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The Complexity of Boolean Functions The Complexity of Boolean Functions Ingo Wegener Johann Wolfgang Goethe-Universit¨at WARNING: This version of the book is for your personal use only. The material is copyrighted and may not be redistributed. Copyright c 1987 by John Wiley & Sons Ltd, and B. G. Teubner, Stuttgart. All rights reserved. No part of this book may be reproduced by any means, or transmitted, or translated into a machine language without the written permission of the publisher. Library of Congress Cataloguing in Publication Data: Wegener, Ingo The complexity of boolean functions. (Wiley-Teubner series in computer science) Bibliography: p. Includes index. 1. Algebra, Boolean. 2. Computational complexity. I. Title. II. Series. AQ10.3.W44 1987 511.3’24 87-10388 ISBN 0 471 91555 6 (Wiley) British Library Cataloguing in Publication Data: Wegener, Ingo The complexity of Boolean functions.—(Wiley-Teubner series in computer science). 1. Electronic data processing—Mathematics 2. Algebra, Boolean I. Title. II. Teubner, B. G. 004.01’511324 QA76.9.M3 ISBN 0 471 91555 6 CIP-Kurztitelaufnahme der Deutschen Bibliothek Wegener, Ingo The complexity of Boolean functions/Ingo Wegener.—Stuttgart: Teubner; Chich- ester; New York; Brisbane; Toronto; Singapore: Wiley, 1987 (Wiley-Teubner series in computer science) ISBN 3 519 02107 2 (Teubner) ISBN 0 471 91555 6 (Wiley) Printed and bound in Great Britain On this version of the “Blue Book” This version of “The Complexity of Boolean Functions,” for some people simply the “Blue Book” due to the color of the cover of the orig- inal from 1987, is not a print-out of the original sources. It is rather a “facsimile” of the original monograph typeset in LATEX. The source files of the Blue Book which still exist (in 1999) have been written for an old version of troff and virtually cannot be printed out anymore. This is because the (strange) standard font used for the text as well as the special fonts for math symbols seem to be nowhere to find today. Even if one could find a solution for the special symbols, the available text fonts yield a considerably different page layout which seems to be undesirable. Things are further complicated by the fact that the source files for the figures have been lost and would have to be redone with pic. Hence, it has been decided to translate the whole sources to LATEX in order to be able to fix the above problems more easily. Of course, the result can still only be an approximation to the original. The fonts are those of the CM series of LATEX and have different parameters than the original ones. For the spacing of equations, the standard mechanisms of LATEX have been used, which are quite different from those of troff. Hence, it is nearly unavoidable that page breaks occur at different places than in the original book. Nevertheless, it has been made sure that all numbered items (theorems, equations) can be found on the same pages as in the original. You are encouraged to report typos and other errors to Ingo Wegener by e-mail: [email protected] Preface When Goethe had fundamentally rewritten his IPHIGENIE AUF TAURIS eight years after its first publication, he stated (with resig- nation, or perhaps as an excuse or just an explanation) that, ˝Such a work is never actually finished: one has to declare it finished when one has done all that time and circumstances will allow.˝ This is also my feeling after working on a book in a field of science which is so much in flux as the complexity of Boolean functions. On the one hand it is time to set down in a monograph the multiplicity of important new results; on the other hand new results are constantly being added. I have tried to describe the latest state of research concerning re- sults and methods. Apart from the classical circuit model and the parameters of complexity, circuit size and depth, providing the basis for sequential and for parallel computations, numerous other models have been analysed, among them monotone circuits, Boolean formu- las, synchronous circuits, probabilistic circuits, programmable (univer- sal) circuits, bounded depth circuits, parallel random access machines and branching programs. Relationships between various parameters of complexity and various models are studied, and also the relationships to the theory of complexity and uniform computation models. The book may be used as the basis for lectures and, due to the inclusion of a multitude of new findings, also for seminar purposes. Numerous exercises provide the opportunity of practising the acquired methods. The book is essentially complete in itself, requiring only basic knowledge of computer science and mathematics. This book I feel should not just be read with interest but should encourage the reader to do further research. I do hope, therefore, to have written a book in accordance with Voltaire’s statement, ˝The most useful books are those that make the reader want to add to v vi them.˝ I should like to express my thanks to Annemarie Fellmann, who set up the manuscript, to Linda Stapleton for the careful reading of the text, and to Christa, whose complexity (in its extended definition, as the sum of all features and qualities) far exceeds the complexity of all Boolean functions. Frankfurt a.M./Bielefeld, November 1986 Ingo Wegener Contents 1. Introduction to the theory of Boolean functions and circuits 1 1.1 Introduction 1 1.2 Boolean functions, laws of computation, normal forms 3 1.3 Circuits and complexity measures 6 1.4 Circuits with bounded fan-out 10 1.5 Discussion 15 Exercises 19 2. The minimization of Boolean functions 22 2.1 Basic definitions 22 2.2 The computation of all prime implicants and reductions of the table of prime implicants 25 2.3 The minimization method of Karnaugh 29 2.4 The minimization of monotone functions 31 2.5 The complexity of minimizing 33 2.6 Discussion 35 Exercises 36 3. The design of efficient circuits for some fundamental functions 39 3.1 Addition and subtraction 39 3.2 Multiplication 51 3.3 Division 67 3.4 Symmetric functions 74 3.5 Storage access 76 3.6 Matrix product 78 3.7 Determinant 81 Exercises 83 vii viii 4. Asymptotic results and universal circuits 87 4.1 The Shannon effect 87 4.2 Circuits over complete bases 88 4.3 Formulas over complete bases 93 4.4 The depth over complete bases 96 4.5 Monotone functions 98 4.6 The weak Shannon effect 1 06 4.7 Boolean sums and quadratic functions 107 4.8 Universal circuits 110 Exercises 117 5. Lower bounds on circuit complexity 119 5.1 Discussion on methods 119 5.2 2 n - bounds by the elimination method 122 5.3 Lower bounds for some particular bases 125 5.4 2.5 n - bounds for symmetric functions 127 5.5 A 3n - bound 133 5.6 Complexity theory and lower bounds on circuit complexity 138 Exercises 142 6. Monotone circuits 145 6.1 Introduction 145 6.2 Design of circuits for sorting and threshold functions 148 6.3 Lower bounds for threshold functions 154 6.4 Lower bounds for sorting and merging 158 6.5 Replacement rules 160 6.6 Boolean sums 163 6.7 Boolean convolution 168 6.8 Boolean matrix product 170 6.9 A generalized Boolean matrix product 173 6.10 Razborov’s method 180 6.11 An exponential lower bound for clique functions 184 6.12 Other applications of Razborov’s method 192 ix 6.13 Negation is powerless for slice functions 195 6.14 Hard slices of NP-complete functions 203 6.15 Set circuits - a new model for proving lower bounds 207 Exercises 214 7. Relations between circuit size, formula size and depth 218 7.1 Formula size vs. depth 218 7.2 Circuit size vs. formula size and depth 221 7.3 Joint minimization of depth and circuit size, trade-offs 225 7.4 A trade-off result 229 Exercises 233 8. Formula size 235 8.1 Threshold - 2 235 8.2 Design of efficient formulas for threshold - k 239 8.3 Efficient formulas for all threshold functions 243 8.4 The depth of symmetric functions 247 8.5 The Hodes and Specker method 249 8.6 The Fischer, Meyer and Paterson method 251 8.7 The Nechiporuk method 253 8.8 The Krapchenko method 258 Exercises 263 9. Circuits and other non uniform computation methods vs. Turing machines and other uniform computation models 267 9.1 Introduction 267 9.2 The simulation of Turing machines by circuits: time and size 271 9.3 The simulation of Turing machines by circuits: space and depth 277 9.4 The simulation of circuits by Turing machines with oracles 279 9.5 A characterization of languages with polynomial circuits 282 9.6 Circuits and probabilistic Turing machines 285 x 9.7 May NP-complete problems have polynomial circuits ? 288 9.8 Uniform circuits 292 Exercises 294 10. Hierarchies, mass production and reductions 296 10.1 Hierarchies 296 10.2 Mass production 301 10.3 Reductions 306 Exercises 318 11. Bounded-depth circuits 320 11.1 Introduction 320 11.2 The design of bounded-depth circuits 321 11.3 An exponential lower bound for the parity function 325 11.4 The complexity of symmetric functions 332 11.5 Hierarchy results 337 Exercises 338 12. Synchronous, planar and probabilistic circuits 340 12.1 Synchronous circuits 340 12.2 Planar and VLSI - circuits 344 12.3 Probabilistic circuits 352 Exercises 359 13. PRAMs and WRAMs: Parallel random access machines 361 13.1 Introduction 361 13.2 Upper bounds by simulations 363 13.3 Lower bounds by simulations 368 13.4 The complexity of PRAMs 373 13.5 The complexity of PRAMs and WRAMs with small communication width 380 13.6 The complexity of WRAMs with polynomial resources 387 13.7 Properties of complexity measures for PRAMs and WRAMs 396 Exercises 411 xi 14.
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