On Time-Constructible Functions in One-Dimensional Cellular Automata

In this paper, we investigate time-constructible functions in one-dimensional cellular automata (CA). It is shown that (i) if a function t(n) is computable by an O(t(n) - n)-time Turing machine, then 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 example for which time-constructible functions are required, we present a time-hierarchy theorem based on CA. It is shown that if t1(n) and t2(n) are time-constructible functions such that limn→∞t1(n)/t2(n)=0, then there is a language which can be recognized by a CA in t2(n) time but not by any CA in t1(n) time.

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