Complexity of reachability problems for finite discrete dynamical systems

AbstractSequential Dynamical Systems (SDSs) are a special type of finite discrete dynamical systems that can be used to model simulation systems. We focus on the computational complexity of testing several phase space properties of SDSs. Our main result is a sharp delineation between classes of SDSs whose behavior is easy to predict and those whose behavior is hard to predict. Specifically, we show the following.1.Several state reachability problems for SDSs are PSPACE-complete, even when restricted to SDSs whose underlying graphs are of bounded bandwidth (and hence of bounded pathwidth and treewidth), and the function associated with each node is symmetric. Moreover, this result holds even when the underlying graph is d-regular for some constant d and all the nodes compute the same symmetric Boolean function. An immediate corollary of this result is a PSPACE-hard lower bound on the complexity of reachability problems for regular generalized 1D-Cellular Automata and undirected systolic networks with Boolean totalistic local transition functions.2.In contrast, the above reachability problems are solvable in polynomial time for SDSs when the Boolean function associated with each node is symmetric and monotone. The PSPACE-completeness results follow as corollaries of simulation results which show for several classes of SDSs, how one class of SDSs can be efficiently simulated by another (more restricted) class of SDSs. We also prove several structural properties concerning the phase space of an SDS. SDSs are closely related to Cellular Automata (CA), concurrent transition systems, discrete Hopfield networks and systolic networks. This observation in conjunction with our lower bounds for SDSs, yields new PSPACE-hard lower bounds on the complexity of state reachability problems for these models, extending some of the earlier results in [K. Culik II, J. Karhumäki, On totalistic systolic networks, Inform. Process. Lett. 26 (5) (1988) 231–236; P. Floréen, E. Goles, G. Weisbuch, Transient length in sequential iterations of threshold functions, Discrete Appl. Math. 6 (1983) 95–98; P. Floréen, P. Orponen, Complexity issues in discrete Hopfield networks, Research Report No. A-1994-4, Department of Computer Science, University of Helsinki, 1994. Also appears in: I. Parberry (Ed.), Comp. and Learning Complexity of Neural Networks: Advanced Topics, 1999; D. Harel, O. Kupferman, M.Y. Vardi, On the complexity of verifying concurrent transition systems, Inform. and Comput. 173 (2002) 143–161; S.K. Shukla, H.B. Hunt III, D.J. Rosenkrantz, R.E. Stearns, On the complexity of relational problems for finite state processes, in: International Colloquium on Automata Programming and Languages, ICALP, 1996, pp. 466–477; A. Rabinovich, Complexity of equivalence problems for concurrent systems of finite agents, Inform. and Comput. 127 (2) (1997) 164–185]