Linear Time Language Recognition on Cellular Automata with Restricted Communication

Abstract

It is well-known that for classical one-dimensional one-way CA (OCA) it is possible to speed up language recognition times from (1 + r)n, r  ∈  R ₊, to (1 + r/2)n. In this paper we show that this no longer holds for OCA in which a cell can comminucate only one bit (or more generally a fixed amount) of information to its neighbor in each step. For arbitrary real numbers r ₂ > r ₁ > 1 in time r ₂ n 1-bit OCA can recognize strictly more languages than those operating in time r ₁ n. Thus recognition times may increase by an arbitrarily large constant factor when restricting the communication to 1 bit. For two-way CA there is also an infinite hierarchy but it is not known whether it is as dense as for OCA. Furthermore it is shown that for communication restricted CA two-way flow of information can be much more powerful than an arbitrary number of additional communication bits.