# LogicMP: A Neuro-symbolic Approach for Encoding First-order Logic Constraints

arXiv: https://arxiv.org/abs/2309.15458 · HTML: https://mlsanity.com/paper/2309.15458/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 52 references that are in the map.
- First posted on arXiv 27 Sep 2023.

## Builds on (most cited first, 26 of 26)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [LayoutLM: Pre-training of Text and Layout for Document Image Understanding](https://mlsanity.com/paper/1912.13318/) (2019)
- [Neural Architectures for Named Entity Recognition](https://mlsanity.com/paper/1603.01360/) (2016)
- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [FUNSD: A Dataset for Form Understanding in Noisy Scanned Documents](https://mlsanity.com/paper/1905.13538/) (2019)
- [A Semantic Loss Function for Deep Learning with Symbolic Knowledge](https://mlsanity.com/paper/1711.11157/) (2017)
- [Conditional Random Fields as Recurrent Neural Networks](https://mlsanity.com/paper/1502.03240/) (2015)
- [Named Entity Recognition with Bidirectional LSTM-CNNs](https://mlsanity.com/paper/1511.08308/) (2015)
- [Logic Tensor Networks](https://mlsanity.com/paper/2012.13635/) (2020)
- [Harnessing Deep Neural Networks with Logic Rules](https://mlsanity.com/paper/1603.06318/) (2016)
- [Semantic Probabilistic Layers for Neuro-Symbolic Learning](https://mlsanity.com/paper/2206.00426/) (2022)
- [Probabilistic Logic Neural Networks for Reasoning](https://mlsanity.com/paper/1906.08495/) (2019)
- [Spatial Dependency Parsing for Semi-Structured Document Information Extraction](https://mlsanity.com/paper/2005.00642/) (2020)
- [Hinge-Loss Markov Random Fields and Probabilistic Soft Logic](https://mlsanity.com/paper/1505.04406/) (2015)
- [SpanNER: Named Entity Re-/Recognition as Span Prediction](https://mlsanity.com/paper/2106.00641/) (2021)
- [Augmenting Neural Networks with First-order Logic](https://mlsanity.com/paper/1906.06298/) (2019)
- [Efficient Probabilistic Logic Reasoning with Graph Neural Networks](https://mlsanity.com/paper/2001.11850/) (2020)
- [MultiplexNet: Towards Fully Satisfied Logical Constraints in Neural Networks](https://mlsanity.com/paper/2111.01564/) (2021)
- [Multi-Label Classification Neural Networks with Hard Logical Constraints](https://mlsanity.com/paper/2103.13427/) (2021)
- [Relational Neural Machines](https://mlsanity.com/paper/2002.02193/) (2020)
- [Injecting Logical Constraints into Neural Networks via Straight-Through Estimators](https://mlsanity.com/paper/2307.04347/) (2023)
- [Structural Knowledge Distillation: Tractably Distilling Information for Structured Predictor](https://mlsanity.com/paper/2010.05010/) (2020)
- [Understanding the Complexity of Lifted Inference and Asymmetric Weighted Model Counting](https://mlsanity.com/paper/1405.3250/) (2014)
- [AIN: Fast and Accurate Sequence Labeling with Approximate Inference Network](https://mlsanity.com/paper/2009.08229/) (2020)
- [DeepPSL: End-to-end perception and reasoning](https://mlsanity.com/paper/2109.13662/) (2021)
- [Lifted Probabilistic Inference for Asymmetric Graphical Models](https://mlsanity.com/paper/1412.0315/) (2014)

## Built on by (most cited first, 3 of 3)

- [Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows](https://mlsanity.com/paper/2503.09069/) (2025)
- [Local-to-Global Logical Explanations for Deep Vision Models](https://mlsanity.com/paper/2601.13404/) (2026)
- [Compositional Benchmark Synthesis for Hierarchical Human Action Recognition](https://mlsanity.com/paper/2608.10765/) (2026)

## Abstract (arXiv)

Integrating first-order logic constraints (FOLCs) with neural networks is a crucial but challenging problem since it involves modeling intricate correlations to satisfy the constraints. This paper proposes a novel neural layer, LogicMP, whose layers perform mean-field variational inference over an MLN. It can be plugged into any off-the-shelf neural network to encode FOLCs while retaining modularity and efficiency. By exploiting the structure and symmetries in MLNs, we theoretically demonstrate that our well-designed, efficient mean-field iterations effectively mitigate the difficulty of MLN inference, reducing the inference from sequential calculation to a series of parallel tensor operations. Empirical results in three kinds of tasks over graphs, images, and text show that LogicMP outperforms advanced competitors in both performance and efficiency.