# Neural Probabilistic Logic Programming in Discrete-Continuous Domains

arXiv: https://arxiv.org/abs/2303.04660 · HTML: https://mlsanity.com/paper/2303.04660/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 65 references that are in the map.
- First posted on arXiv 8 Mar 2023.

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [End-to-End Object Detection with Transformers](https://mlsanity.com/paper/2005.12872/) (2020)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [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](https://mlsanity.com/paper/2012.13635/) (2020)
- [Neural-Symbolic Learning and Reasoning: A Survey and Interpretation](https://mlsanity.com/paper/1711.03902/) (2017)
- [End-to-End Differentiable Proving](https://mlsanity.com/paper/1705.11040/) (2017)
- [Neural-Symbolic Computing: An Effective Methodology for Principled Integration of Machine Learning and Reasoning](https://mlsanity.com/paper/1905.06088/) (2019)
- [Backpropagation through the Void: Optimizing control variates for black-box gradient estimation](https://mlsanity.com/paper/1711.00123/) (2017)
- [Implicit MLE: Backpropagating Through Discrete Exponential Family Distributions](https://mlsanity.com/paper/2106.01798/) (2021)
- [REBAR: Low-variance, unbiased gradient estimates for discrete latent variable models](https://mlsanity.com/paper/1703.07370/) (2017)
- [From Statistical Relational to Neurosymbolic Artificial Intelligence: a Survey](https://mlsanity.com/paper/2108.11451/) (2021)
- [DeepStochLog: Neural Stochastic Logic Programming](https://mlsanity.com/paper/2106.12574/) (2021)
- [VAEL: Bridging Variational Autoencoders and Probabilistic Logic Programming](https://mlsanity.com/paper/2202.04178/) (2022)
- [The Generalized Reparameterization Gradient](https://mlsanity.com/paper/1610.02287/) (2016)
- [Deep Probabilistic Programming](https://mlsanity.com/paper/1701.03757/) (2017)
- [Learning with Algorithmic Supervision via Continuous Relaxations](https://mlsanity.com/paper/2110.05651/) (2021)
- [Design and Implementation of Probabilistic Programming Language Anglican](https://mlsanity.com/paper/1608.05263/) (2016)
- [Declarative Probabilistic Logic Programming in Discrete-Continuous Domains](https://mlsanity.com/paper/2302.10674/) (2023)

## Built on by (most cited first, 25 of 25)

- [A Neuro-Symbolic Benchmark Suite for Concept Quality and Reasoning Shortcuts](https://mlsanity.com/paper/2406.10368/) (2024)
- [KLay: Accelerating Arithmetic Circuits for Neurosymbolic AI](https://mlsanity.com/paper/2410.11415/) (2024)
- [Differentiable Sampling of Categorical Distributions Using the CatLog-Derivative Trick](https://mlsanity.com/paper/2311.12569/) (2023)
- [Neurosymbolic Diffusion Models](https://mlsanity.com/paper/2505.13138/) (2025)
- [A Probabilistic Neuro-symbolic Layer for Algebraic Constraint Satisfaction](https://mlsanity.com/paper/2503.19466/) (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)
- [Automated Theorem Provers Help Improve Large Language Model Reasoning](https://mlsanity.com/paper/2408.03492/) (2024)
- [Semirings for Probabilistic and Neuro-Symbolic Logic Programming](https://mlsanity.com/paper/2402.13782/) (2024)
- [Compositional Generalization Across Distributional Shifts with Sparse Tree Operations](https://mlsanity.com/paper/2412.14076/) (2024)
- [Unveiling LLMs: The Evolution of Latent Representations in a Dynamic Knowledge Graph](https://mlsanity.com/paper/2404.03623/) (2024)
- [A Pattern Language for Machine Learning Tasks](https://mlsanity.com/paper/2407.02424/) (2024)
- [A Fast Convoluted Story: Scaling Probabilistic Inference for Integer Arithmetic](https://mlsanity.com/paper/2410.12389/) (2024)
- [To Neuro-Symbolic Classification and Beyond by Compiling Description Logic Ontologies to Probabilistic Circuits](https://mlsanity.com/paper/2601.14894/) (2026)
- [Neurosymbolic Object-Centric Learning with Distant Supervision](https://mlsanity.com/paper/2506.16129/) (2025)
- [Declarative Design of Neural Predicates in Neuro-Symbolic Systems](https://mlsanity.com/paper/2405.09521/) (2024)
- [CTSketch: Compositional Tensor Sketching for Scalable Neurosymbolic Learning](https://mlsanity.com/paper/2503.24123/) (2025)
- [Independence Is Not an Issue in Neurosymbolic AI](https://mlsanity.com/paper/2504.07851/) (2025)
- [The Theory and Practice of MAP Inference over Non-Convex Constraints](https://mlsanity.com/paper/2602.08681/) (2026)
- [Neural Markov Prolog](https://mlsanity.com/paper/2312.01521/) (2023)
- [DeepProofLog: Efficient Proving in Deep Stochastic Logic Programs](https://mlsanity.com/paper/2511.08581/) (2025)
- [Can VLMs Reason Robustly? A Neuro-Symbolic Investigation](https://mlsanity.com/paper/2603.23867/) (2026)
- [NeSyCat Torch: A Differentiable Tensor Implementation of Categorical Semantics for Neurosymbolic Learning](https://mlsanity.com/paper/2606.19279/) (2026)
- [Foundations of MT-PDCL: Measure-Theoretic Probabilistic Definite Clause Logic](https://mlsanity.com/paper/2608.13018/) (2026)
- [Neurosymbolic Learning of Interpretable World Models](https://mlsanity.com/paper/2610.19955/) (2026)

## Abstract (arXiv)

Neural-symbolic AI (NeSy) allows neural networks to exploit symbolic background knowledge in the form of logic. It has been shown to aid learning in the limited data regime and to facilitate inference on out-of-distribution data. Probabilistic NeSy focuses on integrating neural networks with both logic and probability theory, which additionally allows learning under uncertainty. A major limitation of current probabilistic NeSy systems, such as DeepProbLog, is their restriction to finite probability distributions, i.e., discrete random variables. In contrast, deep probabilistic programming (DPP) excels in modelling and optimising continuous probability distributions. Hence, we introduce DeepSeaProbLog, a neural probabilistic logic programming language that incorporates DPP techniques into NeSy. Doing so results in the support of inference and learning of both discrete and continuous probability distributions under logical constraints. Our main contributions are 1) the semantics of DeepSeaProbLog and its corresponding inference algorithm, 2) a proven asymptotically unbiased learning algorithm, and 3) a series of experiments that illustrate the versatility of our approach.