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

TL;DR

It is demonstrated first that the SO model can solve concrete combinatorial satisfiability problems: The Liars problem and the map coloring problem and it is discussed how under certain 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.

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.

Figures (4)

• Figure 1: Simulation results for (a-c) the Liars problem ($N=50$ people, $34$ statements, learning rate $\alpha=2.5\times10^{-7}$, $20N$ steps), and (d-f) the Checkerboard map coloring problem ($N=2\cdot64=128$ states, $\alpha=8\times10^{-21}$, $10N$ steps). (a),(d) The initial weights $\mathbf{W}_{0}$ derived from the cost function $E_{\lnot\Phi}$ comparison with \ref{['eq:E_HN']}, (b),(e) the weights $\mathbf{W}$ after learning, (c),(f) the energy at the end of convergence for a set without learning (resets 1--1000 and 1--40, blue), during learning (1001--2000 and 41--80, red), and after learning (2001-3000 and 81-120, light blue) for the two problems, respectively.
• Figure 2: Colored checkerboard from the learned state $\mathbf{S}$.
• Figure 3: Self-optimization simulation results for coloring the map of South America ($N=2\cdot16=32$ states, $20N$ steps) for (a-d) $\omega=\left[1,1,1\right]$ (learning rate $\alpha=8\times10^{-7}$), (e-h) $\omega=\left[5,5,1\right]$ ($\alpha=2.1\times10^{-5}$) and (i-l) $\omega=\left[1,1,b_{ii'}^{L}\right]$ ($\alpha=2.1\times10^{-5}$). First and second columns show the weight matrices before and after learning, respectively. Third column shows energy at the end of convergence for a set without learning (resets 1-1000, blue), during learning (1001-2000, red), and after learning (2001-3000, light blue). The right column shows the colored map resulting from the learned state $\mathbf{S}$. Note that while the Hopfield dynamics and learning are computed using modified weights for the lower two rows, the energies in sub-plots g and k are computed from the non-weighted constraints identical to sub-plot c for ease of comparison.
• Figure 4: Self-optimization simulation results for coloring the map of South America ($N=2\cdot16$ states, $20N$ steps) $\omega=\left[1,1,b_{ii'}^{L}\right]$ ($\alpha=6\times10^{-6}$). Sub-plot a shows energy at the end of convergence for a set without learning (resets 1-1000, blue), during learning (1001-2000, red), and after learning (2001-3000, light blue). Subplot b shows the colored map resulting from the learned state $\mathbf{S}$. For this learning rate, the weights on the constraints $\omega=\left[1,1,b_{ii'}^{L}\right]$ result in the system converging to the same energy state as for $\omega=\left[5,5,1\right]$ (compare to Fig. \ref{['fig:3by4Plot']}g-h).