Quasilinear cellular automata

Abstract

Simulating a cellular automaton (CA) for t time-steps into the future requires t 2 serial computation steps or t parallel ones. However, certain CAs based on an Abelian group, such as addition mod 2, are termed linear because they obey a principle of superposition. This allows them to be predicted efficiently, in serial time O( t) (Robinson, 1987) or O(log t) in parallel. In this paper, we generalize this by looking at CAs with a variety of algebraic structures, including quasigroups, non-Abelian groups, Steiner systems, and other structures. We show that in many cases, an efficient algorithm exists even though these CAs are not linear in the previous sense; we term them quasilinear. We find examples which can be predicted in serial time proportional to t, t log t, t log 2 t and t α for α < 2, and parallel time log t, log t log log t and log 2 t. We also discuss what algebraic properties are required or implied by the existence of scaling relations and principles of superposition, and exhibit several novel “vector-valued” CAs.