Algorithmics the sprit of computing 3rd david harel

Harel Cvr.QXD 16/01/2006 09:40 PM Page Algorithmics Algorithmics From a review of the first edition: ‘This book is a veritable tour de force Harel writes with uncommon verve, clarity and imagination ‘Through the use of tantalizing questions and aptly chosen and often amusing examples, the author transmits to the reader the excitement and intellectual satisfaction of computer science research Without the use of formal mathematics and without any sacrifice of intellectual integrity, he conveys to the general reader the profound principles on which computer science is founded and which hitherto were only accessible in abstruse and esoteric textbooks and papers ‘This is scientific writing at its best.’ Dr Stan Scott, Queen's University Belfast The Times Higher Education Supplement THIRD EDITION This book tells the story of the concepts, ideas, methods and results fundamental to computer science, in a form independent of the details of specific computers, languages and formalisms It concerns the true ‘spirit’ of computers; with the ‘recipes’ that make them tick – their algorithms New to this edition ■ ■ ■ ■ www.pearson-books.com David Harel an imprint of with David Harel is Professor and Dean of the Faculty of Mathematics and Computer Science at the Weizmann Institute of Science He is renowned for outstanding research in many areas of the field, and has recently been awarded the Israel Prize in Computer Science Yishai Feldman is on the faculty of the Efi Arazi School of Computer Science at the Interdisciplinary Centre, Herzliya He specializes in the use of artificial–intelligence techniques in software engineering and their real–world applications Yishai Feldman Chapters on software engineering and on reactive systems Thoroughly revised chapter on programming languages New material on quantum and molecular computing Whole text thoroughly updated to include new material on many topics, including abstract data types, the object–oriented paradigm, primality testing, and system verification and validation Algorithmics THIRD EDITION The Spirit of Computing The Spirit of Computing The Spirit of Computing THIRD EDITION David Harel with Yishai Feldman P1: GIG PE002-FM PE002-Harel PE002-Harel-FM-v1.cls March 19, 2004 19:35 Algorithmics The Spirit of Computing i P1: GIG PE002-FM PE002-Harel PE002-Harel-FM-v1.cls March 19, 2004 19:35 We work with leading authors to develop the strongest educational materials in computing, bringing cutting-edge thinking and best learning practice to a global market Under a range of well-known imprints, including Addison-Wesley, we craft high quality print and electronic publications which help readers to understand and apply their content, whether studying or at work To find out more about the complete range of our publishing, please visit us on the World Wide Web at: www.pearsoned.co.uk ii P1: GIG PE002-FM PE002-Harel PE002-Harel-FM-v1.cls March 19, 2004 19:35 Algorithmics The Spirit of Computing THIRD EDITION David Harel The Weizmann Institute with Yishai Feldman The Interdisciplinary Center, Herzliya iii P1: GIG PE002-FM PE002-Harel PE002-Harel-FM-v1.cls March 19, 2004 19:35 Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk First published 1987 Second edition 1992 Third edition published 2004 C C Addison-Wesley Publishers Limited 1987, 1992 Pearson Education Limited 2004 The rights of David Harel and Yishai Feldman to be identified as authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP The programs in this book have been included for their instructional value They have been tested with care but are not guaranteed for any particular purpose The publisher does not offer any warranties or representations nor does it accept any liabilities with respect to the programs All trademarks used herein are the property of their respective owners The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owner ISBN 321 11784 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Harel, David, 1950– Algorithmics : the spirit of computing / David Harel, with Yishai Feldman.–3rd ed p cm Includes bibliographical references and index ISBN 0-321-11784-0 Computer algorithms I Feldman, Yishai A., 1958– II Title QA76.9.A43H37 005.1–dc22 2004 2004041063 10 08 07 06 05 04 Typeset by 59 in 10/12pt Times Printed and bound in Great Britain by Biddles Ltd., Guildford and King’s Lynn iv P1: GIG PE002-FM PE002-Harel PE002-Harel-FM-v1.cls March 19, 2004 19:35 For my dear mother, Joyce Fisch And in memory of my dear father, Harold Fisch Beloved parents and gifted teachers both (D.H.) To the memory of my dear parents, Hadassa and Moshe Feldman For their unfailing love and support (Y.F.) v P1: GIG PE002-FM PE002-Harel PE002-Harel-FM-v1.cls March 19, 2004 19:35 Tell me, I pray thee, in what thy great strength lies JUDGES 16: Behold, this I have found counting one thing to another, to find out the sum ECCLESIASTES 7: 27 vi P1: GIG PE002-FM PE002-Harel PE002-Harel-FM-v1.cls March 19, 2004 19:35 Contents Declare the things that are to come hereafter ISAIAH 41: 23 Preface xi Acknowledgments xvii Part I Preliminaries ■ 1 Introduction and Historical Review or, What’s It All About? ■ Algorithms and Data 19 or, Getting It Done ■ Programming Languages and Paradigms 49 or, Getting It Done by Computer Part II Methods and Analysis ■ 79 Algorithmic Methods 81 or, Getting It Done Methodically ■ The Correctness of Algorithms or, Getting It Done Right vii 99 P1: GIG PE002-FM PE002-Harel viii PE002-Harel-FM-v1.cls March 19, 2004 19:35 Contents ■ The Efficiency of Algorithms 129 or, Getting It Done Cheaply Part III Limitations and Robustness ■ Inefficiency and Intractability 157 159 or, You Can’t Always Get It Done Cheaply ■ Noncomputability and Undecidability 191 or, Sometimes You Can’t Get It Done At All! ■ Algorithmic Universality and Its Robustness 219 or, The Simplest Machines That Get It Done Part IV Relaxing the Rules ■ 10 Parallelism, Concurrency, and Alternative Models 255 257 or, Getting Lots of Stuff Done at Once ■ 11 Probabilistic Algorithms 297 or, Getting It Done by Tossing Coins ■ 12 Cryptography and Reliable Interaction 317 or, Getting It Done in Secret Part V The Bigger Picture ■ 13 Software Engineering 335 337 or, Getting It Done When It’s Large ■ 14 Reactive Systems 357 or, Getting It to Behave Properly Over Time ■ 15 Algorithmics and Intelligence or, Are They Better at It Than Us? 379 P1: GIG PE002-FM PE002-Harel PE002-Harel-FM-v1.cls March 19, 2004 19:35 Contents ix Postscript 401 Selected Solutions 403 Bibliographic Notes 433 Index 495 P1: GDZ/FFX PE002-Index P2: GDZ/FFX PE002-Harel 500 QC: GDZ/FFX PE002-Harel-v1.cls T1: GDZ March 17, 2004 21:3 Index E Eckert, Presper J., economical systems, 358 economics, 167 effective computability, 228, see also computability; decidability effective reductions, 201–202, 210, 235–236, 241–243 efficiency of algorithms, 123, 129–190, 278, 401, see also complexity theory; lower bounds; upper bounds research on, 151–153, 354 EIFFEL, 74, 120 elections in Quebec, 99 electrical engineering, elementary instructions, 10–12, 33, 270 as related to time complexity, 129, 138–139, 146 sequence of, 19 simplifying, 223 unspecified, 182, 211 ELIZA, 65, 383–384 encapsulation, 359, 360 encoding inputs as numbers, 233–234 as strings, see linearizing inputs encription, see also cryptography encryption, 317–324 encyclopedia lookup, 307, see also pattern matching equivalence of algorithmic problems polynomial, see polynomial equivalence of models of computation, 229, 238–240, 248 of syntactic definitions, see syntactic equivalence equivalence of algorithmic problems computational (algorithmic), 210 equivalence problem, 118, 122, 248 error messages, 50, 64 errors, 99–104, 132, 281, 286, 337, see also correctness of algorithms; verification algorithmics, see errors, logical design, 351 in hardware, 103, 302, 354 in software, 100, 338–339, 354 language, 101 logical, 101–103, see also bugs; elections in Quebec; Mariner space probe; salary summation with low probability, 301–309 semantic, 102 stage of discovery, 345, 347, 350 syntax, see errors, language escape clause, 51, 115 Esterel, 359 ethics, code of, 353 Euclid, 6, 271 Euclidean algorithm, 6, 307 Euler, Leonard, 170 Eulerian paths, 169–170, 196 polynomial-time algorithm for, 169–170 evaluating situations, 389–391 event-driven systems, see reactive systems events, 247, 357, 358, 360, 362–365, 369, 370, 375–376 executable models, see model execution executable specifications, see model execution execution of algorithm, see run of algorithm execution sequence, 112, see also run of algorithm exhaustive search, 82 expanding parallelism, 260, see also parallelism expert systems, 393–394 exponential blow-up, 360, 362 exponential function, 160, 162–165 exponential space (EXPSPACE), 184–185 exponential time (EXPTIME), 162–165, 184–185, 267–270, 302, see also intractability nondeterministic, see NEXPTIME robustness of, 240 EXPSPACE, see exponential-space EXPTIME, see exponential time extended Riemann hypothesis, 303 external elements finding, see min&max finding eye, 222–223 F factorial function, 161, 162, 387 factoring numbers, 130, 290–291, 307, 311, 322–323 factors, 302 fairness, 281–282, 286 family tree, 39 FIFO (first-in-first-out), see queue Fifth-Generation Project, 70 file, 36, 62, 63, 220 file structure (in PL/I), 62 fingerprinting, 308–309 finite automata, 243–247, 359, see also transition diagram drawbacks of, 246–247 power of, 245–247 P1: GDZ/FFX PE002-Index P2: GDZ/FFX PE002-Harel QC: GDZ/FFX PE002-Harel-v1.cls T1: GDZ March 17, 2004 21:3 Index research on, 247–249 with stack, see pushdown automata finite certificates, 207–208 finite-state automaton, see finite automata finite-state protocol, 281 first-order arithmetic, 210 fixed parallelism, 260, see also parallelism fixed-connection networks, see networks fixed-width problems, 212 flight reservation system, 44, 274 Floid’s method, see intermediate assertion method flowcharts, 24–27, 100, 110, 111, 134 for statement, 36, 50–53, 55–56, 131 formal language theory, 248 formal language thory, 248 FORTRAN, 60–61 four-color problem, 123–125 four-color theorem, 123–125, 313 function bad, 163, 164 bounded from above, 163 exponential, 160, 162–165 growth rates of, 162–164 linear, 132–133 logarithmic, 135–136 polynomial, 163 super-polynomial, 163–164 function describing program, 76 functional algorithms, 92–94 functional language, 92 functional programming, 59, 65–68, 92 functional queue, 92–94 functional stack, 92–93 fundamental levels of computability, 212–213, see also sphere of algorithmic problems robustness of, 213 G game playing, 385–388 game theory, 167 game tree, 40, 180, 386–391 games, see backgammon; checkers; chess; puzzles; roadblock; tic-tac-toe garbage collector, 67 gates, 264 gcd, see greatest common divisor gearbox, 222, 223 generating primes, 302–304, 322, 323, see also primality testing genetic engineering, 385 genome project, 45 501 geometry analytic, 85 computational, 83, 149 high school, 394 global constraints, 273 G¨odel, Kurt, Goldstine, Herman, goto statement, 23–24, 233, 286 grade normalization, 130 graph, 88 coloring, 171–172, see also coloring directed, 89 directed acyclic, 89 problems, see coloring; Eulerian paths; Hamiltonian paths; minimal spanning tree; shortest paths; traveling salesman problem graph theory, 167 graphical languages, see visual formalisms graphical user interface (GUI), 365, 373–374 graphics, see computer graphics greatest common divisor (gcd), 6, 271, 286, 307 greedy algorithms, 87–89 greedy method, see greedy algorithms guarding conditions, 369, 370 H half-grid, 213, 241–243 halting problem, 199–201, 210, 214, 236, see also termination certificates for, 207–208 proof of undecidability, 202–207 reduction to tiling half-grids, 241–244 reduction to verification, 202, 210 Hamiltonian paths, 169, 172–175, 196, 292 reduction to traveling salesman, 174–175 hardware, 4, 103, 124, 226, 229, 258, 262, 302, 324 HASKELL, 67 head, 223–226, 230, 242 of White Rabbit, 373 head, 110–113 heap, 91–92, 143 vector representation, 92 heapsort, 143, 148 heuristic search, 387–391, 393 heuristics, 136, 179, 186, 380, 387–391, 400–401 hidden assumptions, 337, 344, 347, see also software, errors in spaceflight, 338–339 hierarchical model, 44, 392 P1: GDZ/FFX PE002-Index P2: GDZ/FFX PE002-Harel 502 QC: GDZ/FFX PE002-Harel-v1.cls T1: GDZ March 17, 2004 21:3 Index hierarchy, see trees in databases, see hierarchical model in diagrams, 357 high undecidability, 211–214 highly undecidable problems, 211–214, 240, see also propositional dynamic logic with recursion; recurring dominoes history of computer science, 6–8 Hollerith, Herman, hotels corridors in, as metaphor for vectors, 35, 37 as metaphor for arrays, 36–38 rooms in, as metaphor for variables, 34, 37 taking showers in, 273–274, see also mutual exclusion house building, 257, 287 tiling, 193 I IBM (International Business Machines), 8, 231 IEEE Computer Society, 353 code of ethics, 353 if statement, 20, 222 imperative programming, 58–59 programming languages, 60–64 index, 35–36, 122, 495–513 induction, 116 inductive hypothesis, 116 inductive/coinductive level of undecidability, see high undecidability inefficiency, see intractability infinite computation, 105–106, 279, see also halting problem infinite loop, see infinite computation; loops information hiding, 73, 342 inherent limitations of algorithms, see limitations inherent sequentiality, 259 inheritance, 71 input/output paradigm, 192, 273, 280, 311 inputs, 4, see also input/output framework borderline cases of, 103, 279 distribution of, 136, 143–144 finite number of, 192 infinite number of, 13 legality of, 14 length of, 129, 133, see also size of algorithmic problem linearizing, see linearizing inputs probabilistic distribution of, 312 signals and events as, 247 specification of, 14–15, 106 telltale, 103 undesirable, 15 integer grid, 195, 211, 241–243, see also tiling integer part, 135 intelligence artificial, see artificial intelligence human, 379–402 interactive polynomial time (IP, MIP), 327–328 interactive proofs, 327–332 interactive systems, 76, 104, 121 interactive verification, see verification interface, see modules, interface interference freedom, 280 intermediate assertion method, 107–115, 279 variant for recursion, 115–117 intermediate assertions, 107–115, see also invariants International Business Machines Corporation (IBM), 8, 231 internet, 358 interpolation search, 83, 136 interpreter, 57, 101, 229, 236 interrupts, 361 intractability, 159–191, 269, see also intractable problems intractable problems, 165, 212–213, 267–270, 302, 401, see also exponential time; nonelementary time; provably intractable problems; super-polynomial time invariant assertion method, see intermediate assertion method invariants, 107–109, 111–113 finding, 114–115 iterations, 20–21, 182, 211, see also looping constructs; loops J Jacobi symbol, 306 Jacquard, Joseph M, Jacquard’s loom, JAVA, 72–75, 120, 342 jigsaw puzzles, 167–168, 213 Jim’s phone, 399–400 joint time/space complexity, 153 K keys, 317–323 Kleene, Stephen C., 7, 229, 230 knowledge, 45, 380, 383, 384, 386, 392–396, 398, 402 items, 45 representating, see knowledge representation P1: GDZ/FFX PE002-Index P2: GDZ/FFX PE002-Harel QC: GDZ/FFX PE002-Harel-v1.cls T1: GDZ March 17, 2004 21:3 Index knowledge base, 45, 248, 392 knowledge engineer, 393 knowledge representation, 392–396, 398, 402 L labyrinth traversal, 39 lambda calculus, 67, 229 language errors, see errors language manual, 53 Las Vegas algorithms, 309, 311 Las Vegas algrithms, see also probabilistic algorithms lazy railroad contractor, see minimal spanning tree leaf, 39 learning, 384, 386, 394–395 legal inputs, 14, 192 levels of a tree, 39, 40, 220–221, 386, 387 levels of computability, see fundamental levels of computability levels of detail, 10–11 LIFO (last-in-first-out), see stack limitations of algorithms, 16 of algorithms, 6, 7, 191, see also Church/Turing thesis; intractability; noncomputability of computers, 6, 9, 50, 191, 269 our own, 6, 191–192 of three-dimensional space, 269 line segment, 149 linear planning, 184–185 linear programming, see linear planning linear search, 131–133, 159 linear size, 263, 265 linear time, 132–133, 140, 148–149, 152, 159, 167, 240, 261 linearizing inputs, 200, 219–221, 237, 248 linguistics, 6, 248 linked list, 72, 74 LISP, 65–67, 220, 347–348, 392 list, see linked list; vectors in LISP, 65–67 liveness, 278–280, 370–372 loan taking, 350 locations in an algorithm, 222 locked box metaphor, 318–319, 324–325 lockout, see starvation logarithm, 135, 138, 178 base of, 135, 138 logarithmic time, 135–138, 185, 259–264 503 logic, 6, 122, 167, 214, see also dynamic logics; first-order arithmetic; modal logic; propositional dynamic logic; predicate calculus; Presburger arithmetic; propositional calculus; recursion theory; temporal logic; tense logic; WS1S logic programming, 59, 68–70 logical deduction, see deduction logical errors, 101–103, see also errors logics of programs, 122–123, see also dynamic logics; propositional dynamic logic; temporal logic LOGSPACE, 185 LOGTIME, see logarithmic time loop invariants, 107–109, 111–113, 120 loop-free segment, 112, 114, 122, 136 looping constructs, 20–23, 229, see also bounded iteration; conditional iteration; for statement; while statement indexed versions of, 36 power of, relative to recursion, 33 relationship to vectors, 36 as tools for describing long processes, 20–21 loops, see also looping constructs going to midst of, 24 infinite, 105–106 inner, 21–23 nested, 21–23, 233 time analysis of, 139–140, 151 oscillating, 105 outer, 21–23 lower bounds, 144–146, 401, see also efficiency of algorithms; intractability exponential-time, 166 research on, 152 for searching, 146–148 for sorting, 148 using Turing machines to prove, 241–243 Lustre, 359 M machine code, 56, 64 machine language, 50, 54–56, 139 machine level, 55 magic, 32, 42, 172–173, 230–231, 241, 267, 269, 272, 310 malicious scheduling, see scheduler Mariner space probe, 99 Markov, Andre˘ı A., Mars Climate Orbiter, 338 Mars Polar Lander, 338 matching problems, 170 P1: GDZ/FFX PE002-Index P2: GDZ/FFX PE002-Harel 504 QC: GDZ/FFX PE002-Harel-v1.cls T1: GDZ March 17, 2004 21:3 Index mathematical induction, see induction mathematical logic, see logic mathematical truth absolute, 125, 314 mathematics, 6, see also algorithmics; automata theory; combinatorics; formal language theory; game theory; geometry; graph theory; logic; number theory; probability theory; recursion theory; topology matrix, 36, 266, see also arrays matrix multiplication by vector, 266 Mauchly, John W., maximal polygonal distance, 83–85, 140, 148 medicine, 390, 393 memory in algorithms, 34, 129–130, 153, 184, see also shared memory; space in computers, 55, 58, 59 mergesort, 86–87, 134, 142–144, 148, 260–261 parallel version of, 260–262, 279 merging lists, 86–87, 261 mermaids, 104, 107 mesh-connected array, 267, 269 method postconditions, see design by contract method preconditions, see design by contract methods (in object-oriented programming), 72, 342–343, see also design by contract methods, algorithmic, 81–98, see also divide and conquer; dynamic planning; greedy method; searches research on, 95 min&max finding, 85–86, 134, 140–142, 148, 160 efficient iterative solution, 141–142 recursive solution, 85–86, 140–141 minimal colorability, 179–180, see also coloring minimal paths, see shortest paths minimal spanning tree, 87–89, 148–149, 152, 168 minimax method, 390–392 MIRANDA, 67 modal logic, 392 model checking, 121, 123 model execution, 362–365 modularity, 74, 341–343, see also modules modules, 342–344 clients, 342–343 integration, 344 interface, 342 suppliers, 342 molecular biology, 12 molecular computing, 257, 292–293 money counting, 26–28, 101–102, 119, 140, 233 monkey puzzles, 160–162, 165–167, 176, 193, 241, 269, 301–302 certificates for, 172 trivial, 162 Monte Carlo algorithms, 309, 311, 313, see also probabilistic algorithms Mother Nature, 358 mousseline au chocolat, 10, 14, 16, 19–20, 29, 33 multiplication of matrix by vector, 266, see also weighted average computation of numbers, 10–11, 228, 233, 383 mutual exclusion, 275–278, 280, 281, 285–286, see also dining philosophers; shower taking mutual inverse, 319, 321 N nanosecond, 139, 161, 163, 164 natural languages ambiguity in, 397–400 inability of computers to understand, 50, 397–400 NC, see Nick’s class neighboring processors, 263 nest building, 9, 381 nested loops, see loops networks, 263–270, see also beehive; mesh-connected array; pipeline arrangement boolean, see also boolean circuits sorting, see sorting networks systolic, see systolic arrays neurobiology, 12, 192, 379, 386 newspaper distribution, 16 NEXPTIME, see nondeterministic exponential time Nick’s class (NC), 271–272, 287 node, 39, see also vertex terminal, 386, 389, 390 noncomputability, 191–218, 311, 401, see also undecidability nondestructive algorithms, 92–94 nondeterminism, 173, see also nondeterministic polynomial time nondeterministic exponential time (NEXPTIME), 328, 333 nondeterministic polynomial time (NP, NPTIME), 173, 177–178, 185–186, 207, 240–241, 267–269, 272, 303, 310 defined via Turing machines, 240–241 robustness of, 240 P1: GDZ/FFX PE002-Index P2: GDZ/FFX PE002-Harel QC: GDZ/FFX PE002-Harel-v1.cls T1: GDZ March 17, 2004 21:3 Index nondeterministic polynomial time(NP, NPTIME), see also NP-complete problems nonelementary space, 184 nonelementary time, 183, see also WS1S noted number, 34 noted numbers, 12–13, 19–21, 26, 33, see also counters; variables NP, see nondeterministic polynomial time; NP-complete problems NP problems, 177–178, 292, 333 NP-complete problems, 166–180, 201, 210, 212, 258, 269, 271, 301, 302, 312, see also arrangement problems; coloring; matching problems; satisfiability problems; scheduling problems NP-completeness, 173–180, see also NP-complete problems proving, 174–177 NPC, 167, see also NP-complete problems; NP-completeness NPTIME, 185–186, see also nondeterministic polynomial time number theory, 200, 302 numbers, 33 binary, 33, see also bits composite, 302, see also composite numbers; factoring numbers decimal, see decimal numbers fractional, 219 hexadecimal, 234 integer, 219 length of, 178 non-unary, 238 prime, see prime numbers pseudo-random, 313 random, see random numbers numerical problems, 233 O Object Management Group, 376 object model diagrams, 366 object-oriented databases, 45 object-oriented development, 349–350 object-oriented programming, 34, 59–60, 64, 70–75, 120, 349–350 object-oriented programming languages, 70, 72–75 objects, 59–60, 71–74, 119–120, 349, 366–375 odd-even sorting network, 264–265 off-line algorithms, see omniscient algorithms offspring, 39, 40, 386, 390 505 omniscient algorithms, 94, 95, see also on-line algorithms on-line algorithms, 94–95 one-way trapdoor function, 319, 321, 324 ongoing behavior, 258, 272–287, 311, see also behavioral description operating system, 64, 284, 313, 337 Unix, 64 operational semantics, 76 operations research, 167 optimal algorithm, 144, 262–263, see also closed problems; lower bounds; upper bounds; almost optimal solutions optimizing compiler, 130, 199 oracle, 202, 210 order of, on the, 132, see also big-O notation order-of-magnitude improvements, 132–133, see also efficiency ordered list search, see telephone book search orthogonal state components, 361–362 outdegree, 40, 386–387 output, 4, 198, 227, see also input/output paradigm P P, see polynomial time P vs NC question, 271–272, 401, see also Nick’s class P vs NP question, 177–178, 185–186, 240–241, 271–272, 401, see also NP-completeness beliefs about, 212, 269, 272 P-completeness, 272 pair programming, 351 palindromes, 224–226, 247 parallel algorithms, 258–272, 292, see also parallelism sequentializing, 262–267 parallel computation thesis, 270–271, see also Church/Turing thesis, refined parallel version of parallel computer, 58, 258, 263, 269, 270, 287 parallel mergesort, 260–261, 279 parallel PTIME, 270–272 relationship to PSPACE, 270–271 robustness of, 270 parallelism, 257–272, 311, 380 expanding, 260 fixed, 260 quantum, 289–291 research on, 286–287 parameters, 26, 101, see also subroutines parity, 245–246 P1: GDZ/FFX PE002-Index P2: GDZ/FFX PE002-Harel 506 QC: GDZ/FFX PE002-Harel-v1.cls T1: GDZ March 17, 2004 21:3 Index partial correctness, 106, 108–117, 279, see also verification partial information, 326 partial solution, 162, 167, 168, 172–173 partially decidable problems, 209 particles in the universe, number of, 152, 163, 183, 184, 339, 387 PASCAL, 63 path-finding problems, 168–170 paths, 39 between checkpoints, 112–113 minimal, 89–91, see also shortest paths pattern matching probabilistic, 307–309 PCP (Probabilistically Checkable Proofs), 332–333 PDL, see propositional dynamic logic perpetuity, 258, see also ongoing behavior Petri nets, 359 philosophy, 6, 191–192, 385, see also dining philosophers physics, 6, 12 quantum, see quantum physics Pi, approximate value for, 53 pigeon hole principle, 246–247 pipeline arrangement, 267 PL/I, 61–64 planning, 394–395 pointer, 26–28, 33 poker playing, 324–326 polylogarithmic time, 271 polynomial equivalence, 173–176, 210, 239 polynomial function, 163 polynomial reducability, see also polynomial equivalence polynomial reducibility, 174–175 polynomial space (PSPACE), 185–186 defined via Turing machines, 243–244 relationship to parallel time, 270–271, 287 robustness of, 240 polynomial time (P, PTIME), 164, 177–178, 271–272, see also tractable problems certificates, 172–173 completeness, 272 defined via Turing machines, 240, 243 interactive (IP, MIP), 327–328 low order, 319 nondeterministic, see NP parallel, 267–272, see also parallel-PTIME random, 310, see also random-polynomial time reductions, 174–177, 210 relationship to NC, 271–272 relationship to RP, 310 relevance to breaking cryptographic systems, 321–322 robustness of, 238–240 significance of, 240 popping, 38, 39, 248 Post, Emil L., 7, 229, 230 predicate (in PROLOG), 68–70 predicate calculus, 392 Presburger arithmetic, 183, 210 primality testing, 303–307, 309, 311, 322, 389 polynomial-time algorithm for, 303, see also AKS algorithm prime numbers, 130, 302–307, 388, see also factoring numbers; generating primes; primality testing probabilism, 186, 298, 301, 311, 401, see also probabilistic algorithms probabilistic algorithms, 297–334, 388–389, see also Dining philosophers; pattern matching; primality testing Las Vegas, 309, 311 Monte Carlo, 309, 311, 313 research on, 311–313 use in cryptography, 322–324 verification of, 311 probability (zero), 298–300 (one), 298–301, 311 negligible, 297, 298, 302, 304–307, 323, 327, 331 probability theory, 297 problems, see algorithmic problem process as prescribed by algorithms, 4, 12–13, 19–21, 24 processors, 19, see also Runaround multiple, 257–287 product complexity, 262–263 production rules, 393 production system a la Post, 229 production systems rule-based, 393 professional ethics, 353 program editor, 358 program state, 76 program transformations, 123, 130 program verification, see verification programmer, 50, 351–352, see also pair programming programming, 90, 282, 337 programming environments, 76, 104 programming languages, 49–229, see also assembly language; machine language; P1: GDZ/FFX PE002-Index P2: GDZ/FFX PE002-Harel QC: GDZ/FFX PE002-Harel-v1.cls T1: GDZ March 17, 2004 21:3 Index ALGOL; C; C++; JAVA; COBOL; EIFFEL; FORTRAN; LISP; PASCAL; PROLOG concurrent, 58, see also concurrent programming languages documentation of, 53 downward transformations of, 54–57 high-level, 50, 54–56, 192 object-oriented, 70, 72–75, 342 parallel, see programming languages, concurrent research on, 75–77 semantics of, 52–53, 76, 123, see also semantics syntax of, 50–52, 215 universality of, 77, see also Church/Turing thesis visual, 76, see also visual formalisms programming paradigms, 58–61 programming tricks, 131–132, 139 programs, 5, 49 breaking up into modules, see modules clarity in, see clarity in algorithms and programs errors in, 99–103, see also errors modification, see software, modification punctuation in, 52, see also semicolon syndrome statements in, 50–52 transformations of, 123, 130–131 PROLOG, 68–70, 392 proof, see also verification by contradiction, 202–205 by induction, 116–117 circular-looking, 205 interactive, 123 probabilistically checkable, 123 proof checking, 120–121 proof of correctness, see verification propositional calculus, 170–171 certificates for, 172 satisfiability problem for, 170–171, 181–183 propositional dynamic logic (PDL), 181–182, 215 with recursion, 211 protein folding, 58 protocols, 272, 274–280, 401 cryptographic, 317, see also cryptography finite-state, 281 fully distributed, 284, 298–301 fully symmetric, 284, 298–301 interactive, 327–332 probabilistic, 298–301 protons, 163, 183, 184, 387 provably intractable problems, 180–183, 311, see also checkers; chess; propositional 507 dynamic logic; Presburger arithmetic; roadblock; WS1S pseudo-random numbers, 313 PSPACE, see polynomial space psychology, 12 PTIME, see polynomial time public-key cryptography, 318–324, see also poker playing; RSA cryptosystem punctuation, 52, see also semicolon syndrome pushdown automata, 248 equivalence problem for, 248 pushing, 38, 39, 248 puzzles airline, 167 jigsaw, 167–168 monkey, 160–162, see also monkey puzzles moving rings, 30–32, see also towers of Hanoi Q Q.E.D (quod erat demonstrandum), 124, 313–314 QP (quantum-PTIME), 289 quadratic time, 139–140, 159 quantifiers, 183 quantum algorithms, 289–291 interference, 290 Shor’s factoring algorithm, 290–292 quantum computer, 288, 291–292 decoherence, 291 quantum computing, 257, 287–292 quantum parallelism, 289–291 quantum physics, 287 qubits, 288–289, 291–292 basis states, 288 entanglement, 288 superposition, 288 Queen of Hearts, 373 query languages, 76, see also databases queue, 38 as a class, 119–120 defined as a class, 71–74 in JAVA, 72–74 functional, 92–94 implemented as a linked list, 73–75 implemented as two stacks, 92–94 in a supermarket, 70–71 quicksort, 144 R r.e problems, see partially decidable problems random numbers, 305, 313, 364, see also pseudo-random numbers random polynomial time (RP), 310–311 P1: GDZ/FFX PE002-Index P2: GDZ/FFX PE002-Harel 508 QC: GDZ/FFX PE002-Harel-v1.cls T1: GDZ March 17, 2004 21:3 Index randomization, 298, 301, 310, 312–313, 388, see also probabilistic algorithms randomized algorithms, see probabilistic algorithms reactive systems, 26, 357–378, 401 anti-scenarios, 370, 371 behavior, 357–378 aircraft, 364 cellular phone, 369, 371 football game, 367 multi-telephone exchange, 372 office, 367 VCR, 367 development of, 358–359, 364, 367, 374–375 implementation of, 367 inter-object behavior, 367–375, 377, 378 intra-object behavior, 367, 368, 377 live sequence charts (LSCs), 368–375, 377 cold conditions, 370–371, 373 existential, 370 hot conditions, 370–372 main chart, 370–374 play-engine, 373, 374 play-in, 373–374, 378 play-out, 373–374, 378 prechart, 370–374 smart play-out, 374, 378 universal, 370 message sequence charts (MSCs), 368–370, 376 modeling, 366–375 of nature, 378 object-oriented specification and analysis, 366–375 reachability, 364–365 research on, 376–378 implementation, 377–378 verification, 377 scenarios, 363–365, 367–375 vs transformational systems, 357–358, 364 reading, 263, 270 real-time systems, 130, 282–283, 375–376 aircraft collision-avoidance system, 375 anti-braking system, 375 anti-missile system, 375 development of, 375 MASS (Marionettes Activation Scheme Specification Language), 375–376 reactions, 375–376 specification of, 375–376 reasonable time, 162–165, 267, 270, see also polynomial time; tractable problems for parallel algorithms, see Nick’s class reasoning, 65 commonsense, 58 recipes, 4–6, 10–11, 14, 16, 19–20, 29, 103, 381, see also mousseline au chocolat record, 12, 220 recurrence, 211, see also recurring dominos recurrence relations, 141 recurring dominos, 211–213 recursion, 30–32, 222, see also self-reference depth of, 32, 115 implementing with a stack, 38–39 in LISP, 66 power of, relative to iteration, 33 in propositional dynamic logic, 211 relationship to trees, 43 time analysis of, 140–143 verification of, 115–117, 119 recursive function theory, 214, 249 recursive functions, 229 recursively enumerable problems (r.e.), see partially decidable problems reductions effective, 201–202, 210, 235–236, 241–243 polynomial-time, 174–177, 210 relational model, 44, 392 release, 285–286 request, 285–286 requirements analysis, see software, development of, requirements acquisition requirements document, 343 research on algorithmic methods, 95 on complexity classes, 184–186, 286–287, 311–312 on concurrency, 286–287 on correctness, 121–123, 354–355 on efficiency, 151–153, 354 on intractability, 184–186 on models of computation, 247–249 on parallelism, 286–287 on probabilistic algorithms, 311–313 on programming languages, 75–77 on reactive systems, 376–378 on system development, 353–355 on undecidability, 214–215 resort area, as metaphor for three-dimensional array, 38 reverse, 110–113 reversing a string, 110–113 P1: GDZ/FFX PE002-Index P2: GDZ/FFX PE002-Harel QC: GDZ/FFX PE002-Harel-v1.cls T1: GDZ March 17, 2004 21:3 Index Riemann hypothesis, extended, 303 roadblocks, 180–181 robotics, 380, 389, 394–395 robots, 9, 358, 380–381, 395, 396 robustness of big-O notation, 138–139 of complexity classes, 238–240 of computability, 229–230, see also Church/Turing thesis of parallel complexity classes, 270, see also parallel computation thesis of tractability, 238–240, see also sequential computation thesis root, 39 rountangles, 223 routines, see subroutines RP, see random polynomial time RSA cryptosystem, 321–324, 326 different version of, 323 rule-based systems, 393, see also production system rules breaking, 351 Grand Book of, 374 in logic programming, 68–70 of thumb, 380, 387–389 run of algorithm, 14, 22–23, 41–42, 55, 89, 112, 137, 225, 227–228, 268 run of model, 362–365 interactive mode, 363 non-interactive mode, 363–364 run-time errors, 104 Runaround, Little R., 19, 23, 24, 27, 50, 204, 222, 257, 297, 357, 358 running time, see time Russian roulette, 297, 301, 302, 305 S safety, 278–280 salary summation, 12–15, 20–21, 24–25, 49, 65–67, 81–82, 129, 148 erroneous version of, 100, 102–104 parallel algorithm for, 259–261, 271 time analysis of, 140, 144 satisfiability problems, 170–171, see also first-order arithmetic; propositional dynamic logic; Presburger arithmetic; propositional calculus; WS1S scheduler, 282, 284, 298–300 malicious, 284, 298, 300 scheduling problems, 170 SCHEME, 65–68 509 SDL diagrams, 359 searches, 81–85 analytic, 388 binary, see binary search blind, 388 breadth-first, 82 depth-first, 82 heuristic, 388 lazy, 388 linear, see linear search systematic, 388, see also binary search seed, 313 self-adjusting data structures, 43, 142 self-reference, 12, 30, 33, 51, 54, 204–207, 502, 509 semantic errors, see errors semantics of natural language, 397–400 of programs, 52–53, 76, 123 denotational, 76 operational, 76 of visual formalisms, 362 semaphore, 285–286 binary, 286 semicolon syndrome, 52 sequence diagrams, 376 sequence of instructions, 19 sequencing, 19, 182, 211, 221 sequential computation thesis, 239–240, see also Church/Turing thesis, refined version sequentializing parallel algorithms, 262–267 sets, 182 Shakespeare, 288 shared memory, 263, 270, 275, 284, 300, see also blackboard short certificates, 172–173, 181 shortest paths, 89–91, 168, 233 shower taking, 273–278, 300, see also mutual exclusion N-processor solution, 277–278 proof of correctness, 276 two-processor solution, 275–276 signals, 247 signatures, 317–321, 333 simplex method, 184 SIMULA, 70 simulations, 231, 235–238, 262, 270, 311 simultaneity, 241, 272, see also concurrency simultaneous state components, see also orthogonal state components size, 262, 401, see also product complexity exponential, 269 P1: GDZ/FFX PE002-Index P2: GDZ/FFX PE002-Harel 510 QC: GDZ/FFX PE002-Harel-v1.cls T1: GDZ March 17, 2004 21:3 Index size (Cont.) linear, 263, 265 multiplied by time, see product complexity reasonable, 263 unreasonable, 269 size of algorithmic problem, 133, 178, 303 reasonable, 185 ski rental, 94–95 SMALLTALK, 72 social systems, 358 software, 4, 191, see also programming development of, see software engineering errors, see errors, in software gradual development of, 118–119 modification, 340, 344–345, 350–351 transition diagram as, 226 software engineering, 337–355 deliverables, 347, 349 design, 343–344, 346, 349–351 documentation, 347, 349, 351 implementation, 344, 350–351 integration, 344, 351 life-cycle models, 343–351 maintenance, 107, 344–345 methodologies, 337–353 agile, 352–353 evolutionary development, 347–348, 351 exploratory programming, 347 extreme programming, 350–352 rapid prototyping, 347, 348 the spiral model, 348–349 the waterfall model, 345–347, 350 process control, 347, 349 prototype, 347 psychology of, 351–353 refactoring, 350–351 requirements acquisition, 343, 349 research on, 353–355 reuse, 349–350 risk management, 348 validation, 344, see also verification, as-you-go sorting, 21–22, 106, 108, 159, see also bubblesort; mergesort; odd-even sorting network; parallel mergesort; quicksort; sorting networks; treesort in parallel, 260–265, 271 time complexity of, 139–140, 142–143, 148, 262–263 unexpected use of, 149–151 sorting networks, 264–265 odd-even, 264–265 optimal, 262, 264–265 space, 129, 144, 185–186, 243–244, 308, 401 exponential, 184–185 joint with time, 153, see also time/space tradeoff research on, 153 unlimited amount of, 230, 235 unreasonable, 184 spaceflight, 120, 338–339 spanning tree, 88 finding minimal, 87–89 specification formalisms, see specification languages specification languages, 121–122, 199, 355 specification of inputs, 14–15, 106 specifications, 107, 122, 311 behavioral, see behavioral description erroneous, 107 speech recognition, 395, 397 spell-checking, 12 sphere of algorithmic problems, 165, 167, 194–195, 212, 240, see also fundamentals levels of computability version I, 165 version II, 195 version III, 213 spirit of computing, 5, 9, see also algorithmics square roots, 260, 321 stack, 38–39 functional, 92–93 implemented as a linked list, 92–93 in pushdown automata, 248 as used to implement recursion, 38–39, 248 starvation, 274–276, 283–285, 299 state-transition diagram, see statecharts; transition diagram statecharts, 359–362, 371–373, see also states statements, 50–52, see also programming languages states in finite automaton, 245–247 in program, 76 in Turing machine, 222–228 halting, 224 start, 224 step, 362 storage space, see memory story understanding, 58 strong AI, 381, 385 structured programming, see clarity in algorithms and programs submarines, 379 subrecipe, 29, see also subroutines P1: GDZ/FFX PE002-Index P2: GDZ/FFX PE002-Harel QC: GDZ/FFX PE002-Harel-v1.cls T1: GDZ March 17, 2004 21:3 Index subroutines, 26–30, 120, 203, 222, 229 elimination of, 33 interface of, 119 parameters of, 26–28, 101, 203 recursive, see recursion virtues of, 28–30, 118 subscript, see index substates, 360 subtree, 42–43 summing numbers, 50 summing salaries, see salary summation super-computer, 229, 231 super-polynomial function, 163–164 super-polynomial time, 163–164, 167, 269, 303, 322, see also exponential time symbol strings, 26–28, 33, 110–113, 219–221 symbolic computation, 65 symmetry breaking, 298–299 synchronization, 299, see also concurrency syntactic equivalence, 197–198, 215, 354 syntax, 50–52, 101, 197 syntax diagrams, 51 syntax errors, see errors syntax rules, 51, 101 synthesis of algorithms, 123 system development, see software engineering system software, 57, 64, 124 systolic arrays, 265–269 T table-driven algorithm, 192 tables, see arrays tail, 65, 110–113 tape, 220–224, 242–245, 248 folding over, 231 multiple, 230 two-dimensional, 230 telephone book search, 132–136, 144–148, 159 temporal logic, 280–281, 374 tense logic, 280 termination, 106, 109–110, 114–116, 199, 279, see also halting problem test sets, 104 testing, 104, 121, 279 integration testing, 344 unit testing, 344 text, 219 theorem proving, 122, 354, 394 three-colorability, 172, 327, 329–332 Tibet, 30, 160 511 tic-tac-toe, 386–387, 390 tiling problems (domino problems), 193–195, 198, 202, 210 bounded, 212 certificates for, 208 co-r.e.-ness, 209 fixed-width bounded, 212 half-grid version, 241–244 non, 208, 210 proving undecidability of, 241–244 recurring, 211–213, see also recurring dominoes reduction from halting problem, 241–244 snakes, see domino snakes unbounded, 193–195, 202, 208, 210–212 variants of, 213 time, 129–218, 401 almost linear, 152 almost polynomial, 303 average-case, see average-case complexity bounds on, 144–146, see also lower bounds; upper bounds constant, 136–138, 308 cubic, 144 definition of, 129 depicted as progress of tiling, 243, 244 double-exponential, 183, see also double-exponential time exponential, 162–165, see also exponential time interactive, 327–328 joint with space, 153, see also time/space tradeoff linear, 132–133, see also linear time logarithmic, 135–138, see also logarithmic time multiplied by size, see product comlexity nondeterministic, see nondeterministic polynomial time nonelementary, 183, see also nonelementary time of nested loops, 139–140 of recursion, 140–143 parallel, 260–261, 267–272, see also parallel-PTIME polylogarithmic, 271, see also polylogarithmic time polynomial, 164, see also polynomial time quadratic, 139–140, see also quadratic time random, 310–311 reasonable, 162–165, see also polynomial time; tractable problems unlimited amount of, 230 P1: GDZ/FFX PE002-Index P2: GDZ/FFX PE002-Harel 512 QC: GDZ/FFX PE002-Harel-v1.cls T1: GDZ March 17, 2004 21:3 Index time (Cont.) unreasonable, 162–165, see also exponential time; intractablity; nonelementary time; super-polynomial time; unreasonable amounts of time worst-case, see worst-case analysis time analysis, see complexity theory; efficiency of algorithms; time time efficiency, see complexity theory; efficiency of algorithms; time Time Magazine, 191 time/space tradeoff, 153 timetables, 170, 176, 269 timing issues, 282 TL, see temporal logic tomography, computerized, 9, 381 top-down design, 29 topology, 124, 394 total correctness, 106, 114–115 totality problem, 210, 211 tour, 168–170, see also Eulerian paths; Hamiltonian paths; traveling salesman problem Towers of Hanoi, 30–32, 38, 43, 85, 115–118, 159–160, 180 iterative solution, 32, 118 proof of correctness, 115–117 recursive solution, 31–32, 69 time analysis of, 159–160 trace, 208 tractability, see polynomial time; reasonable time; tractable problems robustness of, 238–240 tractable problems, 165, 212–213, see also polynomial time transition, 223 transition diagram of finite automaton, 245, 247, see also statecharts linearizing, 237 of Turing machine, 223–225, 235 traveling salesman problem, 130, 168–169, 178, 269 approximation algorithms for, 179 reduction from Hamiltonian paths, 174–175 traversals, 40–43, 81–85, see also searches trees, 39–43 balanced, 142 binary, 40, 147–148, see also binary search tree comparison, 147–148 empty, 42 family, 39 full, 43 game, 40, 180, 386–391 imbalanced, 142 levels of, 40, 220–221, 386 linearizing, 220–221 relationship to recursion, 43 spanning, see minimal spanning tree traversing, 40–43 treesort, 40–43, 82, 142–144, 148 trigger, 223 triple-exponential function, 183 truck fitting, 170 truth assignment, 171, see also satisfiability problems truth propagation, 112–113 Turing machines, 223–243, 311 as computers, 226–228 with counter, 240 deterministic, 224 equivalence to counter programs, 233–235, 308 nondeterministic, 230–231, 240–241 one-way, see finite automata probabilistic, 310–311 programming, 226 resource-bounded, 243–244 time-bounded, 238–241, 243 universal, 237–238 use in proving lower bounds, 241–243 variants of, 230–231 Turing test, 382–385, 392, 394 Turing, Alan M., 7, 223, 228–230, 382, 392 two-way certificates, see certificates typewriters, 228 U unbounded iteration, see conditional iteration unboundedness of problem, 195–196, 211, 212 undecidability, 191–218, 267, 311, see also noncomputability degrees of, 211, 214, see also high undecidability; inductive/coinductive level of undecidability; partially decidable problems hierarchies of, 211, 214 high, 211–214 proving, 201–207, 241–244 research on, 214–215 undecidable problems, 193–201, 211–213, 230, 248, see also undecidability P1: GDZ/FFX PE002-Index P2: GDZ/FFX PE002-Harel QC: GDZ/FFX PE002-Harel-v1.cls T1: GDZ March 17, 2004 21:3 Index unified modeling language (UML), 376–377 universal algorithms, 236–238 universality of programming languages, 77, see also Church/Turing thesis unlimited intractability, see nonelementary time unreasonable amounts of time, 130, 159–165, see also intractability upper bounds, 144–146, 241, see also efficiency of algorithms use-cases, 368 V values, see variables variable, 58–59 variables, 34–35, 129 centralized, see variables, shared in counter programs, 232–233 courtesy, 275 distributed, 275, 284 shared, 263, 270, 275, 300, see also blackboard; shared memory values of, 12–13, 34–35 vectors, 35–36, 131, see also list; queue; stack in FORTRAN, 60 indices into, see index linearizing, 220 of vectors, 37 relationship to loops, 35, 36 verification, 99–128, 354–355, see also correctness of algorithms; errors after-the-fact, 114–115, 118–119 algorithmic, see verification, automatic as algorithmic problem, 198–199, 210 reduction from halting problem, 202, 210 undecidability of, 107, 199, 341 as-you-go, 115, 118–119 automatic, 107, 114, 120–121, 123, 198–199, 281, 341, 365 of concurrency, 276, 278–283 examples of proofs, 110–117, 276 interactive, 120–121 the need for, 107, 124–125, 279 of probabilistic algorithms, 311 of reactive behavior, 364–365 of recursion, 115–117 research on, 121–123, 311 verification conditions, 109, 112–114, 121 feasibility of checking, 114, 122 513 vertex, 84–85 neighboring, 84 virtual reality, 58 viruses, 64 vision, 381, 389, 395 visual algorithmics, see also diagrammatic techniques visual formalisms, 24, 26, 76, 358–378 research on, 376–377 synthsis, 377 visual languages, see visual formalisms von Neumann model, 58 von Neumann, John, W weary travelers, see shortest paths weighted average computation, 265–268, 271 Where’s Waldo, 328–329, see zero-knowledge while statement, 20, 222, see also looping constructs winning strategy, 180–181 witness, 304–307 word correspondences bounded version of, 212 fixed-width version of, 212 recurring version of, 212 word corrspondences, 197–198, 202, 210 certificates for, 207 word-processing program, 343–345, 358 words, see symbol strings worst-case analysis, 94, 132, 161, see also time worst-case complexity, 132, 143–144, 307, see also complexity classes worst-case running time, see worst-case complexity writing, 263, 270 WS1S, 183 Y Y2K bug, see Year 2000 Problem Year 2000 Problem, 99, 340–341 yes/no problems, see decision problems Z zero-knowledge, 328–332 zero-probability, 298–300 ZPP, 311 zupchocks, 384–385 P1: GDZ/FFX PE002-Index P2: GDZ/FFX PE002-Harel QC: GDZ/FFX PE002-Harel-v1.cls T1: GDZ March 17, 2004 21:3 514 ... available from the British Library Library of Congress Cataloging-in-Publication Data Harel, David, 1950– Algorithmics : the spirit of computing / David Harel, with Yishai Feldman. 3rd ed p cm Includes... to the recipe The ingredients are the inputs to the process, the cake is its output, and the recipe is the algorithm In other words, the algorithm prescribes the activities that constitute the. .. important parts of the science of computing are not sufficiently known, not only to members of the first group, but to members of the second group as well It is one of the purposes of this book