# VAEL: Bridging Variational Autoencoders and Probabilistic Logic Programming

arXiv: https://arxiv.org/abs/2202.04178 · HTML: https://mlsanity.com/paper/2202.04178/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 64 of its 74 references that are in the map.
- First posted on arXiv 7 Feb 2022.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 30 of 64)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Generating Diverse High-Fidelity Images with VQ-VAE-2](https://mlsanity.com/paper/1906.00446/) (2019)
- [Object-Centric Learning with Slot Attention](https://mlsanity.com/paper/2006.15055/) (2020)
- [Conditional Image Generation with PixelCNN Decoders](https://mlsanity.com/paper/1606.05328/) (2016)
- [Generative Adversarial Text to Image Synthesis](https://mlsanity.com/paper/1605.05396/) (2016)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [StackGAN: Text to Photo-realistic Image Synthesis with Stacked Generative Adversarial Networks](https://mlsanity.com/paper/1612.03242/) (2016)
- [Semi-Supervised Learning with Deep Generative Models](https://mlsanity.com/paper/1406.5298/) (2014)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [MONet: Unsupervised Scene Decomposition and Representation](https://mlsanity.com/paper/1901.11390/) (2019)
- [Multi-Object Representation Learning with Iterative Variational Inference](https://mlsanity.com/paper/1903.00450/) (2019)
- [Image Generation from Scene Graphs](https://mlsanity.com/paper/1804.01622/) (2018)
- [Ladder Variational Autoencoders](https://mlsanity.com/paper/1602.02282/) (2016)
- [Neural-Symbolic VQA: Disentangling Reasoning from Vision and Language Understanding](https://mlsanity.com/paper/1810.02338/) (2018)
- [VAE with a VampPrior](https://mlsanity.com/paper/1705.07120/) (2017)
- [Conditional Object-Centric Learning from Video](https://mlsanity.com/paper/2111.12594/) (2021)
- [StackGAN++: Realistic Image Synthesis with Stacked Generative Adversarial Networks](https://mlsanity.com/paper/1710.10916/) (2017)
- [GENESIS: Generative Scene Inference and Sampling with Object-Centric Latent Representations](https://mlsanity.com/paper/1907.13052/) (2019)
- [Attend, Infer, Repeat: Fast Scene Understanding with Generative Models](https://mlsanity.com/paper/1603.08575/) (2016)
- [Neural-Symbolic Learning and Reasoning: A Survey and Interpretation](https://mlsanity.com/paper/1711.03902/) (2017)
- [DRAW: A Recurrent Neural Network For Image Generation](https://mlsanity.com/paper/1502.04623/) (2015)
- [SPACE: Unsupervised Object-Oriented Scene Representation via Spatial Attention and Decomposition](https://mlsanity.com/paper/2001.02407/) (2020)
- [Learning Shape Abstractions by Assembling Volumetric Primitives](https://mlsanity.com/paper/1612.00404/) (2016)
- [Variational Lossy Autoencoder](https://mlsanity.com/paper/1611.02731/) (2016)
- [CvxNet: Learnable Convex Decomposition](https://mlsanity.com/paper/1909.05736/) (2019)
- [Generating Images from Captions with Attention](https://mlsanity.com/paper/1511.02793/) (2015)

## Built on by (most cited first, 30 of 31)

- [Concept-based Explainable Artificial Intelligence: A Survey](https://mlsanity.com/paper/2312.12936/) (2023)
- [Not All Neuro-Symbolic Concepts Are Created Equal: Analysis and Mitigation of Reasoning Shortcuts](https://mlsanity.com/paper/2305.19951/) (2023)
- [Interpreting CLIP with Hierarchical Sparse Autoencoders](https://mlsanity.com/paper/2502.20578/) (2025)
- [Compositional Generative Modeling: A Single Model is Not All You Need](https://mlsanity.com/paper/2402.01103/) (2024)
- [BEARS Make Neuro-Symbolic Models Aware of their Reasoning Shortcuts](https://mlsanity.com/paper/2402.12240/) (2024)
- [Neuro-Symbolic Continual Learning: Knowledge, Reasoning Shortcuts and Concept Rehearsal](https://mlsanity.com/paper/2302.01242/) (2023)
- [How Realistic Is Your Synthetic Data? Constraining Deep Generative Models for Tabular Data](https://mlsanity.com/paper/2402.04823/) (2024)
- [Neural Probabilistic Logic Programming in Discrete-Continuous Domains](https://mlsanity.com/paper/2303.04660/) (2023)
- [Interpretability is in the Mind of the Beholder: A Causal Framework for Human-interpretable Representation Learning](https://mlsanity.com/paper/2309.07742/) (2023)
- [A Neuro-Symbolic Benchmark Suite for Concept Quality and Reasoning Shortcuts](https://mlsanity.com/paper/2406.10368/) (2024)
- [Beyond the convexity assumption: Realistic tabular data generation under quantifier-free real linear constraints](https://mlsanity.com/paper/2502.18237/) (2025)
- [Neural Concept Binder](https://mlsanity.com/paper/2406.09949/) (2024)
- [Neurosymbolic Diffusion Models](https://mlsanity.com/paper/2505.13138/) (2025)
- [Counterfactual Concept Bottleneck Models](https://mlsanity.com/paper/2402.01408/) (2024)
- [GenCO: Generating Diverse Designs with Combinatorial Constraints](https://mlsanity.com/paper/2310.02442/) (2023)
- [ABS: Enforcing Constraint Satisfaction On Generated Sequences Via Automata-Guided Beam Search](https://mlsanity.com/paper/2506.09701/) (2025)
- [Right for the Right Reasons: Avoiding Reasoning Shortcuts via Prototypical Neurosymbolic AI](https://mlsanity.com/paper/2510.25497/) (2025)
- [Logic Tensor Network-Enhanced Generative Adversarial Network](https://mlsanity.com/paper/2601.03839/) (2026)
- [Unsupervised Decomposition and Recombination with Discriminator-Driven Diffusion Models](https://mlsanity.com/paper/2601.22057/) (2026)
- [Generating by Understanding: Neural Visual Generation with Logical Symbol Groundings](https://mlsanity.com/paper/2310.17451/) (2023)
- [Neurosymbolic Object-Centric Learning with Distant Supervision](https://mlsanity.com/paper/2506.16129/) (2025)
- [A Neuro-Symbolic Approach for Probabilistic Reasoning on Graph Data](https://mlsanity.com/paper/2507.21873/) (2025)
- [Differentiable Rule Induction from Raw Sequence Inputs](https://mlsanity.com/paper/2602.13583/) (2026)
- [A Survey on the Integration of Generative AI for Critical Thinking in Mobile Networks](https://mlsanity.com/paper/2404.06946/) (2024)
- [Declarative Design of Neural Predicates in Neuro-Symbolic Systems](https://mlsanity.com/paper/2405.09521/) (2024)
- [Deep Generative Models with Hard Linear Equality Constraints](https://mlsanity.com/paper/2502.05416/) (2025)
- [Independence Is Not an Issue in Neurosymbolic AI](https://mlsanity.com/paper/2504.07851/) (2025)
- [Towards Better Generalization and Interpretability in Unsupervised Concept-Based Models](https://mlsanity.com/paper/2506.02092/) (2025)
- [On Improving Neurosymbolic Learning by Exploiting the Representation Space](https://mlsanity.com/paper/2602.07973/) (2026)
- [Parallel Noising in Neural Markov Logic Networks](https://mlsanity.com/paper/2607.19126/) (2026)

## Abstract (arXiv)

We present VAEL, a neuro-symbolic generative model integrating variational autoencoders (VAE) with the reasoning capabilities of probabilistic logic (L) programming. Besides standard latent subsymbolic variables, our model exploits a probabilistic logic program to define a further structured representation, which is used for logical reasoning. The entire process is end-to-end differentiable. Once trained, VAEL can solve new unseen generation tasks by (i) leveraging the previously acquired knowledge encoded in the neural component and (ii) exploiting new logical programs on the structured latent space. Our experiments provide support on the benefits of this neuro-symbolic integration both in terms of task generalization and data efficiency. To the best of our knowledge, this work is the first to propose a general-purpose end-to-end framework integrating probabilistic logic programming into a deep generative model.