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The Beauty of Mathematics in Computer Science THE BEAUTY OF MATHEMATICS IN COMPUTER SCIENCE Jun Wu A Chapman & Hall Book The Beauty of Mathematics in Computer Science The Beauty of Mathematics in Computer Science Jun Wu Translated from the Chinese edition by Rachel Wu and Yuxi Candice Wang CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-1-138-04960-4 (Paperback) 978-1-138-04967-3 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or here- after invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Wu, Jun, 1967- author. Title: The beauty of mathematics in computer science / Jun Wu. Description: Boca Raton, FL : Taylor & Francis Group, 2019. Identifiers: LCCN 2018035719| ISBN 9781138049604 paperback | ISBN 9781138049673 hardback Subjects: LCSH: Computer science--Mathematics. | Machine learning. Classification: LCC QA76.9.M35 W84 2019 | DDC 004.01/51--dc23 LC record available at https://lccn.loc.gov/2018035719 Visit the Taylor & Francis Web site at http:==========www.taylorandfrancis.com and the CRC Press Web site at http:==========www.crcpress.com Contents Foreword . xi Preface . xiii Acknowledgments . xv 1 Words and languages, numbers and information . 1 1.1 Information . .................................................... 1 1.2 Words and numbers . ........................................... 2 1.3 The mathematics behind language . ............................... 8 1.4 Summary.......................................................11 2 Natural language processing—From rules to statistics . 13 2.1 Machine intelligence . ..........................................13 2.2 From rules to statistics ..........................................19 2.3 Summary.......................................................22 3 Statistical language model . 23 3.1 Describing language through mathematics . ......................23 3.2 Extended reading: Implementation caveats ......................27 3.2.1 Higher order language models ............................27 3.2.2 Training methods, zero-probability problems, and smoothing . ........................................28 3.2.3 Corpus selection . ........................................32 3.3 Summary.......................................................33 Bibliography . .......................................................33 4 Word segmentation . 35 4.1 Evolution of Chinese word segmentation . ......................35 4.2 Extended reading: Evaluating results . ..........................39 4.2.1 Consistency . ............................................39 4.2.2 Granularity . ............................................40 4.3 Summary.......................................................41 Bibliography . .......................................................41 v vi Contents 5 Hidden Markov model . 43 5.1 Communication models . ......................................43 5.2 Hidden Markov model ..........................................45 5.3 Extended reading: HMM training . ..............................48 5.4 Summary.......................................................50 Bibliography . .......................................................50 6 Quantifying information . 53 6.1 Information entropy . ..........................................53 6.2 Role of information . ..........................................55 6.3 Mutual information . ..........................................57 6.4 Extended reading: Relative entropy . ..........................59 6.5 Summary.......................................................61 Bibliography . .......................................................62 7 Jelinek and modern language processing . 63 7.1 Early life .......................................................64 7.2 From Watergate to Monica Lewinsky . ..........................66 7.3 An old man’s miracle . ..........................................68 8 Boolean algebra and search engines . 71 8.1 Boolean algebra . ...............................................72 8.2 Indexing . .......................................................74 8.3 Summary.......................................................76 9 Graph theory and web crawlers . 77 9.1 Graph theory ...................................................77 9.2 Web crawlers ...................................................79 9.3 Extended reading: Two topics in graph theory . .................79 9.3.1 Euler’s proof of the Königsberg bridges ...................79 9.3.2 The engineering of a web crawler . ........................80 9.4 Summary.......................................................82 10 PageRank: Google’s democratic ranking technology . 83 10.1 The PageRank algorithm . ..................................83 10.2 Extended reading: PageRank calculations .....................86 10.3 Summary . ..................................................87 Bibliography . .......................................................87 11 Relevance in web search . 89 11.1 TF-IDF ......................................................90 Contents vii 11.2 Extended reading: TF-IDF and information theory . .........92 11.3 Summary . ..................................................93 Bibliography . .......................................................93 12 Finite state machines and dynamic programming: Navigation in Google Maps . 95 12.1 Address analysis and finite state machines .....................96 12.2 Global navigation and dynamic programming .................98 12.3 Finite state transducer . .....................................101 12.4 Summary . .................................................102 Bibliography . ......................................................103 13 Google’s AK-47 designer, Dr. Amit Singhal . .105 14 Cosines and news classification . .109 14.1 Feature vectors for news .....................................109 14.2 Vector distance . .............................................111 14.3 Extended reading: The art of computing cosines . ............114 14.3.1 Cosines in big data .................................. 114 14.3.2 Positional weighting . .............................. 115 14.4 Summary . .................................................116 15 Solving classification problems in text processing with matrices . .117 15.1 Matrices of words and texts . .................................117 15.2 Extended reading: Singular value decomposition method and applications .....................................120 15.3 Summary . .................................................121 Bibliography . ......................................................121 16 Information fingerprinting and its application . .123 16.1 Information fingerprint . .....................................123 16.2 Applications of information fingerprint . ....................125 16.2.1 Determining identical sets . .......................... 125 16.2.2 Detecting similar sets . .............................. 126 16.2.3 YouTube’s anti-piracy .............................. 127 16.3 Extended reading: Information fingerprint’s repeatability andSimHash ................................................128 16.3.1 Probability of repeated information fingerprint . ..... 128 16.3.2 SimHash . .......................................... 129 16.4 Summary . .................................................131 Bibliography . ......................................................132 viii Contents 17 Thoughts inspired by the Chinese TV series Plot: The mathematical principles of cryptography . .133 17.1 The spontaneous era of cryptography ........................133 17.2 Cryptography in the information age . ........................136 17.3 Summary . .................................................140 18 Not all that glitters is gold: Search engine’s anti-SPAM problem and search result authoritativeness question . .141 18.1 Search engine
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