And certainly it is now very hard for us to imagine just what range of purposes the first known stone tools from 2.6 million years ago might have been put to—or what purpose the arrays of dots or handprints in cave paintings from 30,000 BC might have had.

(Generically there are 2 n solutions for v , and even for integer coefficients in the range -r to +r already in 95% of cases there are 4 solutions with n = 2 as soon as r ≥ 6 .)

across a vast range of physical, biological and other systems we are continually confronted with what seems to be immense complexity.

And in more recent years, I have discovered a vast range of new phenomena as a result of easily being able to set up large numbers of computer experiments in Mathematica.

For in the course of the chapter we have discussed a whole range of different kinds of perception and analysis, yet in essentially all cases we have found that the overall capabilities they exhibit are rather similar.

For in the late 1950s a whole hierarchy of systems with so-called intermediate degrees were constructed with the property that questions about the ultimate output from their evolution could not in general be answered by finite computation, but for which the actual form of this output was not flexible enough to be able to emulate a full range of other systems, and thus support universality.

But what the Principle of Computational Equivalence implies is that there are actually a vast range of very different kinds of rules that all lead to exactly the same computational capabilities—and so can all in principle be used as a basis for making computers.

But usually these turn out to lead only to modest speedups, and despite various hopes over the years there seem in the end to be no techniques that work well across any very broad range of systems.

Higher-order logics In ordinary predicate—or so-called first-order—logic the objects x that ∀ x and ∃ x range over are variables of the kind used as arguments to functions (or predicates) such as f[x] .

Almost all electronic devices also exhibit a third kind of noise, whose main characteristic is that its spectrum is not flat, but instead goes roughly like 1/f over a wide range of frequencies.