Logical Boltzmann Machines

TL;DR

Logical Boltzmann Machines (LBM) are a neurosymbolic system that efficiently represent propositional logic and perform sound reasoning, demonstrating superior learning performance over existing systems on multiple datasets.

Contribution

This paper introduces LBM, a novel neurosymbolic model that encodes propositional logic in a form suitable for efficient reasoning and learning, bridging symbolic logic and neural networks.

Findings

LBM can find all satisfying assignments of certain logical formulas efficiently.

LBM outperforms existing systems in learning performance on most datasets.

LBM demonstrates sound reasoning by equivalence with logical satisfiability.

Abstract

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.