Technical Report Number
This paper considers the problem of expressing predicate calculus in connectionist networks that are based on energy minimization. Given a first-order-logic knowledge base and a bound k, a symmetric network is constructed (like a Boltzman machine or a Hopfield network) that searches for a proof fora given query. If a resolution-based proof is length no longer k exists, then the global minima of the energy function that is associated with the network represent such proofs. If no proof exist then the global minima indicate the lack of a proof. The network that is generated is of size polynomial in the bound k and the knowledge size. There are no restrictions on the type of logic formulas that can be represented. An extension enables the mechanism to copy with inconsistency in the knowledge base; i.e. a query is entailed if there exists a proof supporting the query and no "better" (or equally "good") proof exists supporting its negation. Fault tolerance is obtained since symbolic roles are dynamically assigned to units and many units are competing for those roles.
Pinkas, Gadi, "First-Order Logic Proofs using Connectionist Constraints Relaxation" Report Number: WUCS-91-54 (1991). All Computer Science and Engineering Research.