Simulating quadratic dynamical systems is PSPACE-complete (preliminary version)

Abstract

Quadratic Dynamical Systems (QDS), whose definition extends that of Markov chains, are used to model phenomena in a variety of fields like statistical physics and natural evolution. Such systems also play a role in genetic algorithms, a widelyused class of heuristics that are notoriously hard to analyze. Recently Rabinovich et al. took an important step in the study of QDS’s by showing, under some technical assumptions, that such systems converge to a stationary distribution (similar theorems for Markov Chains are well-known). We show, however, that the following sampling problem for QDS’s is PSPACE-hard: Given an initial distribution, produce a random sample from the t’th generation. The hardness result continues to hold for very restricted classes of QDS’s with very simple initial distributions, thus suggesting that QDS’s are intrinsically more complicated than Markov chains. ∗Supported by an IBM Graduate Fellowship and partly under NSF grant CCR-9310214. Email: arora@cs.berkeley.edu. †Work done while at ICSI, Berkeley, and supported in part by a Rothschild postdoctoral fellowship. Email: rabani@theory.lcs.mit.edu. ‡Supported by NSF grant CCR-9310214. Email: vazirani@cs.berkeley.edu.