Switching Machines: Volume 2 Sequential Systems

433 Pages·1972·26.226 MB·English

by  J.-P. Perrin, M. Denouette, E. Daclin (auth.)

SWITCHING MACHINES
VOLUME 2
1.-P. PERRIN, M. DENOUETTE, AND E. DACLIN
SWITCHING MACHINES
VOLUME 2
SEQUENTIAL SYSTEMS
D. REIDEL PUBLISHING COMPANY/DORDRECHT-HOLLAND
SYSTEMES LOGIQUES, TOME 2
First published by Dunod, Paris, 1967
Translatedfrom the French
Library of Congress Catalog Card Number 70-118379
ISBN-13; 978-94-010-2869-1 e-ISBN-13: 978-94-010-2867-7
001: 10.1007/978-94-010-2867-7
All Rights Reserved
Copyright© 1972 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover 1s t edition 1972
No part of this book may be reproduced in any form, by print, photoprint, microfilm,
or any other means without written permission from the publisher
CONTENTS
CHAPTER 7/SYNTHESIS OF THE TABLES
7.1. Generalizations
7.1.1. Introduction
7.1.2. Review of a sequential system's general equations I
7.1.3. Normal form of the hypotheses 2
7.2. Natural methods 4
7.2.1. Ginsburg method-first case 4
7.2.1.2. The method in the general case 7
7.2.2. Ginsburg method-second case 8
7.2.2.1. Introductory examples 8
7.2.2.2. General statement of the method 12
7.2.3. Aizerman's method 13
7.2.3.1. Introductory example 13
7.2.3.2. General statement of the Aizerman
method 18
7.2.3.3. Other examples of application 20
7.2.4. Asynchronous machines-Moisil-Ioanin method 24
7.3. Algebraic methods-Notion of a regular expression 28
7.3.1. Introduction 28
7.3.2. The algebra of regular expressions 29
7.4. Gloushkov method 32
7.4.1. Generalizations. Indexation of regular expressions 32
7.4.2. Examples of synthesis starting from regular
expressions 35
7.4.2.1. Firstexample 35
7.4.2.2. Second example of synthesis by the
Gloushkov method 39
7.4.3. Statement of the Gloushkov method 45
7.4.4. Application of regular expression to the
synthesis of asynchronous systems 45
7.4.4.1. Representation of asynchronous controls
in terms of regular expressions 45
7.4.4.2. Example of synthesis of an
asynchronous system 47
7.5. Conclusion 52
Appendix 55
7.A. Brzozowski method 55
VI CONTENTS
7.A.l. Basic definitions. The derivative of a regular
expression with respect to a sequence of unity
length 55
7.A.2. Use of the derivative to obtain the table of a
machine 58
Bibliography 64
Exercises 65
CHAPTER 8/REDUCTION OF THE NUMBER
OF STATES IN A TABLE
8.1. Introduction-Statement of the problem 67
8.2. Equivalence of states 69
8.3. Reduction of complete tables 71
8.3.1. Construction of the table of equivalent pairs 71
8.3.2. Grouping of equivalent pairs 75
8.3.3. Formation of the minimal flow table 76
8.3.4. Another example of the minimization of a table 78
8.4. Reduction of incomplete tables 79
8.4.1. Basic definitions 80
8.4.2. Determination of compatible pairs 82
8.4.3. Grouping compatible terms 82
8.4.4. Choice of the M.e. and construction of the
minimal flow table 85
8.4.5. Second example of reduction of an incomplete
flow table 88
8.4.6. Third example of reduction 92
8.5. Programming of flow table reduction on digital
computers 95
8.6. Reduction of a phase table 97
8.6.1. Equivalent states-pseudo-equivalent states 98
8.6.2. Row merging 99
8.7. Application of the method of compatible pairs to
asynchronous systems 103
8.7.1. Synthesis ofthe reduction by Huffman's
method 103
8.7.2. Example of the reduction of an asynchronous
sequential system 104
8.7.2. 1. Reduction by the method of compatible
pairs 104
CONTENTS VO
8.7.2.2. Reduction by the Huffman method 107
8.8. Conclusion 109
Bibliography 110
Exercises III
CHAPTER 9/ ASSIGNMENT OF THE INTERNAL STATES
(ASYNCHRONOUS SEQUENTIAL SYSTEMS)
9.1. Introduction 115
9.1.1. Generalizations 115
9.1.2. Asynchronous systems 116
9.1.3. Introductory example 116
9.1.4. Diverse methods and solutions 118
9.2. Connected sets 119
9.2.1. Connected sets and sequences 119
9.2.2. Application to the problem of asynchronous
assignment 120
9.3. Huffman numbers 120
9.4. The influence of essential connections on the
density of the assignment table 121
9.5. Reduction of the system's number of connections 124
9.5.1. Example 1 124
9.5.2. Example 2 127
9.5.3. Example 3 129
9.5.4. General principles of the method 130
9.5.5. Case oftables having 'don't cares' 132
9.6. Creation of supplementary unstable states 134
9.6.1. Example4 134
9.6.2. Example 5 139
9.6.3. Remarks about the method 140
9.7. Incomplete merging of the primitive phase table 140
9.8. General remarks about assignment 142
9.9. Assignments and universal circuits 143
9.9.1. Universal assignments 144
9.9.2. Circuit with 2so+ 1 relays (assignment by 2so + 1
variables) 145
9.9.3. Circuits with one relay per row (assignment by
one variable per row) 149
Bibliography 154
Exercises 155
VIII CONTENTS
CHAPTER 10/ ASSIGNMENT OF INTERNAL STATES
(SYNCHRONOUS SYSTEMS)
10. I. Introduction 160
10.2. Distinct assignments-valid assignments 161
10.3. Example of the different assignments of a same
table 162
lOA. Assignment from adjacency study 168
10.5. General concepts concerning partitions 170
10.5.1. Relations of order, sums, products 170
10.5.2. Use of p.s. p. for assignment 171
10.6. Search for the p.s.p. In
In
10.6.1. Study of the pairs
10.6.2. Maximal partitions 177
10.7. Properties connected with partitions p.s.p. 180
10.7.1. Systems having a 2 block p.s.p. 180
10.7.2. Systems having p.s.p. of more than 2 blocks 182
10.8. Use of the p.s.p. in assignment 183
10.9. Decomposition of sequential machines 187
10.9.1. Definitions 187
10.9.2. Decomposition theorem 189
10.9.3. Examples 192
10.9.4. Remarks concerning circuit realization 194
10.10. Partition pairs 195
10. 10. I. Definition 195
10. 10.2. Properties and particular partitions 196
10.10.3. Method for finding partition pairs 196
10. 1004. Properties connected with partition pairs 197
10.10.5. Conclusion 200
10. I I. Assignment of the uncompletely specified tables 200
10.12. Extension methods 202
10. 12. I. Examples of application extension of a given
flow table by adding equivalent states 203
10.12.2. Example2 205
10.12.3. Importantcomments 208
10. 13. Assignment of internal states by taking into account
the output 2 I 0
10. 14. Conclusion 213
Bibliography 2 I 5
Exercises 2 I 7
CONTENTS IX
CHAPTER II/EXAMPLES OF APPLICATIONS
11.1. Introduction 219
11.2. Applications on computers 220
11.2.1. Shift register-logical flip-flop 220
11.2.2. Algebraic binary adder-deducter 224
11.2.3. Transfer authorization from one register to
another 228
11.2.4. Reduction of a microprogram's length 232
11.2.4.1. Statement of the problem 232
11.2.4.2. Generalizations-inputs, outputs,
states 234
11.2.4.3. Application to the example 235
11.2.4.4. Points of interest 236
11.3. Sequentially controlled machines 238
11.3.1. Complex automaton 238
11.3.1.1. Preliminaries 238
11.3.1.2. Phase table 242
11.3.1.3. Conclusions 246
11.3.2. Sequential functioning in a cement's oven
control 246
11.3.2.1. Statement of the problem 246
11.3.2.2. Definitions ofthe different quantities 248
11.3.2.3. Study ofthe sequential functioning 249
11.3.2.4. The problem put into equations 252
11.3.2.5. Note relative to the section 255
11.3.2.6. Conclusions 257
11.3.3. Marshalling trains 257
11.4. Analysis of a system of electrical airplane
generation 262
11.4.1. Generalizations 262
11.4.2. Logical functioning of the system 264
11.4.2.1. Notation 264
11.4.2.2. Equations 264
11.4.2.3. Excitation matrix 265
11.4.2.4. Study of the transitions 267
11.4.3. Study of logical failures 268
11.4.3.1. Generalizations 268
11.4.3.2. Functioning without the ground
generator 269
x CONTENTS
I 1.4.3.3. Functioning with the ground
generator 27 I
11.4.4. Defaults in the protection boxes or phase
order 272
1 1.4.5. Conclusions 273
Appendix 274
I I .A I. Example of memory synthesis by the phase table
method 274
II.AI.1. Phase table 274
I LA 1.2. Reduction of the phase table 275
I LA 1.3. Output matrix-excitation matrix 275
11.A2. Sequential switching system on an analog computer
analac A I 10 277
I I.A2.1. Statement of the problem 277
II.A2.2. Realization ofthe scheme 280
Il.A2.3. Generalization 282
I I.A3. Automaton with securities 284
II.A3.1. Generalities 284
1I.A3.2. Equations-primitive phase table 284
II.A3.3. Excitation table-output table 285
I I.A3.4. Introduction of the first type of
security-use of delayed signals 286
II.A3.5. Introduction of the second type security
lock control 288
1I.A3.6. Conclusion 289
II.A.4. Simulation of sequential systems 290
II.A4.1. Logical simulators 290
II.A4.2. Computer simulation 290
Bibliography 295
Exercises 296
CHAPTER 12/LINEAR SEQUENTIAL SYSTEMS
12.1. Introduction 304
12.1.1. Example 304
12.2. Review of algebra 307
12.2.1. Matrices and determinants 307
12.2.2. Polynomial forms-Galois theorem 309
12.2.3. Eigen values-Eigen vectors of a matrix 314
12.3. Transition of linear sequential systems 316
CONTENTS XI
12.3.1. OInput-non-singularmatrixA 316
12.3.2. Any input-non-singular matrix A 318
12.3.3. MatrixA+1 320
12.4. General configurations of linear machines 324
12.5. Discrete Laplace transform 330
12.5.1. Introduction 330
12.5.2. p-Transforms 331
12.5.2.1. Introduction: example ofa non-
periodical sequence-representation 332
12.5.2.2. Example ofa periodical sequence 333
12.5.2.3. Generalization-inverse transform 333
12.5.2.4. Initial conditions 334
12.6. Study of linear systems by the discrete Laplace
transform 336
12.6.1. Transfer function 337
12.6.2. Elementary linear operators 338
12.6.2. 1. Transfer function of the shift 338
12.6.2.2. Representation oftransfer-syste-
matic 339
12.6.2.3. Composition of systems 340
12.6.3. Variables of state 342
12.6.3.1. Passage from the transfer function to
the state representation 342
12.6.3.2. Passage from the state representation
to the transfer representation 345
12.6.3.3. Comparison of the representations 347
12.7. Application 347
12.7.1. Determination of sequences from a pulse
generator 347
12.7.2. Eigen functions and their applications 348
12.7.2.1. Introduction 348
12.7.2.2. Functions shifted to the left 353
12.7.2.3. Functions shifted to the right 357
12.7.2.4. Shifting to the right with conditioned
memories 358
12.7.3. Transferences 361
12.8. Conclusion 366
Bibliography 368
Exercises 369