Introduction to theoretical computer science

121 Pages·1990·36.359 MB·English

by  Ma, Xiwen

World Scientific Series in Computer Science — Vol. 23
Introduction
to
A
Theoretical
Computer
Science
Xiwen MA
World Scientific
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Introduction to
Theoretical Computer Science
World Scientific Series in Computer Science
Published
1: Computer-Aided Specification Techniques (J Demetrovics, E Knuth
& P Rado)
2: Proceedings of the 2nd RIKEN International Symposium on Symbolic
and Algebraic Computation by Computers (Eds. N Inada & TSoma)
3: Computational Studies of the Most Frequent Chinese Words and
Sounds (Ching YSuen)
4: Understanding and Learning Statistics by Computer (MCK Yang &
D H Robinson)
5: Visualizing Abstract Objects and Relations - A Constraint-Based
Approach (TKamada)
6: DC Flux Parametron - A New Approach to Josephson Junction Logic
(E Goto & K F Loe)
7: Syntactic and Structural Pattern Recognition - Theory and Applications
(Eds. H Bunke & A Sanfeliu)
8: Information, Randomness & Incompleteness - Papers on Algorithmic
Information Theory, Second Edition (G J Chaitin)
9: P-Prolog - A Parallel Logic Programming Language (Rong Yang)
10: Intelligent Chinese Language Pattern and Speech Processing
(Ed. PSP Wang)
11: Control Theory of Robotic Systems (J M Skowronski)
12: An Introduction to Chinese, Japanese, and Korean Computing
(JKTHuang&TD Huang)
13: Mathematical Logic for Computer Science (ZIV Lu)
14: Computer Vision and Shape Recognition (Eds. A Krzyzak, T Kasvand
& C YSuen)
15: Stochastic Complexity in Statistical Inquiry (J Rissanen)
15: Stochastic Complexity in Statistical Inquiry (J Rissanen)
16: A Perspective in Theoretical Computer Science - Commemorative
16: A Perspective in Theoretical Computer Science - Commemorative
Volume for Gift Siromoney (Ed. R Narasimhan)
Volume for Gift Siromoney (Ed. R Narasimhan)
17: Computer Transformation of Digital Images and Patterns (Z C Li,
17: Computer Transformation of Digital Images and Patterns (Z C Li,
TDBui, YYTang&CY Suen)
TDBui, YYTang&CY Suen)
18: Array Grammars, Patterns and Recognizers (Ed. PSP Wang)
18: Array Grammars, Patterns and Recognizers (Ed. PSP Wang)
19: Structural Pattern Analysis (Eds. R Mohr. Th Pavlidis & A Sanfeliu)
19: Structural Pattern Analysis (Eds. R Mohr, Th Pavlidis & A Sanfeliu)
20: A Computational Model of First Language Acquisition (A/ Satake)
21: The Design and Implementation of ConcurrentSmalltalk (/ Yokote)
23: I ntrod uction to Theoretical Computer Science (XIV Ma)
Forthcoming
22: From Humans to Computers - Cognition Through Visual Perception
(VVAIexandrov&ND Gorsky)
World Scientific Series in Computer Science — Vol. 23
Introduction
to
Theoretical
Computer
Science
Xiwen Ma
Peking University
World Scientific
Singapore • New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farrei Road, Singapore 9128
USA office: 687 Hartwell Street, Teaneck, NJ 07666
UK office: 73 Lynton Mead, Totteridge, London N20 8DH
Library of Congress Cataloging-in-Publication data is available.
INTRODUCTION TO THEORETICAL COMPUTER SCIENCE
Copyright © 1990 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced
in any form or by any means, electronic or mechanical, including photo­
copying, recording or any information storage and retrieval system now
known or to be invented, without written permission from the Publisher.
ISBN 981-02-0193-1
Printed in Singapore by JBW Printers & Binders Pte. Ltd.
PREFACE
Theoretical computer science is the mathematical foundation of computer science and this book is an intro­
duction to it. The content of this book is concentrated on the following question; what is the essence of
computation. To answer this question, we need mathematical models for computation. In the history of
computing machines, Babbage proposed a computing device before the systematic investigation of computa­
tion, but his project was uncompleted and gave little influence to his succesors. Many years later, in the
30's of this century, several models were proposed, such as X-calculus, Turing machines and recursive
functions, etc. , and the essentials of computation becomes clear. In particular, Turing machine gave deep
influence to the invention of modern computer. For example, the concept of "general-proposed computer"
came from the theory of Turing machines. Later, the theory of Turing machines became the axis and the
basis of theoretical computer science, and fromhere were developed many fruitful theories such as theory of
computability, theory of equivalence of mathematical models, theory of deterministic and non-deterministic
computation, theory of complexity, and many others.
However, the computer science has been highly developed since then, and it became more and more obvious
that, as the axis and basis of whole computer science, the theory of Turing machine is out of time.
First of all, it is because this theory is too far from the practice of computers. If we want to write a book on
mathematical physics, but begin with set theory and construction of number systems, the book must be fire
some. This is exactly the case we meet in the theory of Turing machines. To the readers who have suffi­
cient knowledge in computer practice, it is not necessary to make the models of computation so simple. An
axiomatic approach might be more acceptable.
Moreover, Turing machine is a mathematical model of a particular kind of computing mechanism, the se­
quential computation. The natural implementation of such mechanism is the von Neumann computer,
which is provoking more and more criticisms. Different kinds of computing mechanisms are proposed to
complete for the new generation computers, some of them, such as logical computation, are hardly mod-
elized in the framework of Turing machines.
In addition, Turing machine also has difficulties on formalism, that is, in discussions of Turing machines,
we need a special language which is very distant to the traditional mathematical languge, as well as to the
ones used in other areas of computer science.
With this consideration, we adopt the generalized recursive functions of symbolic expressions as the axis of
our theory. This is partially motivated by John McCarthy's pioneer works. We hope that this approach is
easier to the students, because they may feel familiar with our methodology and results, and, in fact, there
are no complecated mathematical theorems in this book.
To study the theoretical computer science, the students should have good mathematical accomplishments.
Unfortunately, as the computer science becomes more and more technical, its theoretical stricktness be-
v
comes less and less attentional. Most of the students in computer science are taking lessons from professors
having very good mathematical background, but rarely have chances to improve their own mathematical a-
bilities. Lessons on theoretic computer science could be such a chance. Hence, we paid great attention to the
mathematical strictness and we hope our readers will be conscious of this point.
The first chapter is an outline of computability, chapter 2 gives an introduction to the theory of S-expres-
sions, and the remainding chapters discuss models of computation. The contents of this book are self-suffi­
cient in the sense that no preliminary knowledge other than elementary set theory is needed. However, this
book is not prepared for those who have no experience in computer practice.
This book arose from courses the author taught in the Graduate School of Peking University. Thanks are
due to my colleague Mr. Zongyan Qiu, who gave much advice during the course. Thanks also due to Mrs.
Zhihua Meng and her collegues, who helped me to prepare the manuscripts by using their typesetting system
Huaguang IV.
Xiwen Ma
Peking University
June 1989
VI