The Phenomenon of Universality
In the previous section we saw that it is possible to get cellular automata to perform some fairly sophisticated computations. But for each specific computation we wanted to do, we always set up a cellular automaton with a different set of underlying rules. And indeed our everyday experience with mechanical and other devices might lead us to assume that in general in order to perform different kinds of tasks we must always use systems that have different underlying constructions.
But the remarkable discovery that launched the computer revolution is that this is not in fact the case. And instead, it is possible to build universal systems whose underlying construction remains fixed, but which can be made to perform different tasks just by being programmed in different ways.
And indeed, this is exactly how practical computers work: the hardware of the computer remains fixed, but the computer can be programmed for different tasks by loading different pieces of software.
The idea of universality is also the basis for computer languages. For in each language, there are a certain set of primitive operations, which are then strung together in different ways to create programs for different tasks.
The details of a particular computer system or computer language will certainly affect how easy it is to perform a particular task. But the crucial fact that is by now a matter of common knowledge is that with appropriate programming any computer system or computer language can ultimately be made to perform exactly the same set of tasks.
One way to see that this must be true is to note that any particular computer system or computer language can always be set up by appropriate programming to emulate any other one.
Typically the way this is done is by having each individual action in the system that is to be emulated be reproduced by some sequence of actions in the other system. And indeed this is ultimately how, for example, Mathematica works. For when one enters a command such as Log, what actually happens is that the program which implements the Mathematica language interprets this command