# Logical Boltzmann Machines

arXiv: https://arxiv.org/abs/2112.05841 · HTML: https://mlsanity.com/paper/2112.05841/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 20 references that are in the map.
- First posted on arXiv 10 Dec 2021.

## Builds on (most cited first, 5 of 5)

- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [Neurosymbolic AI: The 3rd Wave](https://mlsanity.com/paper/2012.05876/) (2020)
- [Deep Learning: A Critical Appraisal](https://mlsanity.com/paper/1801.00631/) (2018)
- [Learning Explanatory Rules from Noisy Data](https://mlsanity.com/paper/1711.04574/) (2017)
- [TensorLog: Deep Learning Meets Probabilistic DBs](https://mlsanity.com/paper/1707.05390/) (2017)

## Built on by (most cited first, 2 of 2)

- [Neurosymbolic AI and its Taxonomy: a survey](https://mlsanity.com/paper/2305.08876/) (2023)
- [Learning State Transition Rules from Hidden Layers of Restricted Boltzmann Machines](https://mlsanity.com/paper/2212.03374/) (2022)

## Abstract (arXiv)

The idea of representing symbolic knowledge in connectionist systems has been a long-standing endeavour which has attracted much attention recently with the objective of combining machine learning and scalable sound reasoning. Early work has shown a correspondence between propositional logic and symmetrical neural networks which nevertheless did not scale well with the number of variables and whose training regime was inefficient. In this paper, we introduce Logical Boltzmann Machines (LBM), a neurosymbolic system that can represent any propositional logic formula in strict disjunctive normal form. We prove equivalence between energy minimization in LBM and logical satisfiability thus showing that LBM is capable of sound reasoning. We evaluate reasoning empirically to show that LBM is capable of finding all satisfying assignments of a class of logical formulae by searching fewer than 0.75% of the possible (approximately 1 billion) assignments. We compare learning in LBM with a symbolic inductive logic programming system, a state-of-the-art neurosymbolic system and a purely neural network-based system, achieving better learning performance in five out of seven data sets.