and system identification all in effect revolve around finding useful summaries of data.

In traditional science statistical analysis has been the most common way of trying to find summaries of data. And in general perception and analysis can be viewed as equivalent to finding models that reproduce whatever aspects of data one considers relevant.

Perception and analysis correspond in many respects to the inverse of most of what we have studied in this book. For typically what we have done is to start from a simple computer program, and then seen what behavior this program produces. But in perception and analysis we start from behavior that we observe, then try to deduce what procedure or program will reproduce this data.

So how easy is it to do this? It turns out that for most of the kinds of rules used in traditional mathematics, it is in fact fairly easy. But for the more general rules that I discuss in this book it appears to often be extremely difficult. For even though the rules may be simple, the behavior they produce is often highly complex, and shows absolutely no obvious trace of its simple origins.

As one example, the pictures on the facing page were all generated by starting from a single black cell and then applying very simple two-dimensional cellular automaton rules. Yet if one looks just at these final pictures, there is no easy way to tell how they were made. Our standard methods of perception and analysis can certainly determine that the pictures are for example symmetrical. But none of these methods typically get even close to being able to recognize just how simple a procedure can in fact be used to produce the pictures.

One might think that our inability to find such a procedure could just be a consequence of limitations in the particular methods of perception and analysis that we, as humans, happen to have developed. And one might therefore suppose that an alien intelligence could exist which would be able to look at our pictures and immediately tell that they were produced by a very simple procedure.

But in fact I very much doubt that this will ever be the case. For I suspect that there are fundamental limitations on what perception and analysis can ever be expected to do. For there seem to be many kinds of