An Introduction to Computational Geometry
It is the author's view that although the time is not yet ripe for developing a really general theory of automata and computation, it is now possible and desirable to move more explicitly in this direction. This can be done by studying in an extremely thorough way well-chosen particular situations that embody the basic concepts. This is the aim of the present book, which seeks general results from the close study of abstract versions of devices known as perceptrons.
A perceptron is a parallel computer containing a number of readers that scan a field independently and simultaneously, and it makes decisions by linearly combining the local and partial data gathered, weighing the evidence, and deciding if events fit a given “pattern,” abstract or geometric. The rigorous and systematic study of the perceptron undertaken here convincingly demonstrates the authors' contention that there is both a real need for a more basic understanding of computation and little hope of imposing one from the top, as opposed to working up such an understanding from the detailed consideration of a limited but important class of concepts, such as those underlying perceptron operations. “Computer science,” the authors suggest, is beginning to learn more and more just how little it really knows. Not only does science not know much about how brains compute thoughts or how the genetic code computes organisms, it also has no very good idea about how computers compute, in terms of such basic principles as how much computation a problem of what degree of complexity is most suitable to deal with it. Even the language in which the questions are formulated is imprecise, including for example the exact nature of the opposition or complementarity implicit in the distinction “analogue” vs. “digital,” “local” vs. “global,” “parallel” vs. “serial,” “addressed” vs. “associative.” Minsky and Papert strive to bring these concepts into a sharper focus insofar as they apply to the perceptron. They also question past work in the field, which too facilely assumed that perceptronlike devices would, automatically almost, evolve into universal “pattern recognizing,” “learning,” or “self-organizing” machines. The work recognizes fully the inherent impracticalities, and proves certain impossibilities, in various system configurations. At the same time, the real and lively prospects for future advance are accentuated.
The book divides in a natural way into three parts – the first part is “algebraic” in character, since it considers the general properties of linear predicate families which apply to all perceptrons, independently of the kinds of patterns involved; the second part is “geometric” in that it looks more narrowly at various interesting geometric patterns and derives theorems that are sharper than those of Part One, if thereby less general; and finally the third part views perceptrons as practical devices, and considers the general questions of pattern recognition and learning by artificial systems.