Constructible functions in cellular automata and their applications to hierarchy results

Abstract

We investigate time-constructible functions in one-dimensional cellular automata (CA). It is shown that (i) if a function $\text{t(n)}$ is computable by an $\text{O}\text{(t(n)−n)}$-time Turing machine, then $\text{t(n)}$ is time constructible by CA and (ii) if two functions are time constructible by CA, then the sum, product, and exponential functions of them are time constructible by CA. As an application, it is shown that if $\text{t}{}_1\text{(n)}$ and $\text{t}{}_2\text{(n)}$ are time constructible functions such that $\text{lim}{}_{\mathrm{n\to \infty }}\text{t}{}_1\text{(n)/t}{}_2\text{(n)}\text{=}\text{0}$ and $\text{t}{}_1\text{(n)⩾n}$, then there is a language which can be recognized by a CA in $\text{t}{}_2\text{(n)}$ time but not by any CA in $\text{t}{}_1\text{(n)}$ time.