Automatic sequences: theory, applications, generalizations

587 Pages·2003·2.896 MB·English

by  Jean-Paul Allouche, Jeffrey Shallit

AUTOMATIC SEQUENCES
Unitingdozensofdisparateresultsfromdifferentfields,thisbookcombinescon-
ceptsfrommathematicsandcomputersciencetopresentthefirstintegratedtreat-
mentofsequencesgeneratedbythesimplemodelofcomputationcalledthefinite
automaton.
Theauthorsdevelopthetheoryofautomaticsequencesandtheirgeneralizations,
suchasSturmianwordsandk-regularsequences.Further,theydiscussapplications
to number theory (particularly formal power series and transcendence in finite
characteristic),physics,computergraphics,andmusic.
Results are presented from first principles wherever feasible, and the book is
supplemented by a collection of 460 exercises, 85 open problems, and over 1600
citations to the literature. Thus this book is suitable for graduate students or ad-
vanced undergraduates, as well as for mature researchers wishing to know more
aboutthisfascinatingsubject.
Jean-PaulAlloucheisDirecteurdeRechercheatCNRS,LRI,Orsay.Hehaswritten
some90papersinnumbertheoryandcombinatoricsonwords.Heisontheeditorial
boardofAdvancesinAppliedMathematicsandonthescientificcommitteeofthe
JournaldeThe´oriedesNombresdeBordeaux.
JeffreyShallitisProfessorofComputerScienceattheUniversityofWaterloo.He
haswritten80articlesonnumbertheory,algorithms,formallanguages,combina-
toricsonwords,computergraphics,historyofmathematics,algebraandautomata
theory. He is the editor-in-chief of the Journal of Integer Sequences and coauthor
ofAlgorithmicNumberTheory.
AUTOMATIC SEQUENCES
Theory, Applications, Generalizations
JEAN-PAUL ALLOUCHE
CNRS,LSI,Orsay
JEFFREY SHALLIT
UniversityofWaterloo
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , United Kingdom
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521823326
© Jean-Paul Allouche & Jeffrey Shallit 2003
This book is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2003
- 
isbn-13 978-0-511-06208-7 eBook (NetLibrary)
- 
isbn-10 0-511-06208-7 eBook (NetLibrary)
- 
isbn-13 978-0-521-82332-6 hardback
- 
isbn-10 0-521-82332-3 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of
s for external or third-party internet websites referred to in this book, and does not
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Nousde´dionscelivrea` MichelMende`sFrance
ensignedegratitudeetd’amitie´

Contents
Preface pagexiii
1 Stringology 1
1.1 Words 1
1.2 TopologyandMeasure 5
1.3 LanguagesandRegularExpressions 7
1.4 Morphisms 8
1.5 TheTheoremsofLyndonandSchu¨tzenberger 10
1.6 RepetitionsinWords 14
1.7 Overlap-FreeBinaryWords 16
1.8 AdditionalTopicsonRepetitions 23
1.9 Exercises 24
1.10 OpenProblems 30
1.11 NotesonChapter1 31
2 NumberTheoryandAlgebra 39
2.1 DivisibilityandValuations 39
2.2 RationalandIrrationalNumbers 39
2.3 AlgebraicandTranscendentalNumbers 41
2.4 ContinuedFractions 44
2.5 BasicsofDiophantineApproximation 48
2.6 TheThree-DistanceTheorem 53
2.7 AlgebraicStructures 55
2.8 VectorSpaces 56
2.9 Fields 56
2.10 Polynomials,RationalFunctions,andFormalPowerSeries 58
2.11 p-adicNumbers 62
2.12 AsymptoticNotation 63
2.13 SomeUsefulEstimates 63
2.14 Exercises 64
2.15 OpenProblems 67
2.16 NotesonChapter2 67
vii
viii Contents
3 NumerationSystems 70
3.1 NumerationSystems 70
3.2 SumsofDigits 74
3.3 BlockCountingandDigitalSequences 77
3.4 RepresentationofRealNumbers 84
3.5 SumsofSumsofDigits 86
3.6 Base-k RepresentationwithAlternateDigitSets 101
3.7 RepresentationsinNegativeBases 103
3.8 FibonacciRepresentation 105
3.9 Ostrowski’sα-NumerationSystem 106
3.10 RepresentationsinComplexBases 107
3.11 Exercises 112
3.12 OpenProblems 118
3.13 NotesonChapter3 119
4 FiniteAutomataandOtherModelsofComputation 128
4.1 FiniteAutomata 128
4.2 ProvingLanguagesNonregular 136
4.3 FiniteAutomatawithOutput 137
4.4 Context-FreeGrammarsandLanguages 143
4.5 Context-SensitiveGrammarsandLanguages 146
4.6 TuringMachines 146
4.7 Exercises 148
4.8 OpenProblems 150
4.9 NotesonChapter4 150
5 AutomaticSequences 152
5.1 AutomaticSequences 152
5.2 RobustnessoftheAutomaticSequenceConcept 157
5.3 Two-SidedAutomaticSequences 161
5.4 BasicPropertiesofAutomaticSequences 165
5.5 NonautomaticSequences 166
5.6 k-AutomaticSets 168
5.7 1-AutomaticSequences 169
5.8 Exercises 170
5.9 OpenProblems 171
5.10 NotesonChapter5 171
6 UniformMorphismsandAutomaticSequences 173
6.1 FixedPointsofUniformMorphisms 173
6.2 TheThue–MorseInfiniteWord 173
6.3 Cobham’sTheorem 174
6.4 TheTowerofHanoiandIteratedMorphisms 177
6.5 PaperfoldingandContinuedFractions 181
6.6 Thek-Kernel 185
Contents ix
6.7 Cobham’sTheoremfor(k,l)-NumerationSystems 187
6.8 BasicClosureProperties 189
6.9 UniformTransductionofAutomaticSequences 192
6.10 SumsofDigits,Polynomials,andAutomaticSequences 197
6.11 Exercises 201
6.12 OpenProblems 207
6.13 NotesonChapter6 208
7 MorphicSequences 212
7.1 TheInfiniteFibonacciWord 212
7.2 FiniteFixedPoints 213
7.3 MorphicSequencesandInfiniteFixedPoints 215
7.4 Two-SidedInfiniteFixedPoints 218
7.5 MoreonInfiniteFixedPoints 226
7.6 ClosureProperties 228
7.7 MorphicImagesofMorphicWords 231
7.8 LocallyCatenativeSequences 237
7.9 TransductionsofMorphicSequences 240
7.10 Exercises 242
7.11 OpenProblems 244
7.12 NotesonChapter7 245
8 FrequencyofLetters 247
8.1 SomeExamples 247
8.2 TheIncidenceMatrixAssociatedwithaMorphism 248
8.3 SomeResultsonNon-negativeMatrices 249
8.4 FrequenciesofLettersandWordsinaMorphicSequence 266
8.5 AnApplication 276
8.6 Gaps 278
8.7 Exercises 280
8.8 OpenProblems 282
8.9 Notes 282
9 CharacteristicWords 283
9.1 DefinitionsandBasicProperties 283
9.2 GeometricInterpretationofCharacteristicWords 290
9.3 Application:UnusualContinuedFractions 291
9.4 Exercises 293
9.5 OpenProblems 295
9.6 NotesonChapter9 295
10 Subwords 298
10.1 Introduction 298
10.2 BasicPropertiesofSubwordComplexity 300
10.3 ResultsforAutomaticSequences 304
10.4 SubwordComplexityforMorphicWords 306
x Contents
10.5 SturmianWords 312
10.6 SturmianWordsandkth-Power-Freeness 320
10.7 SubwordComplexityofFiniteWords 323
10.8 Recurrence 324
10.9 UniformRecurrence 328
10.10 Appearance 333
10.11 Exercises 334
10.12 OpenProblems 340
10.13 NotesonChapter10 340
11 Cobham’sTheorem 345
11.1 SyndeticandRightDenseSets 345
11.2 ProofofCobham’sTheorem 347
11.3 Exercises 350
11.4 NotesonChapter11 350
12 FormalPowerSeries 351
12.1 Examples 352
12.2 Christol’sTheorem 354
12.3 FirstApplicationtoTranscendenceResults 359
12.4 FormalLaurentPowerSeriesandCarlitzFunctions 359
12.5 TranscendenceofValuesoftheCarlitz–GossGammaFunction 362
12.6 ApplicationtoTranscendenceProofsoverQ(X) 365
12.7 Furstenberg’sTheorem 367
12.8 Exercises 371
12.9 OpenProblems 375
12.10 NotesonChapter12 376
13 AutomaticRealNumbers 379
13.1 BasicPropertiesoftheAutomaticReals 379
13.2 Non-closurePropertiesof L(k,b) 382
13.3 Transcendence:AnAdHocApproach 385
13.4 TranscendenceoftheThue–MorseNumber 387
13.5 TranscendenceofMorphicRealNumbers 391
13.6 TranscendenceofCharacteristicRealNumbers 393
13.7 TheThue–MorseContinuedFraction 394
13.8 Exercises 400
13.9 OpenProblems 402
13.10 NotesonChapter13 403
14 MultidimensionalAutomaticSequences 405
14.1 TheSierpin´skiCarpet 405
14.2 FormalDefinitionsandBasicResults 408
14.3 SubwordComplexity 412
14.4 FormalPowerSeries 413
14.5 AutomaticSequencesinBase−1+i 414