Instruction Sequences for Computer Science

241 Pages·2012·1.534 MB·English

by  J A Bergstra, Cornelis A Middelburg

ATLANTIS STUDIESINCOMPUTING
VOLUME2
SERIESEDITORS: JANA.BERGSTRA, MICHAELW.MISLOVE
Atlantis Studies in Computing
SeriesEditors:
JanA.Bergstra MichaelW.Mislove
InformaticsInstitute DepartmentofMathematics
UniversityofAmsterdam TulaneUniversity
Amsterdam,TheNetherlands NewOrleans,USA
(ISSN:2212-8565)
Aimsandscopeoftheseries
Theseriesaimsatpublishingbooksintheareasofcomputerscience,computerandnetwork
technology,ITmanagement,informationtechnologyandinformaticsfromthetechnologi-
cal,managerial,theoretical/fundamental,socialorhistoricalperspective.
Wewelcomebooksinthefollowingcategories:
Technicalmonographs:thesewillbereviewedastotimeliness,usefulness,relevance,com-
pletenessandclarityofpresentation.
Textbooks.
Booksofamorespeculativenature: thesewillbereviewedastorelevanceandclarityof
presentation.
Formoreinformationonthisseriesandourotherbookseries,pleasevisitourwebsiteat:
www.atlantis-press.com/publications/books
AMSTERDAM–PARIS –BEIJING
(cid:2)c ATLANTISPRESS
Instruction Sequences for
Computer Science
Jan A. Bergstra and Cornelis A. Middelburg
InstituteofInformatics,FacultyofScience, UniversityofAmsterdam
Amsterdam,theNetherlands
AMSTERDAM–PARIS –BEIJING
AtlantisPress
8,squaredesBouleaux
75019Paris,France
ForinformationonallAtlantisPresspublications,visitourwebsiteat:www.atlantis-press.com
Copyright
Thisbook,oranypartsthereof,maynotbereproducedforcommercialpurposesinanyformorby
anymeans,electronicormechanical,includingphotocopying,recordingoranyinformationstorage
andretrievalsystemknownortobeinvented,withoutpriorpermissionfromthePublisher.
AtlantisStudiesinComputing
Volume1:CodeGenerationwithTemplates-B.J.Arnoldus,M.G.J.VandenBrand,A.Serebrenik
ISBNs
Print: 978-94-91216-64-0
E-Book: 978-94-91216-65-7
ISSN: 2212-8565
(cid:2)c 2012ATLANTISPRESS
Preface
Theconceptofaninstructionsequenceisakeyconceptinpractice,butstrangelyenough
it has as yet not come prominentlyinto the picture in theoreticalcircles. In much work
oncomputerarchitecture,instructionsequencesareunderdiscussion. Inspiteofthis,the
notionofaninstructionsequencehasneverbeensubjectedtosystematicandpreciseanal-
ysis. Moreover, in work on computer architecture, the viewpoint is usually taken that a
programisinessenceaninstructionsequence. Bycontrast,inthetheoryofcomputation,
differentviewpointsonwhatisaprogramareusuallytaken. Thisstateofaffairsbrought
usto definea generalnotionofaninstructionsequence,to subjectit toa systematic and
preciseanalysis, andtoprovideevidenceforthehypothesisthatthenotionofaninstruc-
tionsequenceisrelevanttodiversesubjectsfromthetheoryofcomputationandthearea
ofcomputerarchitecture.Manyresultsoftheworkinquestionarebroughttogetherinthis
bookwiththeaimtobringinstructionsequencesasathemeincomputersciencebetterinto
thepicture.
Toputitotherwise,thisbookconcernsinstructionsequences,thebehavioursproduced
by instruction sequences under execution, the interaction between these behaviours and
componentsof the execution environmentconcerning the processing of instructions, the
expressiveness of instruction sequences, and various issues relating to well-known sub-
jects from computer science where we found that the notion of an instruction sequence
is relevant. Most of the issues in question are of a computation-theoretic or computer-
architecturalkind. Theyrelatetosubjectssuchasthehaltingproblem,non-uniformcom-
putationalcomplexity,instruction sequence performanceand instruction set architecture.
Someoftheissuesconsideredaresomehowrelatedtoprocessalgebra,namelyremotein-
struction processing and instruction sequence producible processes. Some variations on
instructionsequencesoftheusualkind,suchasinstructionsequenceswithoutadirectional
biasandprobabilisticinstructionsequences,arealsoconsidered.
v
vi InstructionSequencesforComputerScience
This book is primarily intended for researchers in computer science interested in in-
struction sequences as a theme in computer science. It is also meant to be suitable as
supplementaryreadingin coursesforgraduatestudentsand advancedundergraduatestu-
dentsincomputerscience.Chapters5and6mayasmuchappealtothosewhoareprimarily
interestedinthesubjectsfromthetheoryofcomputationandtheareaofcomputerarchitec-
ture,respectively,thatcomeupinthesechapters. Chapter7mayasmuchappealtothose
whoareprimarilyinterestedinprocessalgebra.
Throughoutthe book, some familiarity with equational logic, universal algebra, and
elementaryset theory is assumed. In Sect. 5.2, some familiarity with non-uniformcom-
putational complexity is assumed. In Sect. 5.1, Sect. 6.2 and Chap. 7, some familiarity
with computability,instruction set architecturesand processalgebra, respectively, would
behelpful. Chapter2isaprerequisiteforChap.3,andbothchaptersareprerequisitesfor
allsubsequentchapters.
Chapter 2 introducesan algebraic theorySPISA of single-pass instructionsequences
and an algebraic theory BTA of mathematical objects that represent in a direct way the
behavioursproducedbyinstructionsequencesunderexecution.Theobjectsconcernedare
called threads. Itis madeprecise in thesetting of the latter theorywhich behavioursare
producedbytheinstructionsequencesconsideredintheformertheory.Theinstructionse-
quencesinquestionincludebothfiniteandinfiniteones,butthetheoryprovidesanotation
by means of which all of them can be represented finitely. This chapter also introduces
alternative notations ISNR and ISNA by means of which all these instruction sequences
canberepresentedfinitelyaswell,butwhichareclosertoexistingassemblylanguages.
Chapter 3 introducesso-called services, which representthe behavioursexhibitedby
thecomponentsofanexecutionenvironmentthatarecapableofprocessingparticularin-
structions and doing so independently, and extends BTA with an operator meant for the
compositionoffamiliesofnamedservicesandoperatorsthathaveadirectbearingonthe
processing of instructions by services from such service families. In addition, the con-
cept of a functionalunit, which is an abstractmodelof a machine, is introduced. In the
frequently occurring case that the behaviours represented by services can be viewed as
thebehavioursofamachinein itsdifferentstates, theservicesconcernedarecompletely
determinedbyafunctionalunit. SomeextensionsofISNRandISNAwithadditionalin-
structionsareexplainedwiththehelpofsomesimplefunctionalunits.
Chapter4givesanswerstobasicexpressivenessissuesregardingSPISA. Inthiscase,
expressiveness is basically about which behaviours can be produced by instruction se-
Preface vii
quencesunderexecution,whichinstructionscanberemovedwithoutreducingtheclassof
behavioursthatcanbeproducedbyinstructionsequencesunderexecution,howtoenlarge
theclassofbehavioursthatcanbeproducedbyinstructionsequencesunderexecution,et
cetera. Thischapter is also concernedwith some issues thatarise fromthe investigation
ofexpressivenessissuesregardingSPISA. Forexample,itisshownthatafinite-stateexe-
cutionmechanismforasetofinstructionsequencesthatbyitselfcanproduceeachfinite-
statebehaviourfromaninstructionsequencebelongingtothesetofinstructionsequences
inquestionisunfeasible.
Chapter 5 concernstwo subjectsfrom the theoryof computation, namely the halting
problemandnon-uniformcomputationalcomplexity.PositioningTuring’sresultregarding
theundecidabilityofthehaltingproblemasaresultaboutprogramsratherthanmachines,
andtakingsingle-passinstructionsequencesasconsideredinSPISAasprograms,theau-
tosolvabilityrequirementthataprogramofacertainkindmustsolvethehaltingproblem
for all programs of that kind is analysed. Thinking in terms of single-pass instruction
sequencesas consideredin SPISA, counterpartsof the classical non-uniformcomplexity
classesP/polyandNP/polyaredefined,anotionofcompletenessforthecounterpartof
NP/polyisintroduced,severalcomplexityhypothesesareformulated,anditisshownthat
aproblemcloselyrelatedto3SATisNP-completeaswellascompleteforthecounterpart
ofNP/poly.
Chapter 6 concerns two subjects from the area of computer architecture, namely in-
struction sequence performanceand instruction set architectures. We study the effect of
eliminatingindirectjumpinstructionsfrominstructionsequenceswithdirectandindirect
jumpinstructionsontheinteractiveperformanceofinstructionsequences.Astrictversion
oftheconceptofaload/storeinstructionsetarchitectureisproposedfortheoreticalwork
relevant to the design of instruction set architectures, and it is studied how the transfor-
mationsonthestatesofthemainmemoryofastrictload/storeinstructionsetarchitecture
that can be achieved by executing instruction sequences on it depend on the parameters
involved.
Chapter7 concernstwo subjectsrelated to processalgebra, namelyprotocolsto deal
with remote instructionprocessing and instructionsequence producibleprocesses. If in-
struction processing takes place remotely, this means that a stream of instructions to be
processedarisesatoneplaceandtheprocessingofthatstream ofinstructionsishandled
atanotherplace. Processalgebraisusedtodescribetwoprotocolstodealwiththisphe-
nomenon. Becauseprocessalgebraisconsideredrelevanttocomputerscience,theremust
viii InstructionSequencesforComputerScience
beprogrammedsystemswhosebehavioursaretakenforprocessesasconsideredinprocess
algebra. Itisshownthatallfinite-stateprocessescanbeproducedbysingle-passinstruc-
tionsequencesasconsideredinSPISA,providedthattheclusterfairabstractionruleknown
fromthealgebraictheoryofprocessescalledACPisvalid.
Chapter8introducesthreevariationsofinstructionsequencesasconsideredinSPISA,
namely polyadic instruction sequences, instruction sequences without a directional bias,
andprobabilisticinstructionsequences. Apolyadicinstructionsequenceisapossiblypa-
rameterizedinstructionsequencefragmentthatcanproduceajointbehaviourtogetherwith
other such fragmentsbecause the fragmentbeing executed can switch over executionto
another. Instructionsequenceswithoutadirectionalbiasrequirethatforeachinstruction
whoseeffectinvolvesthatexecutionproceedsintheforwarddirection,thereisacounter-
partwhoseeffectinvolvesthatexecutionproceedsinthebackwarddirection.Probabilistic
instructionsequencesareinstructionsequencesthatcontaininstructionsthatarethemselves
probabilisticbynature.
Therearealso fourappendices. InAppendixA, fivechallengesforthepointofview
fromwhichtheapproachtosemanticsfollowedinChaps.2and3originatesaresketched.
InAppendixB,someresultsaboutfunctionalunitsfornaturalnumbersaregiven,whichare
exceptonecomputabilityresultsthatarenotdirectlyrelatedtoexistingresultsthatweknow
of. InAppendixC,theusefulnessofthedynamicallyinstantiatedinstructionsintroduced
inChap.3isillustratedbymeansofanexample.InAppendixD,amodelofahypothetical
executionenvironmentfor instruction sequences, designedfor the purpose of explaining
howinstructionsequencesasconsideredinSPISAmaybeexecuted,isdiscussed.
Aglossaryofthenotationsintroducedinthisbookandthegeneralmathematicalno-
tationsusedin thisbookcan befoundfrompage221onward. Atthispoint, onefurther
remarkaboutnotationmaybeuseful:bold-faceditalicletters,withorwithoutdecorations,
willbeusedassyntacticalvariablesinthisbook.
Acknowledgements
Thisbookbringstogetherandstreamlinesworkdonebyagroupofpeoplewhichincludes,
in addition to the authors, Inge Bethke, Marijke Loots, Alban Ponse and Mark van der
Zwaag. Theworkinquestionwaspartlycarriedoutintheframeworkofprojectsfunded
bytheNetherlandsOrganisationforScientificResearch(NWO).
Amsterdam,April2012 J.A.BergstraandC.A.Middelburg
Contents
Preface v
ListofTables xv
1. Introduction 1
2. InstructionSequences 5
2.1 SinglePassInstructionSequenceAlgebra . . . . . . . . . . . . . . . . . 5
2.1.1 Primitiveinstructions . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Constants,operatorsandequationalaxioms . . . . . . . . . . . 7
2.1.3 Theinitialmodel. . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.4 Structuralcongruence . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 BasicThreadAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Constants,operatorsandequationalaxioms . . . . . . . . . . . 12
2.2.2 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.3 Regularthreads . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.4 Theprojectivelimitmodel . . . . . . . . . . . . . . . . . . . . 18
2.2.5 Threadextractionforinstructionsequences . . . . . . . . . . . . 21
2.2.6 Behaviouralequivalenceofinstructionsequences . . . . . . . . 23
2.3 InstructionSequenceNotations . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 TheinstructionsequencenotationISNR . . . . . . . . . . . . . 26
2.3.2 TheinstructionsequencenotationISNA . . . . . . . . . . . . . 28
2.3.3 Inter-translatabilityofISNRandISNA . . . . . . . . . . . . . . 30
2.3.4 Additionalinstructionsequencenotations. . . . . . . . . . . . . 30
ix