Graph theory and interactive protocols for Reachability Problems on finite Cellular automata

Abstract

Studying the computational complexity of the Configuration Reachabilty Problem (CREP) is a good way to investigate properties of a given discrete deterministic dynamical system like a Toroidal Cellular Automaton (TCA). We study CREP for two natural weakly predictable classes of TCA: the booleandisjunctive class and the additive one. For the first class, we reduce CREP to a path problem on a strongly connected digraph and we show a polynomialtime algorithm for this problem. Some consequences of this result on arbitrary TCA are also analysed. For the second class, we show that CREP is not easier than computing the vectorial version of the Discrete Log Problem (DLP). However, we also show CREP is unlikely to be N P-complete. To do this, we prove that CREP is in Co-AM[2], where AM[2] is the class of problems with a constant round interactive protocol [1, 13]. CREP is unlikely to be N P-complete (unless the polynomial time hierarchy collapses), then follows by the results of [3]. All such results hold even when multidimensional and/or non homogeneous TCA arise. As a global consequence, we argue that the structure of the weakly predictable class resembles the N P one.