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

TL;DR

This paper explores how Hopfield networks with associative memory and Hebbian learning can solve propositional satisfiability problems, revealing both their potential and limitations in handling complex combinatorial tasks.

Contribution

It demonstrates the application of the Self-Optimization model to SAT problems and analyzes the risks of information loss during learning.

Findings

SO model can solve SAT problems like Liars and map coloring.

Critical information may be lost, leading to inappropriate solutions.

Loss of information can provide insights into the model's behavior.

Abstract

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.