Speeding-Up Cellular Automata by Alternations
There are two simple models of cellular automata: a semiinfinite array (with left boundary) of cells with sequential input mode, called an iterative array (IA), and a finite array (delimited at both ends) of n cells with parallel input mode, called a bounded cellular array (BCA). This paper presents a quadratic speedup theorem for IAs and an exponential speedup theorem for BCAs by using alternations. It is shown that for any computable functions s(n), t(n) ≥ n, every s(n)t(n)- time deterministic IA can be simulated by an O(s(n))-space O(t(n))- time alternating IA. Since any t(n)-time IA is t(n)-space bounded, every (t(n)) 2-time deterministic IA can be simulated by an O(t(n))-time alternating IA. This leads to a separation result: There is a language which can be accepted by an alternating IA in O(t(n)) time but not by any deterministic IA in O(t(n)) time. It is also shown that every t(n)- time nondeterministic BCA can be simulated by a linear-time alternating BCA.
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