Counting stable states and sizes of attraction domains in Hopfield nets is hard

It is shown that, given a Hopfield net, it is NP-hard to count either the number of stable states or the number of states converging to a given stable state. The latter result holds even when the interconnection weights between neurons are restricted to 0 or +or-1. The authors show that the stable state counting problem and the problem of counting states converging to a given stable state in a single parallel update step are both Hash P-complete. A remaining open problem is to show that computing the size of the full attraction domain of a given stable state is also Hash P-complete.<>