Mathematics in Computing: An Accessible Guide to Historical, Foundational and Application Contexts

293 Pages·2012·5.687 MB·English

by  Gerard O’Regan

Mathematics in Computing
Gerard O’Regan
Mathematics in Computing
An Accessible Guide to Historical,
Foundational and Application Contexts
2123
GerardO’Regan
Mallow,Ireland
ISBN978-1-4471-4533-2 ISBN978-1-4471-4534-9(eBook)
DOI10.1007/978-1-4471-4534-9
SpringerLondonHeidelbergNewYorkDordrecht
LibraryofCongressControlNumber:2012951294
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MaryRose,Donal,FrancisandMarguerita
Preface
Overview
The objective of this book is to give the reader a flavour of mathematics used in
thecomputingfield.Thegoalistoshowhowmathematicsisappliedincomputing,
ratherthanthestudyofmathematicsforitsownsake.
OrganizationandFeatures
Thefirstchapterdiscussesthecontributionsmadebyearlycivilisationstocomputing.
ThisincludesworkdonebytheBabylonians,EgyptiansandGreeks.TheEgyptians
appliedmathematicstosolvepracticalproblemssuchastheconstructionofpyramids.
The Greeks made a major contribution to mathematics and geometry, and most
studentsarefamiliarwiththeworkofEuclid.
Chapter2providesanintroductiontofundamentalbuildingblocksinmathematics
includingsets,relationsandfunctions.Asetisacollectionofwell-definedobjects
and it may be finite or infinite. A relation between two sets A and B indicates a
releationship between members of the two sets, and is a subset of the Cartesian
product of the two sets.A function is a special type of relation such that for each
elementinAthereisatthemostoneelementintheco-domainB.Functionsmaybe
partialortotalandinjective,surjectiveorbijective.
Chapter3providesanintroductiontologicincludingpropositionalandpredicate
logic.Thenatureofmathematicalproofisdiscussed.
Chapter4providesanintroductiontotheimportantfieldofsoftwareengineering.
The birth of the discipline was at the Garmisch conference in Germany in the late
1960s.Theextenttowhichmathematicsshouldbeemployedinsoftwareengineering
isdiscussed,andthisremainsatopicofactivedebate.
Chapter 5 discusses formal methods, which consist of a set of mathematical
techniquestospecifyandderiveaprogramfromitsspecification. Formalmethods
maybeemployedtorigorouslystatetherequirementsoftheproposedsystem;they
maybeemployedtoderiveaprogramfromitsmathematicalspecification;andthey
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viii Preface
provide a rigorous proof that the implemented program satisfies its specification.
Theyhavebeenmainlyappliedtothesafetycriticalfield.
Chapter6presentstheZspecificationlanguage,whichisoneofthemostwidely
usedformalmethods.ItwasdevelopedatOxfordUniversityintheUK.
Chapter7presentsthefundamentalsofnumbertheory,anddiscussesprimenum-
ber theory and the greatest common divisor and least common multiple of two
numbers.
Chapter8discussescryptography, whichisanimportantapplicationofnumber
theory.ThecodebreakingworkdoneatBletchleyParkinEnglandduringtheSecond
WorldWarisdiscussed,andthefundamentalsofcryptography,includingprivateand
publickeycryptosystems,arediscussed.
Chapter9presentscodingtheoryandisconcernedwitherrordetectionanderror
correctioncodes.Theunderlyingmathematicsisdiscussed,andthisincludesabstract
mathematicssuchasgrouptheory,rings,fields,andvectorspaces.
Chapter 10 discusses language theory and includes a discussion on grammar,
parse trees, and derivations from a grammar. The important area of programming
languagesemanticsisdiscussed, includinganoverviewofaxiomatic, denotational
andoperationalsemantics.
Chapter11discussescomputabilityanddecideability.TheChurch-Turingthesis
statesthatanythingthatiscomputableiscomputablebyaTuringmachine.Church
andTuringshowedthatmathematicsisnotdecideable. Inotherwords, thereisno
mechanicalprocedure(i.e.,algorithm)todeterminewhetheranarbitrarymathemati-
calpropositionistrueorfalse,andsotheonlywayistodeterminethetruthorfalsity
ofastatementistrytosolvetheproblem.
Chapter12discussesprobabilityandstatisticsandincludesadiscussionondis-
crete and continuous random variables, probability distributions, sample spaces,
sampling, the abuse of statistics, variance and standard deviation, and hypothesis
testing.Theapplicationofprobabilitytothesoftwarereliabilityfieldisdiscussed.
Chapter13discussesmatricesincluding2×2andgeneraln×mmatrices.Various
operations such as the addition and multiplication of matrices are considered, and
thedeterminantandinverseofamatrixisdiscussed.Theapplicationofmatricesto
solveasetoflinearequationsusingGaussianeliminationiscosidered.
Chapter 14 discusses complex numbers and quaternions. Complex numbers of
the form a+bi where a and b are real numbers, and i2 = −1. Quaternions are a
generalizationofcomplexnumberstoquadruplesthatsatisfythequaternionformula
i2=j2=k2 =−1.
Chapter15providesaveryshortintroductiontocalculus,andprovidesahigh-level
overview of limits, continuity, differentiation, integration, and numerical analysis.
Fourierseries,Laplacetransformsanddifferentialequationsarebrieflydiscussed.
Chapter16discussesgraphtheorywhereagraphG=(V,E)consistsofvertices
andedges.Itisapracticalbranchofmathematicsthatdealswiththearrangements
of vertices and the edges between them. It has been applied to practical problems
suchasthemodelingofcomputernetworks,determiningtheshortestdrivingroute
betweentwocities,andthetravelingsalesmanproblem.
Audience
Theaudienceofthisbookincludescomputersciencestudentswhowishtoobtainan
overviewofmathematicsusedincomputing, andmathematicianswhowishtoget
an overview of how mathematics is applied in the computing field. The book will
alsobeofinteresttothemotivatedgeneralreader.
Acknowledgments
I am deeply indebted to my family and friends who supported my efforts in this
endeavour.
Cork,Ireland GerardO’Regan
ix
Contents
1 MathematicsinCivilization ..................................... 1
1.1 Introduction ............................................... 1
1.2 TheBabylonians ........................................... 3
1.3 TheEgyptians ............................................. 6
1.4 TheGreeks................................................ 8
1.5 TheRomans ............................................... 16
1.6 IslamicInfluence ........................................... 19
1.7 ChineseandIndianMathematics.............................. 20
1.8 ReviewQuestions .......................................... 21
1.9 Summary ................................................. 21
2 Sets,RelationsandFunctions ................................... 23
2.1 Introduction ............................................... 23
2.2 SetTheory ................................................ 24
2.2.1 SetTheoreticalOperations............................. 26
2.2.2 PropertiesofSetTheoreticalOperations ................. 28
2.2.3 Russell’sParadox .................................... 29
2.3 Relations.................................................. 30
2.3.1 Reflexive,SymmetricandTransitiveRelations............ 32
2.3.2 CompositionofRelations ............................. 34
2.3.3 BinaryRelations ..................................... 35
2.4 Functions ................................................. 36
2.5 ReviewQuestions .......................................... 40
2.6 Summary ................................................. 41
3 Logic ......................................................... 43
3.1 Introduction ............................................... 43
3.2 PropositionalLogic......................................... 45
3.2.1 TruthTables......................................... 47
3.2.2 PropertiesofPropositionalCalculus..................... 49
3.2.3 ProofinPropositionalCalculus......................... 50
3.2.4 ApplicationsofPropositionalCalculus .................. 54
3.2.5 LimitationsofPropositionalCalculus ................... 55
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xii Contents
3.3 PredicateCalculus.......................................... 55
3.3.1 FormalisationofPredicateCalculus..................... 58
3.3.2 InterpretationandValuationFunctions................... 59
3.3.3 PropertiesofPredicateCalculus ........................ 60
3.3.4 ApplicationsofPredicateCalculus...................... 60
3.4 UndefinedValues........................................... 61
3.4.1 LogicofPartialFunctions ............................. 62
3.4.2 ParnasLogic ........................................ 63
3.4.3 DijkstraandUndefinedness............................ 65
3.5 OtherLogics............................................... 66
3.6 ToolsforLogic............................................. 68
3.7 ReviewQuestions .......................................... 69
3.8 Summary ................................................. 69
4 SoftwareEngineering........................................... 71
4.1 Introduction ............................................... 71
4.2 WhatisSoftwareEngineering? ............................... 73
4.3 EarlySoftwareEngineering .................................. 78
4.4 SoftwareEngineeringMathematics............................ 81
4.5 FormalMethods ........................................... 82
4.6 SoftwareInspectionsandTesting.............................. 83
4.7 ProcessMaturityModels .................................... 85
4.8 ReviewQuestions .......................................... 86
4.9 Summary ................................................. 86
5 FormalMethods ............................................... 89
5.1 Introduction ............................................... 89
5.2 WhyShouldWeUseFormalMethods?......................... 91
5.3 ApplicationsofFormalMethods .............................. 92
5.4 ToolsforFormalMethods ................................... 93
5.5 ApproachestoFormalMethods ............................... 94
5.5.1 Model-OrientedApproach............................. 94
5.5.2 AxiomaticApproach.................................. 95
5.6 ProofandFormalMethods................................... 95
5.7 TheFutureofFormalMethods................................ 96
5.8 TheViennaDevelopmentMethod ............................. 97
5.9 VDM♣,theIrishSchoolofViennaDevelopmentMethod(VDM)... 98
5.10 TheZSpecificationLanguage ................................ 99
5.11 TheB-Method ............................................. 100
5.12 PredicateTransformersandWeakestPre-Conditions ............. 101
5.13 TheProcessCalculi......................................... 102
5.14 FiniteStateMachines ....................................... 103
5.15 TheParnasWay ............................................ 104
5.16 UsabilityofFormalMethods ................................. 105
5.16.1 WhyAreFormalMethodsDifficult? .................... 105
5.16.2 CharacteristicsofaUsableFormalMethod............... 106