Simulation of d′-dimensional cellular automata on d-dimensional cellular automata

Abstract

In this paper a fast and space efficient method for simulating a d′-dimensional cellular automaton (CA) on a d-dimensional CA (d < d′) is introduced. For d′ -2 and d = 1 this method is optimal (under certain assumptions) with respect to time as well as space complexity. Let in this case t(n) be the time complexity and r(n) the side length of the smallest square enclosing all used cells. Then the simulation does not need more than O(r2) cells and has a running time of O(r· t). In the general case d′ = d +1 a version with the time and space complexity of O(t 2) will be presented. Finally it will be shown, how it is possible to simulate a 2d-dimensional CA on a d-dimensional CA in a similarly efficient way. © Springer-Verlag Berlin Heidelberg 2006.