On the use of associative memory in Hopfield networks designed to solve propositional satisfiability problems

Hopfield networks are an attractive choice for solving many types of computational problems because they provide a biologically plausible mechanism. The Self-Optimization (SO) model adds to the Hopfield network by using a biologically founded Hebbian learning rule, in combination with repeated network resets to arbitrary initial states, for optimizing its own behavior towards some desirable goal state encoded in the network. In order to better understand that process, we demonstrate first that the SO model can solve concrete combinatorial problems in SAT form, using two examples of the Liars problem and the map coloring problem. In addition, we show how under some conditions critical information might get lost forever with the learned network producing seemingly optimal solutions that are in fact inappropriate for the problem it was tasked to solve. What appears to be an undesirable side-effect of the SO model, can provide insight into its process for solving intractable problems.

Scaling up the self-optimization model by means of on-the-fly computation of weights

The Self-Optimization (SO) model is a useful computational model for investigating self-organization in "soft" Artificial life (ALife) as it has been shown to be general enough to model various complex adaptive systems. So far, existing work has been done on relatively small network sizes, precluding the investigation of novel phenomena that might emerge from the complexity arising from large numbers of nodes interacting in interconnected networks. This work introduces a novel implementation of the SO model that scales as $\mathcal{O}\left(N^{2}\right)$ with respect to the number of nodes $N$, and demonstrates the applicability of the SO model to networks with system sizes several orders of magnitude higher than previously was investigated. Removing the prohibitive computational cost of the naive $\mathcal{O}\left(N^{3}\right)$ algorithm, our on-the-fly computation paves the way for investigating substantially larger system sizes, allowing for more variety and complexity in future studies.