Representing symbolic knowledge into a connectionist network is the key element for the integration of scalable learning and sound reasoning. Most of the previous studies focus on discriminative neural networks which unnecessarily require a separation of input output variables. Recent development of generative neural networks such as restricted Boltzmann machines (RBMs) has shown a capability of learning semantic abstractions directly from data, posing a promise for general symbolic learning and reasoning. Previous work on Penalty logic show a link between propositional logic and symmetric connectionist networks, however it is not applicable to RBMs. This paper proposes a novel method to represent propositional formulas into RBMs stack of RBMs where Gibbs sampling can be seen as maximising satisfiability. It also shows a promising use of RBMs to learn symbolic knowledge through maximum likelihood estimation.
From a statistical perspective, representing a propositional formula is similar to representing a uniform distribution over all assignments that satisfy the formula, i.e. make the formula hold. In this paper such assignments are referred to as preferred models . Since RBMs are universal approximators, it is true that any discrete distribution can be approximated arbitrarily well'' @cite_27 , and therefore we can always find an RBM to represent a uniform distribution of preferred models. However, while that work utilises statistical methods over the preferred models of formulas resulting in a very large network, our work focuses on logical calculus to transform and convert formulas directly to the energy function of a more compact RBM.
@cite_27
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Deep belief networks (DBN) are generative neural network models with many layers of hidden explanatory factors, recently introduced by Hinton, Osindero, and Teh (2006) along with a greedy layer-wise unsupervised learning algorithm. The building block of a DBN is a probabilistic model called a restricted Boltzmann machine (RBM), used to represent one layer of the model. Restricted Boltzmann machines are interesting because inference is easy in them and because they have been successfully used as building blocks for training deeper models. We first prove that adding hidden units yields strictly improved modeling power, while a second theorem shows that RBMs are universal approximators of discrete distributions. We then study the question of whether DBNs with more layers are strictly more powerful in terms of representational power. This suggests a new and less greedy criterion for training RBMs within DBNs.