World-famous mathematician John H. Conway based this classic text on a 1966 course he taught at Cambridge University. Geared toward graduate students of mathematics, it will also prove a valuable guide to researchers and professional mathematicians. His topics cover Moore's theory of experiments, Kleene's theory of regular events and expressions, Kleene algebras, the differential calculus of events, factors and the factor matrix, and the theory of operators. Additional subjects include event classes and operator classes, some regulator algebras, context-free languages, communicative regular algebra, axiomatic questions, the strength of classical axioms, and logical problems. Complete solutions to problems appear at the end.
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The maintenance man in charge of a real machine M often has occasion to experiment with M. Sometimes he will be presented with M in some unaccustomed (and probably unknown) state and required to return it with a note specifying its new state. Occasionally he might suspect that some malfunction has transformed M into a new (and probably useless) machine M′, and will need to devise an experiment to determine whether this has in fact occurred. In both cases his method will probably be to apply certain inputs to the machine (each input depending on the previous outputs) until the resulting sequence of outputs in some sense contains the information he requires. The outcome of the experiment will be a member of some set of answers, for instance the set {yes, no, don’t know}.
Formally, we define an experiment on M as a function e: O* → I
A, where O* is the set of all output words (i.e., finite formal sequences of outputs), and A is some set of answers, disjoint from I. We perform e on M as follows. At any stage we will already have applied a word w = ab … k, and observed the resulting output word
If e(r(
, w)) is an input l, say, we extend w by applying l to the machine and observing the new output o(
ab … kl), so proceeding to the next state. If not, e(r(
, w)) is an answer called the outcome of e at
, and the experiment terminates.
We can perform the experiment at any state of any machine with the same console as M, and its outcome, if any, will probably depend on both machine and state. If in all intended performances it terminates in a bounded number of stages, we call if finite, and define its length as the greatest length of any of the corresponding input words w. (Note that the length of an input word w is one less than that of the resulting output word r(
, w), since the first output ‘comes free’.) We shall mostly be conce...
Table of contents
Cover Page
Title Page
Copyright Page
Contents
Preface
Preliminaries to the Moore Theory
1. Moore’s theory of experiments
2. Bombs and detonators
3. Kleene’s theory of regular events and expressions
