Cellular Automata and Continuous Functions: Negative Results
Yuri Ozhigov
Department of Mathematics,
Moscow State Technological University "Stankin,"
Vadkovsky per. 3a, 101472, Moscow, Russia
Abstract
Let 
be a finite alphabet, 
, and 
.
A configuration is a function of the form: 
, and 
is the set of all configurations. Two configurations 
and 
are near if 
is small, where 
.
The following results are proved.
- There is no sequence of functions
such that 
and 
uniformly converge to continuous functions in such a topology. - Evolutions of cellular automata (CA) cannot be approximated by the superpositions of real continuous functions.
In the proofs of these results advantage was taken of some CA acting in 
and in 
with a stationary boundary condition.
