Grounding vs. Compositionality: On the Non-Complementarity of Reasoning in Neuro-Symbolic Systems](https://mlsanity.com/paper/2604.26521/) (2026)
- [Soft Symbol Grounding for Prototypical Concepts](https://mlsanity.com/paper/2609.12247/) (2026)

## Abstract (arXiv)

Neuro-symbolic learning generally consists of two separated worlds, i.e., neural network training and symbolic constraint solving, whose success hinges on symbol grounding, a fundamental problem in AI. This paper presents a novel, softened symbol grounding process, bridging the gap between the two worlds, and resulting in an effective and efficient neuro-symbolic learning framework. Technically, the framework features (1) modeling of symbol solution states as a Boltzmann distribution, which avoids expensive state searching and facilitates mutually beneficial interactions between network training and symbolic reasoning;(2) a new MCMC technique leveraging projection and SMT solvers, which efficiently samples from disconnected symbol solution spaces; (3) an annealing mechanism that can escape from %being trapped into sub-optimal symbol groundings. Experiments with three representative neuro symbolic learning tasks demonstrate that, owining to its superior symbol grounding capability, our framework successfully solves problems well beyond the frontier of the existing proposals.