Denotational Semantics: A Methodology for Language Development

304 Pages·2009·1.128 MB·

by  David A. Schmidt

Denotational
Semantics
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A METHODOLOGY FOR LANGUAGE DEVELOPMENT
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David A. Schmidt
Copyright notice: Copyright  1997 by David A. Schmidt. Permission to reproduce this
materialmustbeobtainedfromtheauthor.
Author’s address: David Schmidt, Department of Computing and Information Sciences, 234
NicholsHall,KansasStateUniversity,Manhattan,KS66506. [email protected]
Contents
Preface viii
Chapter0
INTRODUCTION ________________________________________________________________________________________________ 1
MethodsforSemanticsSpecification 2
SuggestedReadings 3
Chapter1
SYNTAX ______________________________________________________________________________________________________________ 5
1.1 AbstractSyntaxDefinitions 9
1.2 MathematicalandStructuralInduction 12
SuggestedReadings 15
Exercises 15
Chapter2
SETS, FUNCTIONS,ANDDOMAINS ______________________________________________________________ 17
2.1 Sets 17
2.1.1 ConstructionsonSets 18
2.2 Functions 20
2.2.1 RepresentingFunctionsasSets 21
2.2.2 RepresentingFunctionsasEquations 24
2.3 SemanticDomains 25
2.3.1 SemanticAlgebras 25
SuggestedReadings 27
Exercises 27
Chapter3
DOMAINTHEORYI:SEMANTICALGEBRAS ____________________________________________ 30
3.1 PrimitiveDomains 30
3.2 CompoundDomains 34
3.2.1 Product 34
3.2.2 DisjointUnion 35
3.2.3 FunctionSpace 39
3.2.4 LiftedDomainsandStrictness 42
iii
iv Contents
3.3 RecursiveFunctionDefinitions 44
3.4 RecursiveDomainDefinitions 46
3.5 Summary 46
SuggestedReadings 48
Exercises 48
Chapter4
BASICSTRUCTUREOFDENOTATIONALDEFINITIONS ________________________ 54
4.1 TheValuationFunction 54
4.2 FormatofaDenotationalDefinition 57
4.3 ACalculatorLanguage 59
SuggestedReadings 63
Exercises 63
Chapter5
IMPERATIVELANGUAGES ____________________________________________________________________________ 66
5.1 ALanguagewithAssignment 66
5.1.1 ProgramsareFunctions 72
5.2 AnInteractiveFileEditor 73
5.2.1 InteractiveInputandPartialSyntax 78
5.3 ADynamicallyTypedLanguagewithInputandOutput 80
5.4 AlteringthePropertiesofStores 82
5.4.1 DelayedEvaluation 82
5.4.2 RetainingMultipleStores 86
5.4.3 NoncommunicatingCommands 87
SuggestedReadings 88
Exercises 88
Chapter6
DOMAINTHEORYII:RECURSIVELYDEFINEDFUNCTIONS ________________ 94
6.1 SomeRecursivelyDefinedFunctions 95
6.2 PartialOrderings 98
6.3 ContinuousFunctions 102
6.4 LeastFixedPoints 103
6.5 DomainsareCpos 104
6.6 Examples 109
6.6.1 FactorialFunction 109
6.6.2 CopyoutFunction 110
6.6.3 DoubleRecursion 111
6.6.4 SimultaneousDefinitions 111
v
6.6.5 TheWhile-Loop 112
6.6.6 SoundnessofHoare’sLogic 115
6.7 ReasoningaboutLeastFixedPoints 117
SuggestedReadings 119
Exercises 120
Chapter7
LANGUAGESWITHCONTEXTS ____________________________________________________________________ 125
7.1 ABlock-StructuredLanguage 127
7.1.1 Stack-ManagedStorage 134
7.1.2 TheMeaningsofIdentifiers 136
7.2 AnApplicativeLanguage 137
7.2.1 ScopingRules 140
7.2.2 Self-Application 143
7.2.3 RecursiveDeclarations 145
7.3 CompoundDataStructures 148
SuggestedReadings 154
Exercises 154
Chapter8
ABSTRACTION,CORRESPONDENCE,ANDQUALIFICATION ________________ 160
8.1 Abstraction 161
8.1.1 RecursiveBindings 164
8.2 Parameterization 165
8.2.1 Polymorphism andTyping 167
8.3 Correspondence 170
8.4 Qualification 171
8.5 Orthogonality 173
SuggestedReadings 174
Exercises 174
Chapter9
CONTROLASASEMANTICDOMAIN __________________________________________________________ 178
9.1 Continuations 178
9.1.1 OtherLevelsofContinuations 181
9.2 ExceptionMechanisms 182
9.3 BacktrackingMechanisms 183
9.4 CoroutineMechanisms 183
9.5 UnrestrictedBranchingMechanisms 187
vi Contents
9.6 TheRelationshipbetweenDirectandContinuationSemantics 193
SuggestedReadings 195
Exercises 195
Chapter10
IMPLEMENTATIONOFDENOTATIONALDEFINITIONS ________________________ 199
10.1 AGeneralMethodofImplementation 199
10.1.1 TheSISandSPS Systems 200
10.2 StaticSemanticsProcessing 201
10.3 TheStructureoftheEvaluator 203
10.3.1 AStack-BasedEvaluator 204
10.3.2 PSP andAppel’sSystem 208
10.4 Combinator-BasedSemanticNotations 209
10.4.1 ThePlumbandCERESSystems 211
10.5 TransformationsontheSemanticDefinition 212
10.5.1 First-OrderDataObjects 212
10.5.2 GlobalVariables 214
10.5.3 ControlStructures 217
10.6 ImplementationofContinuation-BasedDefinitions 218
10.6.1 TheCGPandVDMMethods 222
10.7 CorrectnessofImplementationandFullAbstraction 223
SuggestedReadings 225
Exercises 226
Chapter11
DOMAINTHEORYIII:RECURSIVEDOMAINSPECIFICATIONS __________ 230
11.1 ReflexiveDomainsHaveInfiniteElements 230
11.2 TheInverseLimitConstruction 234
11.3 Applications 241
11.3.1 LinearLists 242
11.3.2 Self-ApplicativeProcedures 243
11.3.3 RecursiveRecordStructures 244
SuggestedReadings 245
Exercises 245
Chapter12
NONDETERMINISMANDCONCURRENCY ________________________________________________ 250
12.1 Powerdomains 251
12.2 TheGuardedCommandLanguage 251
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12.3 ConcurrencyandResumptionSemantics 254
12.4 AnAlternativeSemanticsforConcurrency 258
12.5 ThePowerdomainStructure 265
12.5.1 DiscretePowerdomains 266
12.5.2 GeneralPowerdomains 268
SuggestedReadings 274
Exercises 274
Bibliography 217
Preface
Denotational semantics is a methodology for giving mathematical meaning to programming
languages and systems. It was developed by Christopher Strachey’s Programming Research
GroupatOxfordUniversityinthe1960s.Themethodcombinesmathematicalrigor,duetothe
work of Dana Scott, with notational elegance, due to Strachey. Originally used as an analysis
tool, denotational semantics has grown in use as a tool for language design and implementa-
tion.
Thisbookwaswrittento makedenotationalsemanticsaccessibleto awideraudienceand
to update existing texts in the area. I havepresentedthe topic from an engineeringviewpoint,
emphasizing the descriptional and implementational aspects. The relevantmathematics is also
included, for it gives rigor and validity to the method and provides a foundation for further
research.
Thebookis intendedasatutorial forcomputing professionalsandasatext foruniversity
courses at the upper undergraduate or beginning graduate level. The reader should be
acquainted with discrete structures and one or more general purpose programming languages.
Experiencewithanapplicative-stylelanguagesuchasLISP,ML,orSchemeisalsohelpful.
CONTENTSOFTHEBOOK __________________________________________________________________________________________________
The Introduction and Chapters 1 through 7 form the core of the book. The Introduction pro-
vides motivation and a brief survey of semantics specification methods. Chapter 1 introduces
BNF, abstract syntax, and structural induction. Chapter 2 lists those concepts of set theory
thatarerelevanttosemanticdomaintheory.Chapter3coverssemanticdomains,thevaluesets
used in denotational semantics. The fundamental domains and their related operations are
presented. Chapter 4 introduces basic denotational semantics. Chapter 5 covers the semantics
ofcomputerstorageandassignmentasfoundinconventionalimperativelanguages. Nontradi-
tional methods of store evaluation are also considered. Chapter 6 presents least fixed point
semantics, which is used for determining the meaning of iterative and recursive definitions.
The related semantic domain theory is expanded to include complete partial orderings;
‘‘predomains’’ (complete partial orderings less ‘‘bottom’’ elements) are used. Chapter 7 cov-
ersblockstructureanddatastructures.
Chapters8through12presentadvancedtopics. Tennent’sanalysisofproceduralabstrac-
tionandgeneralbindingmechanismsisusedasafocalpointforChapter8.Chapter9analyzes
forms of imperative control and branching. Chapter 10 surveys techniques for converting a
denotational definition into a computer implementation. Chapter 11 contains an overview of
Scott’s inverse limit construction for building recursively defined domains. Chapter 12 closes
thebookwithanintroductiontomethodsforunderstandingnondeterminismandconcurrency.
Throughout the book I have consistently abused the noun ‘‘access,’’treating it as a verb.
Also,‘‘iff’’abbreviatesthephrase‘‘ifandonlyif.’’
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Preface ix
ORGANIZATIONOFACOURSE ________________________________________________________________________________________
Thebookcontainsmorematerialthan what canbe comfortablycoveredin one term. A course
plan should include the core chapters; any remaining time can be used for Chapters 8 through
12, which areindependent of one another and can be read in any order. The core can be han-
dledasfollows:
• Present the Introduction first. You may wish to give a one lecture preview of Chapter 4.
A preview motivates the students to carefully study the material in the background
Chapters1through3.
• CoverallofChapters1and2,astheyareshortandintroducecrucialconcepts.
• Use Chapter 3 as a ‘‘referencemanual.’’ You may wish to start at the summary Section
3.5 and outline the structures of semantic domains. Next, present examples of semantic
algebrasfromthebodyofthechapter.
• Cover all of Chapter 4 and at least Sections 5.1 and 5.4 from Chapter 5. If time allows,
coverallofChapter5.
• Summarize Chapter 6 in one or two lectures for an undergraduatecourse. This summary
canbetakenfromSection6.1. Agraduatecourseshouldcoverallofthechapter.
• CoverasmuchofChapter7aspossible.
REFERENCESANDEXERCISES ________________________________________________________________________________________
Following each chapter is a short list of references that suggests further reading. Each refer-
enceidentifies theauthorandtheyearofpublication. Lettersa, b, c,andsoon,areusedifthe
author has multiple references for a year. The references are compiled in the bibliography in
the back of the book. I have tried to make the bibliography current and complete, but this
appears to be an impossible task, and I apologize to those researchers whose efforts I have
unintentionallyomitted.
Exercises are provided for each chapter. The order of a chapter’s exercises parallels the
order of presentation of the topics in the chapter. The exercises are not graded according to
difficulty; an hour’s effort on a problem will allow the reader to make that judgment and will
alsoaiddevelopmentofintuitions aboutthesignificantproblemsinthearea.
ACKNOWLEDGEMENTS ____________________________________________________________________________________________________
Many people deserve thanks for their assistance, encouragement, and advice. In particular, I
thank Neil Jones for teaching me denotational semantics; Peter Mosses for answering my
questions; Robin Milner for allowing me to undertake this project while under his employ;
Paul Chisholm for encouraging me to write this book and for reading the initial draft; Allen
Stoughton formanystimulating discussions; Colin Stirling forbeing an agreeableofficemate;
andmyparents,family,andfriendsinKansas,foralmosteverythingelse.
John Sulzycki of Allyn and Bacon deserves special thanks for his interest in the project,
x Preface
and Laura Cleveland, Sue Freese, and Jane Schulman made the book’s production run
smoothly. The reviewers Jim Harp, Larry Reeker, Edmond Schonberg, and Mitchell Wand
contributed numerous useful suggestions. (I apologize for the flaws that remain in spite of
theirefforts.)Thoseinstructorsandtheirstudents who used preliminarydraftsas texts deserve
thanks;theyareJim Harp,AustinMelton,Colin Stirling, DavidWise,andtheirstudentsatthe
universities of Lowell, Kansas State, Edinburgh, and Indiana, respectively. My students at
Edinburgh,IowaState,andKansasStatealsocontributedusefulsuggestionsandcorrections.
Finally, the text would not have been written had I not been fortunate enough to spend
several years in Denmark and Scotland. I thank the people at Aarhus University, Edinburgh
University, Heriot-Watt University, and The Fiddler’s Arms for providing stimulating and
congenialenvironments.
Iwouldbepleasedtoreceivecommentsandcorrectionsfromthereadersofthisbook.