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.
The paper presents a connectionist framework that is capable of representing and learning propositional knowledge. An extended version of propositional calculus is developed and is demonstrated to be useful for nonmonotonic reasoning, dealing with conflicting beliefs and for coping with inconsistency generated by unreliable knowledge sources. Formulas of the extended calculus are proved to be equivalent in a very strong sense to symmetric networks (like Hopfield networks and Boltzmann machines), and efficient algorithms are given for translating back and forth between the two forms of knowledge representation. A fast learning procedure is presented that allows symmetric networks to learn representations of unknown logic formulas by looking at examples. A connectionist inference engine is then sketched whose knowledge is either compiled from a symbolic representation or learned inductively from training examples. Experiments with large scale randomly generated formulas suggest that the parallel local search that is executed by the networks is extremely fast on average. Finally, it is shown that the extended logic can be used as a high-level specification language for connectionist networks, into which several recent symbolic systems may be mapped. The paper demonstrates how a rigorous bridge can be constructed that ties together the (sometimes opposing) connectionist and symbolic approaches.