Switching Theory: Insight through Predicate Logic

439 Pages·2004·29.576 MB·English

by  Dr. Shimon P. Vingron (auth.)

S.P. Vingron
Switching Theory
Springer-Verlag Berlin Heidelberg GmbH
ONLINE LlBRARY
Engineering
http://www.springer.de/engine/
Shimon P. Vingron
Switching Theory
Insight through Predicate Logic
With 323 Figures
, Springer
Dr. Shimon P. Vingron
formerly head of the Department of Theoretical Systems AnaIysis,
Institute for Information Processing, Austrian Academy of Sciences, Vienna
Barenkogelweg 21
2371 Hinterbriihl
Austria
ISBN 978-3-642-07318-2
Library of Congress Cataloging-in-Publication-Data
Vingron, Shimon Peter, 1936-
Switching theory : insight through predicate logic I Shimon Peter Vingron.
p.cm.
ISBN 978-3-642-07318-2 ISBN 978-3-662-10174-2 (eBook)
DOI 10.1007/978-3-662-10174-2
1. Switching theory. 2. Predicate (Logic) 1. Title.
TK7868.S9VS62004
621.381S'372--dc22
This work is subject to copyright. AU rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitations,
broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Dupli
cat ion of this publication or parts thereof is permitted only under the provisions of the German
copyright Law of September 9, 1965, in its current version, and permission for use must always be
obtained from Springer-Verlag Berlin Heidelberg GmbH.
Violations are liable for prosecution under the German Copyright Law.
http://www.springer.de
© Springer-Vedag Berlin Heidelberg 2004
Originally published by Springer-Verlag Berlin Heidelberg New York in 2004
Softcover reprint ofthe hardcover Ist edition 2004
The use of general descriptive names, registered names trademarks, etc. in this publication does
not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
Typesetting: camera-ready by author
Cover design: deblik Berlin
Printed on acid free paper 6213020/M -S 4 3 2 1 O
Preface
About the subject. Switching Theory is an applied science used in
analysing, designing, and testing switching circuits irrespective of the tech
nology in which they are realised. Its origin goes back to Shannon's [1938J
remarkable paper in which he introduced 'propositional logic' to describe
switching circuits. This was the first time that propositionallogic was ap
plied to an engineering problem. The method proved successful but was
restricted to circuits that had no memorising ability, so-called combinational
circuits. It was Huffman [1954J who, in his doctor's thesis, showed how to
analyse and design circuits that had a memorising ability, so-called sequential
circuits. At the heart of his breakthrough was a simple way of specifying a
sequential circuit by a table he called a 'fiow table '. It enabled a sequential
circuit to be reduced to a number of combinational circuits, each of which
could then be handled by Shannon's methods.
About this book. Just as propositionallogic was superseded in math
ematics by predicate logic, developed by Frege [1879], this is also done here
for Switching Theory. You will notice that with the use of predicate logic
proofs and derivat ions become possible that hitherto remained out of reach.
Huffman's flow table is used to develop a new tree representation of a
sequential circuit, a so-called word-recognition tree, by which a sequential
circuit is not reduced to a number of combinational circuits but, rather, cal
culated directly as interconnected latches (memory circuits). This approach
required a new look at the theory of latches. But most importantly, the
formidable problems connected with the so-called 'encoding of the internal
state.'J' of an asynchronous sequential circuit are bypassed.
Pointing out what's new. The many new concepts, proofs, and
methods incorporated into this book are not explicitly listed. Those chapters
marked by an asterisk (*) present new concepts. Those sections marked by
an asterisk (*) at least contain new concepts, proofs, or methods.
Whom the book is for. The book is written for the theoretically
inclined practising engineer working in the field of digital circuit design, for
the scientist concerned with developing methods and algorithms for designing
digital ('in'nits, and for the student feeling the nccd to supplement a course in
Switching Theory or Logic Design. While Division One is an undergraduate
level introduction, the remaining parts constitute a sound basis for a graduate
course in Switching Theory or Logic Design.
Expressing my gratitude. When I retired, my wife, Dora, told me
she didn't want me hanging around the house not knowing what to do with
myself. So I went into my study, and took up the research (interrupted more
than a decade earlier) on what was to become this book. This project, that
I had thought would take two to possibly three years, engulfed me for eight.
During these years, my wife unfailingly encouraged me in the many difficult
phases of the work. My gratitude is only surpassed by my love to her.
Deep thanks are due to my friend Dr. Oliver König who worked through
the whole manuscript and whose formidable and creative criticism I simulta
neously feared and awaited. Many major improvements are his due.
In many ways, this book is a tribute to my former teacher Prof. emeritus
Dr. Karl Heinz Fasol.
A elosing remark. I should consider it a compliment if this book
were to stimulate comment, or argument, or contradiction. I should consider
it a success if it were to help start a process of questioning established views
in a subject which, although having lost some of its former vigour, has lost
nothing of its importance.
It is not given to you to compleie the iask,
bnt Y01l are not freed from starting.
Rabbi Tarphon
Ethics of the Fathers iij21
Contents
Division One Fundamental Concepts
1 Outline and Basics
3
1.1 Specifying a Circuit in Plain Prose 3
1.2 Diagram of Successive Events 4
1.3 Table of Asserted Events *1) 5
1.4 Logic Variables and Logic Formulas * 7
1.5 Drawing the Circuits 9
*
1.6 Sequential Circuits 10
*
1. 7 Verifying a Circuit 12
2 Switching Devices
14
2.1 Pneumatic Valves 14
2.2 Electric Relays 18
2.3 CMOS Transistor Circuits 23
3 Functions
27
3.1 Ordered Pairs 27
3.2 Spealdng of Functions. . . 31
3.3 Switching Functions 33
4 Logic Functions and Gates
36
4.1 Elementary Switching Functions 36
*
4.2 Positive versus Negative Logic 37
4.3 Elementary Logic Functions * 38
4.4 The Basic Gates * 40
*
4.5 Derived Gates 43
5 Synthesis and Duality
46
*
5.1 Minterm Functions & Minterms 46
*
5.2 Maxterm Functions & Maxterms 49
5.3 Synthesis via Partial Outputs * 50
*
5.4 Duality 53
6 Karnaugh Maps
57
6.1 Speaking of Sets.. . 57
*
6.2 Introducing the Karnaugh Map 59
6.3 Karnaugh Maps for multiple Inputs 62
*
6.4 Karnaugh Sets 64
1) Those Sections or Chapters marked by an asterisk (*) contain new concepts,
proofs, or methods.
7 Utilising Karnaugh Maps
66
*
7.1 Specifying Switching Circuits in K-maps 66
7.2 Obtaining Disjunctive Formulas 67
*
7.3 Obtaining Conjunctive Formulas 68
7.4 Logically Equivalent Expressions * 70
*
7.5 Logical Implications 72
*
7.6 K-maps of Dual Functions 74
Division Two Logic
8 Tautologies 79
8.1 Logic Expressions 79
8.2 Truth Tables 81
8.3 Speaking of Tautologies. . . 82
8.4 Replacement versus Substitution 85
8.5 Logic Reasoning 86
9 Propositional Logic
89
9.1 Axiomatic Approach to Propositional Logic 89
9.2 Complementation 91
9.3 IMPLICATION and NEGATION 93
9.4 DeMorgan's Theorems 94
9.5 Commutativity of AND and OR 96
9.6 Logic Implications of IMPLICATIONS 97
9.7 Formulas with a single Variable 99
10 Summary of Theorems
101
*
10.1 Commutative and Associative Laws 101
10.2 Single-Variable Formulas 103
10.3 Distributive Laws * 104
10.4 Generalised DeMorgan Theorems 106
10.5 Basic Theorems on AND, OR and NOT 109
11 Algebraic Proofs 110
*
11.1 Min- and Maxterms are Complementary 110
*
11.2 Disjunction of all Minterms 111
11.3 Conjunction of two Mintermms * 113
11.4 Maxterm as the Disjunction of Minterms * 114
11.5 Minterm AND/OR Maxterm * 115
*
11.6 Solving a System of Logic Equations 117
11. 7 DeMorgan on DeMorgan 118
12 On Predicate Logic
121
12.1 Inside a Proposition 121
12.2 Symbolic Notation of Propositions 122
12.3 The Switching Algebra Connection * 124
12.4 Quantifiers 124
*
12.5 Quantification and Replication 126
12.6 Free and Bound Variables 126
13 Predicate Logic
129
*
13.1 Axioms and Rules of Switching Algebra 129
13.2 Theorems on Identity 132
13.3 Theorems on Quantification 135
Division Three Combinational Circuits
14 Canonical Normal Forms
141
14.1 An Overview 141
*
14.2 Direct Derivation of Normal Forms 144
*
14.3 Shannon's Expansion Theorems 147
14.4 Shannon's Expansion to Normal Forms 149
15 Shegalkin Normal Form
151
15.1 An Overview 151
15.2 Developing the Shegalkin Polynomial * 153
15.3 Shegalkin Coefficients 155
15.4 On Combinations of Input Variables 156
15.5 Dual Shegalkin Polynomial 157
15.6 Necessary and Sufficient Connectives 160
16 Synthesis Examples
162
16.1 Multiplexers and Demultiplexers 162
16.2 Binary Coded Decimal Digits 163
16.3 Priority Decoders 165
16.4 Comparators 166
17 Concepts Old and New 171
17.1 Multiple Events* 171
17.2 Karnaugh Sets Revisited * 172
17.3 Generalised Minterms * 173
17.4 Generalised Maxterms * 175
17.5 Partitions and Equivalents 176
17.6 Prime Sets 177
17.7 Covers 177
17.8 Inclusions and Exclusions * 179
17.9 Evaluation Formulas * 180
18 Minimisation Preliminaries
182
18.1 Aspects of Minimisation 182
18.2 Incomplete Specification 183