Simulations between multi-dimensional deterministic and alternating cellular automata

Authors:
Chuzo Iwamoto;Katsuyuki Tateishi;Kenichi Morita;Katsunobu Imai
Affiliations:
Graduate School of Engineering, Hiroshima University;NTT Comware Corporation, Tokyo, 108-8019 Japan and Graduate School of Engineering, Hiroshima University;Graduate School of Engineering, Hiroshima University;Graduate School of Engineering, Hiroshima University
Venue:
Fundamenta Informaticae - Special issue on cellular automata
Year:
2003

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Abstract

We present simulation and separation results between multi-dimensional deterministic and alternating cellular automata (CAs). It is shown that for any integers k ≥ l ≥ 1, every k-dimensional t(n)-time deterministic CA can be simulated by an l-dimensional O(t(n)k-l+1/k-l+2)-time alternating CA. This result is a dimension reduction theorem and also a time reduction theorem: (i) Every multi-dimensional deterministic CA can be simulated by a one-dimensional alternating CA without increasing time complexity. (ii) Every deterministic computation in a multi-dimensional deterministic CA can be sped up quadratically by alternations when the dimension is fixed. Furthermore, it is shown that there is a language which can be accepted by a one-dimensional alternating CA in t(n) time but not by any multi-dimensional deterministic CA in t(n) time.