# KLay: Accelerating Arithmetic Circuits for Neurosymbolic AI

arXiv: https://arxiv.org/abs/2410.11415 · HTML: https://mlsanity.com/paper/2410.11415/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 54 references that are in the map.
- First posted on arXiv 15 Oct 2024.

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

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Concept Bottleneck Models](https://mlsanity.com/paper/2007.04612/) (2020)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [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)
- [Differentiation of Blackbox Combinatorial Solvers](https://mlsanity.com/paper/1912.02175/) (2019)
- [Logic Tensor Networks](https://mlsanity.com/paper/2012.13635/) (2020)
- [The Hardware Lottery](https://mlsanity.com/paper/2009.06489/) (2020)
- [Einsum Networks: Fast and Scalable Learning of Tractable Probabilistic Circuits](https://mlsanity.com/paper/2004.06231/) (2020)
- [Semantic Probabilistic Layers for Neuro-Symbolic Learning](https://mlsanity.com/paper/2206.00426/) (2022)
- [A Differential Approach to Inference in Bayesian Networks](https://mlsanity.com/paper/1301.3847/) (2013)
- [Coherent Hierarchical Multi-Label Classification Networks](https://mlsanity.com/paper/2010.10151/) (2020)
- [Scallop: A Language for Neurosymbolic Programming](https://mlsanity.com/paper/2304.04812/) (2023)
- [DeepStochLog: Neural Stochastic Logic Programming](https://mlsanity.com/paper/2106.12574/) (2021)
- [Safe Reinforcement Learning via Probabilistic Logic Shields](https://mlsanity.com/paper/2303.03226/) (2023)
- [Neural Probabilistic Logic Programming in Discrete-Continuous Domains](https://mlsanity.com/paper/2303.04660/) (2023)
- [Scaling Tractable Probabilistic Circuits: A Systems Perspective](https://mlsanity.com/paper/2406.00766/) (2024)
- [Tractable Boolean and Arithmetic Circuits](https://mlsanity.com/paper/2202.02942/) (2022)
- [DPU: DAG Processing Unit for Irregular Graphs with Precision-Scalable Posit Arithmetic in 28nm](https://mlsanity.com/paper/2112.05660/) (2021)
- [Towards Efficient Neuro-Symbolic AI: From Workload Characterization to Hardware Architecture](https://mlsanity.com/paper/2409.13153/) (2024)
- [Acceleration of probabilistic reasoning through custom processor architecture](https://mlsanity.com/paper/2103.00266/) (2021)

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

- [Neurosymbolic Diffusion Models](https://mlsanity.com/paper/2505.13138/) (2025)
- [Answer Set Networks: Casting Answer Set Programming into Deep Learning](https://mlsanity.com/paper/2412.14814/) (2024)
- [The DeepLog Neurosymbolic Machine](https://mlsanity.com/paper/2508.13697/) (2025)
- [Right for the Right Reasons: Avoiding Reasoning Shortcuts via Prototypical Neurosymbolic AI](https://mlsanity.com/paper/2510.25497/) (2025)
- [To Neuro-Symbolic Classification and Beyond by Compiling Description Logic Ontologies to Probabilistic Circuits](https://mlsanity.com/paper/2601.14894/) (2026)
- [Review of Hallucination Understanding in Large Language and Vision Models](https://mlsanity.com/paper/2510.00034/) (2025)
- [DeepLog: A Software Framework for Modular Neurosymbolic AI](https://mlsanity.com/paper/2605.10279/) (2026)
- [CTSketch: Compositional Tensor Sketching for Scalable Neurosymbolic Learning](https://mlsanity.com/paper/2503.24123/) (2025)
- [ExplainFuzz: Explainable and Constraint-Conditioned Test Generation with Probabilistic Circuits](https://mlsanity.com/paper/2604.06559/) (2026)
- [Concise and Logically Consistent Conformal Sets for Neuro-Symbolic Concept-Based Models](https://mlsanity.com/paper/2605.18202/) (2026)
- [Neurosymbolic Decision Trees](https://mlsanity.com/paper/2503.08762/) (2025)
- [Hardware-efficient tractable probabilistic inference for TinyML Neurosymbolic AI applications](https://mlsanity.com/paper/2507.05141/) (2025)
- [DeepProofLog: Efficient Proving in Deep Stochastic Logic Programs](https://mlsanity.com/paper/2511.08581/) (2025)
- [Learning Tractable Distributions Of Language Model Continuations](https://mlsanity.com/paper/2511.16054/) (2025)
- [REASON: Accelerating Probabilistic Logical Reasoning for Scalable Neuro-Symbolic Intelligence](https://mlsanity.com/paper/2601.20784/) (2026)
- [Does Reasoning Mitigate Backdoor Attacks? A Neuro-Symbolic Perspective](https://mlsanity.com/paper/2609.00464/) (2026)

## Abstract (arXiv)

A popular approach to neurosymbolic AI involves mapping logic formulas to arithmetic circuits (computation graphs consisting of sums and products) and passing the outputs of a neural network through these circuits. This approach enforces symbolic constraints onto a neural network in a principled and end-to-end differentiable way. Unfortunately, arithmetic circuits are challenging to run on modern AI accelerators as they exhibit a high degree of irregular sparsity. To address this limitation, we introduce knowledge layers (KLay), a new data structure to represent arithmetic circuits that can be efficiently parallelized on GPUs. Moreover, we contribute two algorithms used in the translation of traditional circuit representations to KLay and a further algorithm that exploits parallelization opportunities during circuit evaluations. We empirically show that KLay achieves speedups of multiple orders of magnitude over the state of the art, thereby paving the way towards scaling neurosymbolic AI to larger real-world applications.