In fact, about the only restriction is that the scheme itself should not exhibit universality just in setting up initial conditions and decoding output. … And usually the only way to be sure of this is to have a more or less complete analysis of all possible behavior that the system can exhibit.

Intelligence in the Universe Whether or not we as humans are the only examples of intelligence in the universe is one of the great unanswered questions of science. … Yet our experience has still been that to build a computer requires sophisticated engineering that in a sense exists only because of human biological and cultural development.

And indeed one of the things that emerges from this book is that traditional engineering has actually considered only a tiny and quite unrepresentative fraction of all the kinds of systems and processes that are in principle possible. … And with traditional intuition it has normally been assumed that the only way to create systems that show a higher degree of complexity is somehow to build this complexity into their underlying rules.

LZW algorithms Practical implementations of pointer-based encoding can maintain only a limited dictionary of possible repeats.

The only smaller cluster with the same property is the trivial one with just a single node. … But there are some clusters that can overlap only in a few small networks.

History of numbers Numbers were probably first used many thousands of years ago in commerce, and initially only whole numbers and perhaps rational numbers were needed. … Nevertheless, for a very long time, and despite some development of algebra, only numbers that could somehow in principle be constructed mechanically were ever considered. … In fact, it was really only through the development of rigorous mathematical analysis in the late 1800s that this confusion finally began to clear up.

So this means that the complete evolution can be represented as in the picture at the top of the facing page , with each sequence shown explicitly only once, and any sequence generated more than once indicated just by an arrow going back to its first occurrence.

But a little experimentation reveals that the simple repetitive pattern above satisfies the constraint, and in fact it is the only pattern to do so.

In each case they start with the number 1, then successively multiply by the specified multiplier, keeping only the rightmost 31 digits in the base 2 representation of the number obtained at each step.

In a sequential substitution system only the first replacement that is found to apply in a left-to-right scan is ever performed at any step.