Theory of Reversible Computing PDF
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Monographs in Theoretical Computer Science An EATCS Series Kenichi Morita Theory of Reversible Computing Monographs in Theoretical Computer Science An EATCS Series Editors: M. Henzinger J. Hromkovi(cid:254) M. Nielsen G. Rozenberg A. Salomaa Founding Editors:W.Brauer G.Rozenberg A.Salomaa OnbehalfoftheEuropeanAssociation forTheoreticalComputerScience(EATCS) AdvisoryBoard: S. Albers H. Attiya G. Ausiello M. Broy C. Calude A. Condon A. Czumaj P. Degano J. Diaz P. Gastin G. Gottlob D. Harel J. Hartmanis R. Heckel L.A. Hemaspaandra T. Henzinger M. Hermenegildo B. Jonsson J. Karhumäki L. Kari M. Koutny D. Kozen T. Leighton H. Lin G. Mauri M. Nivat D. Niwi(cid:276)ski C. Papadimitriou D. Peleg D. Sannella U. Schöning D. Scott P.G. Spirakis D. Wagner E. Welzl M. Wirsing More information about this series at http://www.springer.com/series/776 Kenichi Morita Theory of Reversible Computing Kenichi Morita Professor Emeritus, Hiroshima University Hiroshima, Japan Series Editors Monika Henzinger Juraj Hromkovi(cid:254) Faculty of Science ETH Zentrum Universität Wien Department of Computer Science Wien, Austria Swiss Federal Institute of Technology Zürich, Switzerland Mogens Nielsen Department of Computer Science Grzegorz Rozenberg Aarhus Universitet Leiden Centre of Advanced Aarhus, Denmark Computer Science Leiden University Arto Salomaa Leiden, The Netherlands Turku Centre of Computer Science Turku, Finland ISSN 1431-2654 ISS N2193-2069 (electronic ) Monographs in Theoretical Computer Science. An EATCS Series ISBN 978-4-431-56604-5 ISBN 978-4-431-56606-9 (eBook) DOI 10.1007/978-4-431-56606-9 Library of Congress Control Number: 2017957823 © Springer Japan KK 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, citation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Japan KK The registered company address is: Chiyoda First Bldg. East, 8-1 Nishi-Kanda Chiyoda-ku, 101-0065 Tokyo, Japan Tomyparents,mywife,mychildren,andmy grandchildren Preface Areversiblecomputingsystemisa“backwarddeterministic”systemsuchthatevery stateofthesystemhasatmostonepredecessor.Hence,thereisnopairofdistinct states that go to the same state. Though its definition is so simple, it is closely re- latedtophysicalreversibility.Thestudyofreversiblecomputingoriginatedfroman investigationofenergydissipationinreversibleandirreversiblecomputingsystems. RolfLandauerinvestigatedtherelationbetweenreversibilityincomputingandre- versibilityinphysicsinhispaper“Irreversibilityandheatgenerationinthecomput- ingprocess”(IBMJ.Res.Dev.,Vol.5,pp.183–191,1961).Hepointedoutthatan irreversiblelogicaloperationinevitablycausesenergydissipationinthecomputing system.Sincethen,reversiblecomputinghasbeenstudiedinrelationtophysicalre- versibility.Besidestheproblemofenergydissipationincomputing,itisimportant toknowhowreversibilitycanbeeffectivelyutilizedincomputing.Thisisbecause futurecomputingdeviceswillsurelybeimplementeddirectlybyphysicalphenom- enainthenano-scalelevel,andreversibilityisoneofthefundamentalmicroscopic physicallawsofNature.Forthispurpose,variousmodelsofreversiblecomputing havebeenproposedandinvestigatedtillnow. In this book, reversible computing is studied from the standpoint of the theory of automata and computing. We deal with various reversible computing models belonging to several different levels, which range from a microscopic level to a macroscopic one. They are reversible physical models, reversible logic elements, reversible functional modules composed of logic elements, reversible computing systemssuchasTuringmachines,cellularautomata,andothers.Thepurposeofthis book is to clarify how computation can be carried out efficiently and elegantly in these reversible computing models. We shall see that even very simple reversible systems have computational universality in spite of the constraint of reversibility. Weshallalsoseevariousreversiblesystemsindifferentlevelsarerelatedeachother, i.e.,areversiblesysteminahigherlevelcanbeconstructedoutofthoseinalower level.Moreover,theconstructionmethodsareoftenveryuniqueanddifferentfrom those in the traditional methods. Thus, these computing models as well as the de- signingmethodswillgiveusnewinsightsforfuturecomputingsystems. vii viii Preface This book is not a comprehensive textbook on reversible computing, but de- scribesmainlytheresultsshowninthepapersbymyselfandmycolleagues,which werepublishedbetween1989and2017.Inthisbook,readerswillseehowtheworld of reversible computing works, how reversible computingsystems are constructed outofsimplereversibleprimitives,howtheyaredifferentfromthetraditionalcom- putingsystems,andhowtheycanbecomputationallyuniversal.Infact,weshallsee evenverysimplereversiblesystemshavehighcapabilityofcomputing,andthusre- versibilityisnotaconstraint,butausefulpropertyforcomputing. This book consists of 14 chapters. Chapter 1 is an introduction. In Chaps. 2–4, reversible logic elements for constructing reversible machines are investigated. In Chaps.5–8,reversibleTuringmachines,astandardmodelofreversiblecomputing systems, are studied. In Chap. 9, some other reversible computing models are in- vestigated. In Chaps. 10–14, reversible cellular automata, a spatiotemporal model ofreversibledynamicalsystems,arestudied. There is no prerequisite knowledge to read this book besides some basics on logic,discretemathematicsandformallanguages.But,itispreferabletohavesome knowledgeonthetheoriesofautomataandcomputing.Fortunately,theframework ofreversiblecomputingitselfisverysimple.Therefore,inmanycases,readerscan easilyunderstandthebasicfunctionandthestructureofeachsuchsystem.However, itsbehaviorcanbeverycomplexevenifthestructureofthesystemissimple.Hence, sometimes,itbecomesquitedifficulttofollowitsbehaviorbyusingonlypaperand pencil.Insomeofthesecases,readerscanfindfilesintheReferencesthatcontain computersimulationresultsofsuchreversiblesystems. Morethan50yearshavepassedsinceLandauer’spaperappeared.Thus,thehis- toryofreversiblecomputingisrelativelylong.But,itisstilldeveloping,andthere remain many problems to be investigated. Also, even at present, it is not so clear which results will become practically useful in the future. However, the world of reversible computing will lead readers to the new ways of thinking that cannot be found in the traditional design methodologies for computing systems. I hope the theoryofreversiblecomputingwillstimulatereaders’interest,andopennewvistas forfuturecomputingsystems. Acknowledgments. Thisworkwouldnothavebeenaccomplishedwithoutthehelp and encouragement given by many people in many countries. I would express my highest gratitude to all of them. In particular, I would express my special thanks toProf.ToshioMitsuiandProf.KazuhiroSugatawhoguidedmeintheearlydays ofmyresearchcareerinOsakaUniversity.Frequentdiscussionswiththemledme to the research fields of automata theory, cellular automata, and finally reversible computing. Since then, I had many good colleagues and students in Hiroshima University, Yamagata University, Osaka University, and other places in the world. Workingwiththemwasagreatpleasureforme.Iamgratefultothem,especially, Susumu Adachi, Andrew Adamatzky, Artiom Alhazov, Yoichi Fujii, Yoshifumi Gono, Masateru Harao, Takahiro Hori, Katsunobu Imai, Teijiro Isokawa, Chuzo Iwamoto, Atsushi Kanno, Hiroko Kato, Jia Lee, Maurice Margenstern, Genaro J. Martinez, Mitsuya Morimoto, Yuta Mukai, Noritaka Nishihara, Masanori Ogino, Preface ix Tsuyoshi Ogiro, Ferdinand Peper, Akihiko Shirasaki, Rei Suyama, Keiji Tanaka, TsuyoshiTanizawa,YasuyukiTojima,RyoichiUeno,SatoshiUeno,HiroshiUmeo, YoshikazuYamaguchi,andYasunoriYamamotofortheirhelpfuldiscussions,coop- eration,andcoauthoringthepapersonreversiblecomputingandrelatedfields. I also express my cordial thanks to the anonymous reviewer for his/her very careful reading and valuable comments. The manuscript was greatly improved by thedetailedcomments. This work on reversible computing was supported by JSPS KAKENHI Grant NumberJP15K00019. September2017 KenichiMorita Contents 1 Introduction................................................... 1 1.1 ReversibilityinPhysicsandComputing........................ 1 1.2 SignificanceofReversibleComputing ......................... 2 1.3 ScopeofThisVolume....................................... 4 1.3.1 Organizationofthisbook ............................. 5 1.3.2 Relatedstudiesandreferences ......................... 6 1.4 TerminologyandNotations .................................. 8 References..................................................... 10 2 ReversibleLogicElementswithMemory.......................... 15 2.1 LogicalPrimitivesforReversibleComputers.................... 15 2.2 ReversibleLogicElementwithMemory(RLEM)................ 16 2.2.1 Rotaryelement(RE),atypicalRLEM................... 17 2.2.2 CircuitcomposedofREs.............................. 19 2.2.3 RealizingREinthebilliardballmodel .................. 22 2.3 MakingReversibleSequentialMachines(RSMs)fromRE ........ 26 2.3.1 RE-column,abuildingmoduleforRSMs ................ 26 2.3.2 ComposingreversiblesequentialmachinesbyRE-columns . 27 2.4 ConcludingRemarks........................................ 30 References..................................................... 30 3 ClassificationofReversibleLogicElementswithMemoryandTheir Universality ................................................... 31 3.1 ClassificationofRLEMs .................................... 31 3.1.1 GraphicalrepresentationofRLEMs..................... 32 3.1.2 EquivalenceinRLEMs ............................... 35 3.1.3 DegeneracyinRLEMs................................ 40 3.1.4 Classificationof2-,3-and4-symbolRLEMs............. 40 3.2 UniversalityofAllNon-degenerate2-StateRLEMswithThreeor MoreSymbols ............................................. 41 3.2.1 RealizingREusingnon-degenerate3-symbolRLEMs ..... 41 xi xii Contents 3.2.2 Makinganon-degenerate(k 1)-symbolRLEMfroma (cid:0) non-degeneratek-symbolRLEMs ...................... 45 3.3 SystematicConstructionofRSMsoutofUniversalRLEMs ....... 51 3.4 CompactRealizationofRSMsUsingRLEMs4-31and3-7 ....... 53 3.5 FrontierBetweenUniversalandNon-universalRLEMs........... 56 3.5.1 DefinitionsonRLEM-circuits.......................... 57 3.5.2 Non-universalityofthreekindsof2-state2-symbolRLEMs 60 3.5.3 Universalityofcombinationsof2-state2-symbolRLEMs .. 67 3.6 Realizing4-SymbolRLEMsintheBilliardBallModel........... 70 3.7 ConcludingRemarks........................................ 74 References..................................................... 74 4 ReversibleLogicGates ......................................... 77 4.1 ReversibleLogicGatesandCircuits........................... 77 4.1.1 Reversiblelogicgates ................................ 78 4.1.2 Reversiblecombinatoriallogiccircuits .................. 80 4.1.3 Logicaluniversalityofreversiblelogicgates ............. 82 4.1.4 Clearinggarbageinformation .......................... 84 4.1.5 Realizationinthebilliardballmodel .................... 88 4.2 Relation Between Reversible Logic Gates and Reversible SequentialMachines........................................ 89 4.2.1 MakingFredkingatefromRE ......................... 90 4.2.2 MakingREfromFredkingate ......................... 91 4.2.3 MakingreversiblesequentialmachinesfromFredkingate .. 93 4.3 ConcludingRemarks........................................100 References.....................................................100 5 ReversibleTuringMachines.....................................103 5.1 TuringMachinesandReversibility ............................103 5.1.1 BasicdefinitionsonreversibleTuringmachines(RTMs) ...104 5.1.2 Notionofsimulationandcomputationaluniversality.......111 5.1.3 Conversionbetweenthequadrupleandquintupleforms ....112 5.1.4 InversereversibleTuringmachines .....................115 5.2 ConvertingIrreversibleTuringMachinestoReversibleOnes ......119 5.2.1 Three-tapeTuringmachines ...........................119 5.2.2 Inversethree-tapeTuringmachines .....................121 5.2.3 Computationaluniversalityofthree-tapeRTMs...........122 5.3 VariationsofReversibleTuringMachines ......................129 5.3.1 ConvertingRTMswithtwo-wayinfinitetapesintoRTMs withone-wayinfinitetapes ............................129 5.3.2 Convertingmulti-tapeRTMsintoone-tapeRTMs .........134 5.3.3 Convertingmany-symbolRTMsintotwo-symbolRTMs ...138 5.3.4 Convertingmany-stateRTMsintofour-stateRTMs........143 5.3.5 Convertingmany-stateRTMsintothree-stateRTMs .......148 5.3.6 ComputationaluniversalityofrestrictedclassesofRTMs...154
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