# Neuro-symbolic Weak Supervision: Theory and Semantics

arXiv: https://arxiv.org/abs/2503.18509 · HTML: https://mlsanity.com/paper/2503.18509/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 38 of its 74 references that are in the map.
- First posted on arXiv 24 Mar 2025.

## Builds on (most cited first, 30 of 38)

- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [A Semantic Loss Function for Deep Learning with Symbolic Knowledge](https://mlsanity.com/paper/1711.11157/) (2017)
- [Neurosymbolic AI: The 3rd Wave](https://mlsanity.com/paper/2012.05876/) (2020)
- [Learning Explanatory Rules from Noisy Data](https://mlsanity.com/paper/1711.04574/) (2017)
- [SATNet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver](https://mlsanity.com/paper/1905.12149/) (2019)
- [End-to-End Differentiable Proving](https://mlsanity.com/paper/1705.11040/) (2017)
- [Harnessing Deep Neural Networks with Logic Rules](https://mlsanity.com/paper/1603.06318/) (2016)
- [Progressive Identification of True Labels for Partial-Label Learning](https://mlsanity.com/paper/2002.08053/) (2020)
- [MIPaaL: Mixed Integer Program as a Layer](https://mlsanity.com/paper/1907.05912/) (2019)
- [NeurASP: Embracing Neural Networks into Answer Set Programming](https://mlsanity.com/paper/2307.07700/) (2023)
- [Inductive logic programming at 30: a new introduction](https://mlsanity.com/paper/2008.07912/) (2020)
- [Leveraged Weighted Loss for Partial Label Learning](https://mlsanity.com/paper/2106.05731/) (2021)
- [Not All Neuro-Symbolic Concepts Are Created Equal: Analysis and Mitigation of Reasoning Shortcuts](https://mlsanity.com/paper/2305.19951/) (2023)
- [DeepStochLog: Neural Stochastic Logic Programming](https://mlsanity.com/paper/2106.12574/) (2021)
- [CombOptNet: Fit the Right NP-Hard Problem by Learning Integer Programming Constraints](https://mlsanity.com/paper/2105.02343/) (2021)
- [Instance-Dependent Partial Label Learning](https://mlsanity.com/paper/2110.12911/) (2021)
- [Neural-Symbolic Integration: A Compositional Perspective](https://mlsanity.com/paper/2010.11926/) (2020)
- [Neural-Guided Deductive Search for Real-Time Program Synthesis from Examples](https://mlsanity.com/paper/1804.01186/) (2018)
- [BEARS Make Neuro-Symbolic Models Aware of their Reasoning Shortcuts](https://mlsanity.com/paper/2402.12240/) (2024)
- [Neuro-Symbolic Inductive Logic Programming with Logical Neural Networks](https://mlsanity.com/paper/2112.03324/) (2021)
- [Structured Prediction with Partial Labelling through the Infimum Loss](https://mlsanity.com/paper/2003.00920/) (2020)
- [Programming with a Differentiable Forth Interpreter](https://mlsanity.com/paper/1605.06640/) (2016)
- [Learning from Label Proportions by Learning with Label Noise](https://mlsanity.com/paper/2203.02496/) (2022)
- [Differentiable Inductive Logic Programming for Structured Examples](https://mlsanity.com/paper/2103.01719/) (2021)
- [Neurosymbolic AI -- Why, What, and How](https://mlsanity.com/paper/2305.00813/) (2023)
- [Softened Symbol Grounding for Neuro-symbolic Systems](https://mlsanity.com/paper/2403.00323/) (2024)
- [Learning from Multiple Noisy Partial Labelers](https://mlsanity.com/paper/2106.04530/) (2021)
- [On Learning Latent Models with Multi-Instance Weak Supervision](https://mlsanity.com/paper/2306.13796/) (2023)
- [Backpropagating through Structured Argmax using a SPIGOT](https://mlsanity.com/paper/1805.04658/) (2018)
- [Disambiguated Attention Embedding for Multi-Instance Partial-Label Learning](https://mlsanity.com/paper/2305.16912/) (2023)

## Abstract (arXiv)

Weak supervision enables machine learning models to learn from limited or noisy labels, but it introduces challenges in reliability and semantic clarity, particularly in multi-instance partial label learning (MI-PLL), where models must resolve both ambiguous supervision signals and uncertain instance-label mappings. This paper proposes a semantics for a neuro-symbolic framework that integrates inductive logic programming (ILP) to structure MI-PLL through relational constraints. In this formulation, ILP defines a hypothesis space over label transitions, formalizes the semantics of per-instance classifiers and provides a relational scaffold for reasoning about weak supervision. Two inductive tasks are studied in this framework: inferring the transition predicate from the observed and classifier predicates, and inferring instance-level classifier assignments from the observed and transition predicates. This formal semantics facilitates constraint specification, consistency checking and the diagnosis of semantic failure modes that bag-level accuracy alone may conceal.