Concise Guide to Computation Theory

300 Pages·2011·4.13 MB·English

by  Akira Maruoka (auth.)

Concise Guide to Computation Theory
Akira Maruoka
Concise Guide
to Computation
Theory
AkiraMaruoka
FacultyofScienceandEngineering
IshinomakiSenshuUniversity
ShinmitoMinamisakai1
Ishinomaki,Japan
[email protected]
ISBN978-0-85729-534-7 e-ISBN978-0-85729-535-4
DOI10.1007/978-0-85729-535-4
SpringerLondonDordrechtHeidelbergNewYork
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Endorsements
ReischukRüdiger
Thisfascinatingtextincludesawealthofideaspresentedtonovicereaderssuchthat
theycanunderstandsubstantialsubjectsofcomputationtheoryinjustthewaythat
experiencedresearchersinthefieldgraspthem.Readerswillfindlearningwiththis
booktobebothpleasurableandrewarding.
November,2010
JohnE.Savage
An unusual book that provides an excellent introduction to theoretical computer
sciencewhilestrikingagoodbalancebetweenmathematicalrigorandintuitiveun-
derstanding.
November,2010
v

Foreword
This book is unusual: ambitious in scope, yet compact. Drawing on his broad re-
searchinterestsandextensiveteachingexperience,ProfessorMaruokadistillssome
ofthemajortopicsinthetheoryofcomputationintoarelativelyslimtextbookac-
cessibletoundergraduates.Hisaimistocombineintuitivedescriptionsandillustra-
tionswithrigorousargumentsanddetailedproofsforkeytopics.Thebookisself-
containedinthatitbrieflyintroducesanymathematicalprerequisites(inPartI),and
itwouldbeidealforaone-ortwo-semesterundergraduatecourseinthetheoryof
computation.Theprincipalmaterialisinthreesections:AutomataandLanguages,
Computability,andComplexityofComputation.Theserepresentthefoundationsof
thetheoryofcomputationasunderstoodtoday,underpinningtherichdevelopments
ofthesubjectwhichhavebecomeimportantinsomanyareasofscience,technol-
ogy,andbusiness.
Early progress toward defining models and establishing interesting correspon-
dences between classes of finite automata and formal languages were made in the
1940s and 1950s through the work of such thinkers as McCulloch, Pitts, Kleene,
Moore, Mealy, Shannon, Chomsky, Rabin, and Scott. Some of the striking high-
lights,inconceptsandtheorems,fromthisareaarepresentedclearlyandconcisely
inChaps.3,4and5(PartII).Thefoundationsofcomputability,coveredinChaps.6
and 7 (Part III), were established earlier, from the 1930s on, by scientists such as
Goedel,Church,Markov,Post,andTuring.Arichliteratureontopicsincomputabil-
ityhadbeendevelopedbythe1960s,withmuchemphasison“reducibility”(intro-
duced in Chap. 7) between computational problems, showing that some “unsolv-
able”problemsweremoreunsolvablethanothers.
Prominentsincethe1970s,studiesofcomputational“complexity”suchasthose
ofCookandLevinfocusedattentiononthedistinctionbetweenproblemsthatwere
solvableina“feasible”amountoftimeandthosewhichhadtobeconsideredinfea-
sible,forexampleifthecomputationtimehadtogrowexponentiallyinthesizeof
theinputinstance.Akeyconcept,discussedinChap.8,istheclassNPofproblems
solvableinpolynomialtimebya“nondeterministic”Turingmachine.Theconcepts
neededtounderstandthisclassareintroducedgraduallythroughouttheearlierchap-
ters.Chapter9introducesthefurthercomputationalmodelof“Booleancircuits,”an
abstraction of the logic circuits at the heart of computers, thoroughly explaining
close links between Boolean function complexity and Turing machine complex-
ity.Chapter10givesthedefinitionofNP-completeness.Karp,Garey,andJohnson
vii
viii Foreword
wereamongthemanyearlyresearcherstoestablishtheabundanceofNP-complete
problems arising in practical applications. Many thousands of apparently difficult
problemsin operations research,mathematics,networkdesign,etc., can be shown
by reductions to be all equivalent in their complexity. The notion of reduction is
keyhere.ItisacelebratedconjecturethatP(cid:2)=NP,i.e.,thatNP-completeproblems
cannotbesolvedinpolynomialtime.
A more superficial treatment could certainly outline some of the main ideas in
thetheoryofcomputation,butthistextbookprovidesbothathoroughfoundationfor
thesubject,andamuchdeeperinsightintoitthroughaselectionofresultsobtained
fromthetheory.WithProfessorMaruoka’sguidance,thestudentcandevelopareal
understandingofthisfascinatingsubjectandasolidbasisforcarryingontopursue
researchinthetheoryofcomputation,maybeeventoprovethatP(cid:2)=NP.
UniversityofWarwick,UK MikePaterson
Preface to the English Edition
I am both gratified and grateful to present this English edition of my book titled
Concise Guide to Computation Theory. The original Japanese edition was written
with the intention of enabling readers to understand, both deeply and intuitively,
the profound results achieved in this field. In translating the book, I revised it ex-
tensivelyandthoroughly,addinganumberofproblemstotheendofeachchapter.
I sincerely hope that this revision makes it both easy and enjoyable to read about
thesefascinatingtopics.Icouldnothavecompletedthetranslationwithoutthehelp
of my friends and colleagues. Rüdiger Reischuk and John Savage advised me to
publish a translated edition of the book that had been written in Japanese. Itsuo
Takanamihelpedmeenormouslyinpreparingthetranslatededition.SatoshiOkawa
readthetranslatedmanuscriptandgavemehelpfulcomments.BradFastnicelyre-
visedapartofthetranslatedmanuscript,improvingitsreadability.Iamindebtedto
KazukoTakimotofortypingalargevolumeofthemanuscript.Iamalsogratefulto
EijiTakimotoforprovidingtechnicalsupportfortypingofthemanuscript.Finally,
IwouldliketothankWayneWheelerandSimonReesofSpringer-Verlag,London,
and Alfred Hofmann and Ronan Nugent of Springer-Verlag, Heidelberg, for their
excellentsuggestionsandthoughtfulguidanceinpreparingthisbook.
Just before the publication of this textbook, this area of Japan was struck by a
disasterofhistoricscale:thedevastatingearthquakeandtsunami.Itwillsurelyoften
cometomymindthatourrecoveryfromthisdevastationhadjuststartedwhenthis
book was published. As we heal and rebuild, all here appreciate the best wishes
extendedtousfromallovertheworld.
Sendai,Japan AkiraMaruoka
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