Logical Boltzmann Machines
This work addresses the problem of combining machine learning with scalable sound reasoning for AI researchers, offering a novel neurosymbolic approach that improves efficiency and performance over existing methods.
The paper introduces Logical Boltzmann Machines (LBM), a neurosymbolic system that represents propositional logic formulas and proves equivalence between energy minimization and logical satisfiability, enabling sound reasoning. Empirically, LBM finds all satisfying assignments for a class of logical formulae by searching fewer than 0.75% of possible assignments (approximately 1 billion) and achieves better learning performance in five out of seven datasets compared to baseline systems.
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.