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 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.