# Generate in the Chart, Not on the Boundary: Function-Symbol Grounding for Hard Constraints in LTN-GANs

arXiv: https://arxiv.org/abs/2608.21605 · HTML: https://mlsanity.com/paper/2608.21605/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 44 references that are in the map.
- First posted on arXiv 21 Aug 2026.

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

- [Modeling Tabular data using Conditional GAN](https://mlsanity.com/paper/1907.00503/) (2019)
- [Reliable Fidelity and Diversity Metrics for Generative Models](https://mlsanity.com/paper/2002.09797/) (2020)
- [TabDDPM: Modelling Tabular Data with Diffusion Models](https://mlsanity.com/paper/2209.15421/) (2022)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [A Semantic Loss Function for Deep Learning with Symbolic Knowledge](https://mlsanity.com/paper/1711.11157/) (2017)
- [Logic Tensor Networks](https://mlsanity.com/paper/2012.13635/) (2020)
- [CTAB-GAN: Effective Table Data Synthesizing](https://mlsanity.com/paper/2102.08369/) (2021)
- [Semantic Probabilistic Layers for Neuro-Symbolic Learning](https://mlsanity.com/paper/2206.00426/) (2022)
- [STaSy: Score-based Tabular data Synthesis](https://mlsanity.com/paper/2210.04018/) (2022)
- [SketchGen: Generating Constrained CAD Sketches](https://mlsanity.com/paper/2106.02711/) (2021)
- [Vitruvion: A Generative Model of Parametric CAD Sketches](https://mlsanity.com/paper/2109.14124/) (2021)
- [Coherent Hierarchical Multi-Label Classification Networks](https://mlsanity.com/paper/2010.10151/) (2020)
- [MultiplexNet: Towards Fully Satisfied Logical Constraints in Neural Networks](https://mlsanity.com/paper/2111.01564/) (2021)
- [Transition1x -- a Dataset for Building Generalizable Reactive Machine Learning Potentials](https://mlsanity.com/paper/2207.12858/) (2022)
- [Alchemy: A Quantum Chemistry Dataset for Benchmarking AI Models](https://mlsanity.com/paper/1906.09427/) (2019)
- [VAEL: Bridging Variational Autoencoders and Probabilistic Logic Programming](https://mlsanity.com/paper/2202.04178/) (2022)
- [Physically Constrained Generative Adversarial Networks for Improving Precipitation Fields from Earth System Models](https://mlsanity.com/paper/2209.07568/) (2022)
- [How Realistic Is Your Synthetic Data? Constraining Deep Generative Models for Tabular Data](https://mlsanity.com/paper/2402.04823/) (2024)
- [Efficient Generation of Structured Objects with Constrained Adversarial Networks](https://mlsanity.com/paper/2007.13197/) (2020)
- [Deep Generative Models with Learnable Knowledge Constraints](https://mlsanity.com/paper/1806.09764/) (2018)
- [A Pseudo-Semantic Loss for Autoregressive Models with Logical Constraints](https://mlsanity.com/paper/2312.03905/) (2023)
- [Beyond the convexity assumption: Realistic tabular data generation under quantifier-free real linear constraints](https://mlsanity.com/paper/2502.18237/) (2025)
- [Injecting Logical Constraints into Neural Networks via Straight-Through Estimators](https://mlsanity.com/paper/2307.04347/) (2023)
- [Neuro-symbolic Learning Yielding Logical Constraints](https://mlsanity.com/paper/2410.20957/) (2024)
- [Constrained Generative Adversarial Networks for Interactive Image Generation](https://mlsanity.com/paper/1904.02526/) (2019)
- [Enforcing Deterministic Constraints on Generative Adversarial Networks for Emulating Physical Systems](https://mlsanity.com/paper/1911.06671/) (2019)
- [Semantic Objective Functions: A distribution-aware method for adding logical constraints in deep learning](https://mlsanity.com/paper/2405.15789/) (2024)
- [GenCO: Generating Diverse Designs with Combinatorial Constraints](https://mlsanity.com/paper/2310.02442/) (2023)
- [Logic Tensor Network-Enhanced Generative Adversarial Network](https://mlsanity.com/paper/2601.03839/) (2026)
- [Generating by Understanding: Neural Visual Generation with Logical Symbol Groundings](https://mlsanity.com/paper/2310.17451/) (2023)

## Abstract (arXiv)

Logic Tensor Network-Enhanced Generative Adversarial Networks (LTN-GANs) inject background knowledge by grounding each logical axiom as a predicate and training the generator to raise its satisfaction, a fuzzy truth value in [0,1]. Previous LTN-GAN work grounded every constraint this way, at the predicate level, and improved constraint satisfaction. A predicate, however, only scores a sample, so it cannot embed hard structural constraints, rules such as orderings, positivity, and definitional identities that must hold in every generated sample. In this work, we investigate grounding each axiom as a function symbol inside the LTN framework. We compare against the state-of-the-art alternative, a constraint layer that clamps each violating sample onto the feasible boundary and so produces outputs that are always valid. Our investigation shows that a valid sample is not always a realistic one. An inequality is not merely satisfied or violated. It holds by a margin, and a faithful generator should also reproduce the margin's real distribution. We find that the resolution ratio R, the data's scale over the margin's spread, is a diagnostic, computable before training, of which constraints a chosen grounding can learn. When R is large, the predicate receives no learning signal, the clamp pushes every sample onto the boundary, and the margin distribution is lost while every standard metric still looks fine. A function symbol avoids both failures, computing the constrained variable rather than scoring it. Together the function symbols form a chart, a coordinate system inside the feasible region, where every sample is valid by construction and the margin is learned like any other quantity.