Table Of ContentThe Haskell Road
to
Logic, Math and Programming
KeesDoetsandJanvanEijck
March4, 2004
Contents
Preface v
1 GettingStarted 1
1.1 StartinguptheHaskellInterpreter . . . . . . . . . . . . . . . . . 2
1.2 ImplementingaPrimeNumberTest . . . . . . . . . . . . . . . . 3
1.3 HaskellTypeDeclarations . . . . . . . . . . . . . . . . . . . . . 8
1.4 IdentifiersinHaskell . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 PlayingtheHaskellGame . . . . . . . . . . . . . . . . . . . . . 12
1.6 HaskellTypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7 ThePrimeFactorizationAlgorithm . . . . . . . . . . . . . . . . . 19
1.8 ThemapandfilterFunctions. . . . . . . . . . . . . . . . . . . 20
1.9 HaskellEquationsandEquationalReasoning . . . . . . . . . . . 24
1.10 FurtherReading . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 TalkingaboutMathematicalObjects 27
2.1 LogicalConnectivesandtheirMeanings . . . . . . . . . . . . . . 28
2.2 LogicalValidityandRelatedNotions. . . . . . . . . . . . . . . . 38
2.3 MakingSymbolicFormExplicit . . . . . . . . . . . . . . . . . . 50
2.4 LambdaAbstraction. . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5 DefinitionsandImplementations . . . . . . . . . . . . . . . . . . 60
2.6 AbstractFormulasandConcreteStructures . . . . . . . . . . . . 61
2.7 LogicalHandlingoftheQuantifiers . . . . . . . . . . . . . . . . 64
2.8 QuantifiersasProcedures . . . . . . . . . . . . . . . . . . . . . . 68
2.9 FurtherReading . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3 TheUseofLogic: Proof 71
3.1 ProofStyle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2 ProofRecipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3 RulesfortheConnectives . . . . . . . . . . . . . . . . . . . . . . 78
3.4 RulesfortheQuantifiers . . . . . . . . . . . . . . . . . . . . . . 90
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ii CONTENTS
3.5 SummaryoftheProofRecipes . . . . . . . . . . . . . . . . . . . 96
3.6 SomeStrategicGuidelines . . . . . . . . . . . . . . . . . . . . . 99
3.7 ReasoningandComputationwithPrimes. . . . . . . . . . . . . . 103
3.8 FurtherReading . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4 Sets,TypesandLists 113
4.1 Let’sTalkAboutSets . . . . . . . . . . . . . . . . . . . . . . . . 114
4.2 Paradoxes,TypesandTypeClasses . . . . . . . . . . . . . . . . . 121
4.3 SpecialSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.4 AlgebraofSets . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.5 PairsandProducts. . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.6 ListsandListOperations . . . . . . . . . . . . . . . . . . . . . . 139
4.7 ListComprehensionandDatabaseQuery . . . . . . . . . . . . . 145
4.8 UsingListstoRepresentSets . . . . . . . . . . . . . . . . . . . . 149
4.9 ADataTypeforSets . . . . . . . . . . . . . . . . . . . . . . . . 153
4.10 FurtherReading . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5 Relations 161
5.1 TheNotionofaRelation . . . . . . . . . . . . . . . . . . . . . . 162
5.2 PropertiesofRelations . . . . . . . . . . . . . . . . . . . . . . . 166
5.3 ImplementingRelationsasSetsofPairs . . . . . . . . . . . . . . 175
5.4 ImplementingRelationsasCharacteristicFunctions . . . . . . . . 182
5.5 EquivalenceRelations. . . . . . . . . . . . . . . . . . . . . . . . 188
5.6 EquivalenceClassesandPartitions . . . . . . . . . . . . . . . . . 192
5.7 IntegerPartitions . . . . . . . . . . . . . . . . . . . . . . . . . . 202
5.8 FurtherReading . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
6 Functions 205
6.1 BasicNotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.2 Surjections,Injections,Bijections . . . . . . . . . . . . . . . . . 218
6.3 FunctionComposition . . . . . . . . . . . . . . . . . . . . . . . 222
6.4 InverseFunction. . . . . . . . . . . . . . . . . . . . . . . . . . . 226
6.5 PartialFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 229
6.6 FunctionsasPartitions . . . . . . . . . . . . . . . . . . . . . . . 232
6.7 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
6.8 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
6.9 FurtherReading . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
7 InductionandRecursion 239
7.1 MathematicalInduction . . . . . . . . . . . . . . . . . . . . . . . 239
7.2 RecursionovertheNaturalNumbers . . . . . . . . . . . . . . . . 246
7.3 TheNatureofRecursiveDefinitions . . . . . . . . . . . . . . . . 251
CONTENTS iii
7.4 InductionandRecursionoverTrees . . . . . . . . . . . . . . . . 255
7.5 InductionandRecursionoverLists . . . . . . . . . . . . . . . . . 265
7.6 SomeVariationsontheTowerofHanoi . . . . . . . . . . . . . . 273
7.7 InductionandRecursionoverOtherDataStructures . . . . . . . . 281
7.8 FurtherReading . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
8 WorkingwithNumbers 285
8.1 AModuleforNaturalNumbers. . . . . . . . . . . . . . . . . . . 286
8.2 GCDandtheFundamentalTheoremofArithmetic . . . . . . . . 289
8.3 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
8.4 ImplementingIntegerArithmetic . . . . . . . . . . . . . . . . . . 297
8.5 RationalNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . 299
8.6 ImplementingRationalArithmetic . . . . . . . . . . . . . . . . . 305
8.7 IrrationalNumbers . . . . . . . . . . . . . . . . . . . . . . . . . 309
8.8 TheMechanic’sRule . . . . . . . . . . . . . . . . . . . . . . . . 313
8.9 ReasoningaboutReals . . . . . . . . . . . . . . . . . . . . . . . 315
8.10 ComplexNumbers . . . . . . . . . . . . . . . . . . . . . . . . . 319
8.11 FurtherReading . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
9 Polynomials 331
9.1 DifferenceAnalysisofPolynomialSequences . . . . . . . . . . . 332
9.2 GaussianElimination . . . . . . . . . . . . . . . . . . . . . . . . 337
9.3 PolynomialsandtheBinomialTheorem . . . . . . . . . . . . . . 344
9.4 PolynomialsforCombinatorialReasoning . . . . . . . . . . . . . 352
9.5 FurtherReading . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
10 Corecursion 361
10.1 CorecursiveDefinitions . . . . . . . . . . . . . . . . . . . . . . . 362
10.2 ProcessesandLabeledTransitionSystems . . . . . . . . . . . . . 365
10.3 ProofbyApproximation . . . . . . . . . . . . . . . . . . . . . . 373
10.4 ProofbyCoinduction . . . . . . . . . . . . . . . . . . . . . . . . 379
10.5 PowerSeriesandGeneratingFunctions . . . . . . . . . . . . . . 385
10.6 ExponentialGeneratingFunctions . . . . . . . . . . . . . . . . . 396
10.7 FurtherReading . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
11 FiniteandInfiniteSets 399
11.1 MoreonMathematicalInduction . . . . . . . . . . . . . . . . . . 399
11.2 Equipollence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
11.3 InfiniteSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
11.4 Cantor’sWorldImplemented . . . . . . . . . . . . . . . . . . . . 418
11.5 CardinalNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . 420
iv CONTENTS
TheGreekAlphabet 423
References 424
Index 428
Preface
Purpose
Long ago, when Alexander the Great asked the mathematician Menaechmus for
a crash course in geometry, he got the famous reply “There is no royal road to
mathematics.” Where there was no shortcut for Alexander, there is no shortcut
forus. Still,thefactthatwehaveaccesstocomputersandmatureprogramming
languages means that there are avenues for us that were denied to the kings and
emperorsofyore.
Thepurposeofthisbookistoteachlogicandmathematicalreasoninginpractice,
andtoconnectlogicalreasoningwithcomputerprogramming. Theprogramming
language that will be our tool for this is Haskell, a member of the Lisp family.
Haskellemergedinthelastdecadeasastandardforlazyfunctionalprogramming,
aprogrammingstylewhereargumentsareevaluatedonlywhenthevalueisactu-
allyneeded. Functionalprogrammingisaformofdescriptiveprogramming,very
different from the style of programming that you find in prescriptive languages
likeCorJava. Haskellisbasedonalogicaltheoryofcomputablefunctionscalled
thelambdacalculus.
Lambdacalculusisaformallanguagecapableofexpressingarbitrary
computablefunctions. Incombinationwithtypesitformsacompact
waytodenoteontheonehandfunctionalprogramsandontheother
handmathematicalproofs.[Bar84]
Haskellcanbeviewedasaparticularlyelegantimplementationofthelambdacal-
culus. It is a marvelousdemonstration tool for logic and math because its func-
tionalcharacterallowsimplementationstoremainveryclosetotheconceptsthat
getimplemented,whilethelazinesspermitssmoothhandlingofinfinitedatastruc-
tures.
v
vi
Haskellsyntax is easy to learn, andHaskell programsare constructedand tested
in a modular fashion. This makes the language well suited for fast prototyping.
Programmers find to their surprise that implementation of a well-understood al-
gorithm in Haskell usually takes far less time than implementation of the same
algorithminotherprogramminglanguages. Gettingfamiliarwithnewalgorithms
throughHaskellisalsoquiteeasy. LearningtoprograminHaskellislearningan
extremelyusefulskill.
Throughout the text, abstract concepts are linked to concrete representations in
Haskell. Haskellcomeswithaneasytouseinterpreter,Hugs. Haskellcompilers,
interpretersanddocumentationarefreelyavailablefromtheInternet[HT]. Every-
thingonehastoknowaboutprogramminginHaskelltounderstandtheprograms
inthe bookis explainedaswegoalong, butwe donotcovereveryaspectofthe
language.ForafurtherintroductiontoHaskellwereferthereaderto[HFP96].
Logic in Practice
The subject of this book is the use of logic in practice, more in particular the
useoflogicinreasoningaboutprogrammingtasks. Logicisnottaughthereasa
mathematicaldisciplineperse,butasanaidintheunderstandingandconstruction
of proofs, and as a tool for reasoning about formal objects like numbers, lists,
trees, formulas, and so on. As we go along, we will introduce the concepts and
tools that form the set-theoretic basis of mathematics, and demonstrate the role
of these concepts and tools in implementations. These implementations can be
thoughtofasrepresentationsofthemathematicalconcepts.
Althoughitmaybearguedthatthelogicthatisneededforaproperunderstanding
ofreasoninginreasonedprogrammingwillgetacquiredmoreorlessautomatically
in the process of learning (applied) mathematics and/or programming, students
nowadaysenteruniversitywithoutanyexperiencewhatsoeverwithmathematical
proof,thecentralnotionofmathematics.
TherulesofChapter3representadetailedaccountofthestructureofaproof.The
purposeofthisaccountistogetthestudentacquaintedwithproofsbyputtingem-
phasisonlogicalstructure. Thestudentisencouragedtowrite“detailed”proofs,
witheverylogicalmovespelledoutinfull. Thenextgoalistomoveontowriting
“concise”proofs,in thecustomarymathematicalstyle, whilekeepingthe logical
structureinmind. Oncethestudenthasarrivedatthisstage,mostofthelogicthat
isexplainedinChapter3cansafelybeforgotten,orbetter,cansafelyfadeintothe
subconsciousnessofthematuredmathematicalmind.
PREFACE vii
Pre- and Postconditions of Use
Wedonotassumethatourreadershavepreviousexperiencewitheitherprogram-
mingorconstructionofformalproofs. Wedoassumepreviousacquaintancewith
mathematical notation, at the level of secondary school mathematics. Wherever
necessary,wewillrecallrelevantfacts.Everythingoneneedstoknowaboutmath-
ematical reasoning or programmingis explained as we go along. We do assume
that our readers are able to retrievesoftware from the Internet and install it, and
thattheyknowhowtouseaneditorforconstructingprogramtexts.
After having workedthrough the materialin the book, i.e., after having digested
the text and having carried out a substantial number of the exercises, the reader
willbeabletowriteinterestingprograms,reasonabouttheircorrectness,anddoc-
ument them in a clear fashion. The reader will also have learned how to set up
mathematicalproofsinastructuredway,andhowtoreadanddigestmathematical
proofswrittenbyothers.
How to Use the Book
Chapters 1–7 of the book are devoted to a gradual introduction of the concepts,
toolsandmethodsofmathematicalreasoningandreasonedprogramming.
Chapter 8 tells the story of how the various number systems (natural numbers,
integers, rationals, reals, complex numbers) can be thought of as constructed in
stagesfromthenaturalnumbers.Everythinggetslinkedtotheimplementationsof
thevariousHaskelltypesfornumericalcomputation.
Chapter 9 starts with the question of how to automate the task of finding closed
formsforpolynomialsequences.Itisdemonstratedhowthistaskcanbeautomated
withdifferenceanalysisplusGaussianelimination. Next,polynomialsareimple-
mentedaslistsoftheircoefficients,withtheappropriatenumericaloperations,and
itisshownhowthisrepresentationcanbeusedforsolvingcombinatorialproblems.
Chapter10providesthefirstgeneraltextbooktreatment(asfarasweknow)ofthe
importanttopicofcorecursion.Thechapterpresentstheproofmethodssuitablefor
reasoningaboutcorecursivedatatypeslikestreamsandprocesses,andthengoes
ontointroducepowerseriesasinfinitelistsofcoefficients,andtodemonstratethe
usesofthisrepresentationforhandlingcombinatorialproblems. Thisgeneralizes
theuseofpolynomialsforcombinatorics.
Chapter 11 offers a guided tour through Cantor’s paradise of the infinite, while
providingextrachallengesintheformofawiderangeofadditionalexercises.
viii
The book can be used as a course textbook, but since it comes with solutions to
allexercises(electronicallyavailablefromtheauthorsuponrequest)itisalsowell
suitedforprivatestudy. CoursesbasedonthebookcouldstartwithChapters1–7,
andthenmakeachoicefromtheremainingChapters. Herearesomeexamples:
RoadtoNumericalComputation Chapters1–7,followedby8and9.
RoadtoStreamsandCorecursion Chapters1–7,followedby9and10.
RoadtoCantor’sParadise Chapters1–7,followedby11.
Study of the remaining parts of the book can then be set as individual tasks for
studentsreadyforanextrachallenge. Theguidelinesforsettingupformalproofs
in Chapter 3 should be recalled from time to time while studying the book, for
properdigestion.
Exercises
Partsofthetextandexercisesmarkedbya*aresomewhatharderthantherestof
thebook.
All exercises are solved in the electronically avaible solutions volume. Before
turningtothesesolutions,oneshouldreadtheImportantAdvicetotheReaderthat
thisvolumestartswith.
Book Website and Contact
TheprogramsinthisbookhaveallbeentestedwithHugs98,theversionofHugs
thatimplementstheHaskell98standard. Thefullsourcecodeofallprogramsis
integrated in the book; in fact, each chapter can be viewed as a literate program
[Knu92] in Haskell. The source code of all programs discussed in the text can
be foundon the website devotedto thisbook, ataddress http://www.cwi.nl/
~jve/HR.Hereyoucanalsofindalistoferrata,andfurtherrelevantmaterial.
Readerswhowanttosharetheircommentswiththeauthorsareencouragedtoget
[email protected].
PREFACE ix
Acknowledgments
Remarksfromthepeoplelistedbelowhavesparkedoffnumerousimprovements.
ThankstoJohanvanBenthem,JanBergstra,JacobBrunekreef,ThierryCoquand
(whofoundthelecturenotesontheinternetandsentushiscomments),Timvan
Erven, Wan Fokkink, Evan Goris, Robbert de Haan, Sandor Heman, Eva Hoog-
land, Rosalie Iemhoff, Dick de Jongh, Anne Kaldewaij, Breannda´n O´ Nualla´in,
Alban Ponse, Vincent van Oostrom, Piet Rodenburg, Jan Rutten, Marco Swaen,
JanTerlouw,JohnTromp,YdeVenema,AlbertVisser andStephanieWehnerfor
suggestionsand criticisms. The beautifulimplementation of the sieve of Eratos-
thenesinSection3.7wassuggestedtousbyFer-JandeVries.
The course on which this book is based was developedat ILLC (the Institute of
Logic, Language and Computation of the University of Amsterdam) with finan-
cial support from the Spinoza Logic in Action initiative of Johan van Benthem,
whichisherewithgratefullyacknowledged.WealsowishtothankILLCandCWI
(Centrum voor Wiskunde en Informatica, or Centre for Mathematics and Com-
puter Science, also in Amsterdam), the home institute of the second author, for
providing us with a supportive working environment. CWI has kindly granted
permissiontoreusematerialfrom[Doe96].
ItwasKrzysztofAptwho,perceivingtheneedofadeadline,spurredusontoget
intouchwithapublisherandputourselvesundercontract.
