Reachability problems for sequential dynamical systems with threshold functions

A sequential dynamical system (SDS) over a domain D is a triple (G, F, pi), where (i) G(V,E) is an undirected graph with n nodes with each node having a state value from D, (ii) F = {f(1),f(2),..f(n)} is a set of local transition functions with f(i) denoting the local transition function associated with node v(i) and (iii) pi is a permutation of (i.e., a total order on) the nodes in V. A single SDS transition is obtained by updating the states of the nodes in V by evaluating the function associated with each of them in the order given by pi. We consider reachability problems for SDSs with restricted local transition functions. Our main intractability results show that the reachability problems for SDSs are PSPACE-complete when either of the following restrictions hold: (i) F consists of both simple-threshold-functions and simple-inverted-threshold functions, or (ii) F consists only of threshold-functions that use weights in an asymmetric manner. Moreover, the results hold even for SDSs whose underlying graphs have bounded node degree and bounded pathwidth. Our lower bound results also extend to reachability problems for Hopfield networks and communicating finite state machines. On the positive side, we show that when F consists only of threshold functions that use weights in a symmetric manner, reachability problems can be solved efficiently provided all the weights are strictly positive and the ratio of the largest to the smallest weight is bounded by a polynomial function of the number of nodes. (C) 2002 Elsevier Science B.V. All rights reserved.