Table Of ContentAlgebraiSce manticosf I mperatiPvreo grams
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FoundatioonfsC omputing
MichaGealr eayn dA lbeMrety eerd,i tors
ComplexiItsys ueisn VLSI:O ptimaLla youtfso rt heS huffle-ExchanGgraep ha ndO thre
NetworkFsr,a nTkh omsonL eighto1n9,8 3
EquationLaolg iacs a ProgrammiLnagn guagMei,c haeJl.O 'Donnell1,98 5
GeneraTlh eoroyf D eductiSvyeste ms andI tsA pplicatiSo.nY su, M aslov1,9 87
ResourAclel ocatPiroonb lemAsl:g orithAmpipcr oaches,T oshihiIdbea raaknid N a okiK a
toh1,9 88
AlgebraTiheco ryo fP roceesssM,a ttheHwe nness1y9,8 8
PX:A ComputatioLnogaicl, SusumuH ayashain dH irosNhia kan19o8,9
TheS table MarriagPer obleSmt:r uctuarnedA lgorithDmasn,G usfieladn dR obertI rving,
1989
RealisCtoimcp ileGre neratiPoetne,Lr e e, 1989
Signl-eLayer WireR outing andC ompactino, F.M illeMral ey,1 990
BasiCca tegoTrhye orfyor ComputeSrc ientiBsetnsj,a miCn. P ierc1e9,9 1
CategoriTeysp,e sa,n dS tructurAens I:n trdouctiotnoC ategorThye ory fort heW orking
ComputeSrc ientiAsndtr,e aA spertanid G iuseppLeon go, 1991
SemanotfiPr cgosram ming LanguageSst:r uctuarnedsT echniquCeasr,lA .G unter1,9 92
TheF ormalS emanotfPir cogsram ming LanguageAsn: I ntroductGilyonn,nW inske1l9,93
HilberTte'nst Phr obleYmu, rVi. M atiyasev1i9c9h,. 1
ExploringI nterior-PLoiinnetaP rr ogramminAgl:g orithamnsd ftSoware, AmiA rbel, 1993
TheoerticlaA spectosf O bject-OriePnrtoegdr amminTgy:p es,S emanticasn,d L anguage
Desigend,i tebdy C arlA .G untearn dJ ohnC .M itche1l9l9,4
FromL ogitco L ogicP rgoramming, KeesD oets1,9 94
TheS tructuorfTe ypPerdo gramLamnigunagges , DavidA .S chmid1t9,9 4
Logiacn dI nformatFiloown, editebdy Janv aEni jacnkdA lberVti sse1r9,9 4
CircuCiotm plexiatnydN euralN etworkIasn,P arberr1y9,94
ContrFollo wS emantiJcasc,od eB akkearn dE ridke Vin1k9,9 6
AlgebraSeimca nticso fI mperatPirvoeg ramJsos,e phA .G oguena nGdr antM alcolm1,9 96
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AlgebraiSce manticosfI mperatiPvreo grams
JosepAh.G oguen andG ranMta lcolm
TheM ITP ress
CambridMgaes,s achusetts
LondoEnn,gl and
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© 1996M assachuseItntsst itouftT ee chnology
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LibraroyfC ongresCsa taloging-in-PubDlaitcaa tion
GoguenJ,o sepAh.
Algebrasiemca ntisco fi mperatipvroeg rams/ J oseph A. Goguena ndG rantM alcolm.
p.c m.-(Foundatioonfcs o mputi)ng
Includbeisb liograprhiecfaelr enacnedsi ndex.
ISBN0 -262-07(17h2ae-l:Xk.p ape)r
1.P rogramming languages( Electrocnoimcp ute)r-sSemanticIs..M alcolmG,r antI.I T.i tle.
IISIe.r ies.
QA76.7.G621 996
005.13'1-dc20 95-47440
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Contents
SeriFeosr eword IX
0 Introduction 1
1 Backgrounidn G eneralA lgebraan d OBJ 11
1.1 Signatreus 11
1.2A lgebras 16
1.3T erms 17
1.4V ariables 21
1.5 Equations 24
1.6R ewritianngdE quatioDneadlu ction 29
1.6A.t1tr ibuotfeo sp eratsi on 32
1.6D.e2n otatisoenmaaln tifcosro bjects 36
1.6.T3h eT heoreomf C onstants 38
1.7I mportgi Mnodules 40
1.8L iterature 43
1.9E xercises 43
2 Stores, VariaVballeuse,sa ,n d Assignment 51
2.1S toreVsa,r iablaensdV, a lues 51
2.1O.B1J 'sb uilt-in inequality 55
2.2A sisgnment 60
2.3E xercises 64
3 Compositioann d Conditionals 67
70
3.1S equentCioamlp osition
3.2C onditionals 71
33. Structural Induction 74
76
3.4 Exerceiss
79
4 ProvinPgr ogramC orrectness
4.1E xampleA:b soluVtael ue 83
Copyrighted Material
vi Contents
86
4.2 ExamplCe:o mputitnhge M aximumo fT woV alues
88
4.3 Exercises
91
5 Iteration
92
5.1 Invariants
98
5.1.Ex1a mpleg:rea tetsc ommon divisor
5.2 102
Termination
5.3E xercises 105
109
6 Arrays
6.1 115
SomeS implEex amples
6.2 118
Exercises
6.3 119
SpecificatainodPn rso ofs
6.4 126
Exercises
3
7 Procedures 11
133
7.1 Non-recurPsriovcee dures
7.1.1 133
Proceduwreistn ho parameters
7.1.2 139
Proceduwrietsvh a r-parameters
7.1.Pr3oc edures witehx p-parameters 144
7.2R ecursPirvoec edures 184
7.2P.r1oc eduwrietsnh o p arameters 149
7.2P.ro2ce dures witvha r-parameters 156
7.3 162
Exercises
169
8 Some Compariswoint hO therA pproaches
A Summaryo ft heS emantics 175
185
B FirsOtr derL ogica ndI nduction
C OrderS ortedA lgebra 203
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Contents Vll
209
D OBJ3S yntax
215
E InstructoGrusi'd e
223
Bibliography
227
Index
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SerieFso reword
Theoreticcoamlp utesrc ienhc�se nowu ndergosneev erdaelc adoefsd evelopment.
The "classitcoapli"co sfa utomattah eorfyo,r malla nguagaensd,c omputational
complexhiatvyeb ecomfier mleys tablisahnedtd h,e iirm portatnoco et hetrh eoreti
cawlo rka ndt op ractiiscw ei delrye cogniSzteidm.u labtyet de chnoloagdivcaanlc es,
theoretihcaivabene se nr apidelxyp anditnhgea reausn desrt udayn,d t het imdee lay
betweetnh eoretical apnrdoi gtrpser sasc tiicmapla chta sb eend ecreasdirnagm at
icallMyu.c hp ublichiatsyb eeng iverne centtloby r eakthroiungc hrspy tography
andl ineparro grammianngd,s teadpyr ogreissbs e inmga deo np rogrammilnagn
guagsee manticcosm,p utatigoenoamle trayn,d e fficiendta tas tructurNeesw.e r,
mores peculatairveea,os f s tudiyn clurdeel atiodnaatla basVeLsS,I t heorayn,d
paralalnedld istribcuotmepdu tatiAosnt .h ilsi sotft opiccosn tineuxepsa nding,
iti sb ecominmgo rea ndm ored ifficutlots taayb reaosftt hep rogretshsa its b e
ingm adea ndi ncreasiinmgployr tathnatt t hem osts ignificwaonrtkb ed istilled
andc ommunicaitnea d m annetrh awti lfla cilitate furtahnedra prpelsiecaartciho n
oft hiwso rkB.y publishcionmgp rehenbsoiovkea sn ds pecialmiozneodg rapohns
thet heoretaiscpaelc otfsc omputsecri entchee,s erioensF oundatioofnC so mput
ingp rovidae fso ruimn w hicihm portarnets eartcohp iccasn b ep resentientd h eir
entiraentdyp laceidnp erspecftoirrve es earchsetrusd,e natnsd,p ractitiaolnie.k res
MichaeRl.G arey
AlberRt. M eyer
CopyrighMtaetde rial
AlgebraiSce manticosfI mperatiPvreo grams
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o
Introduction
Thibso oiksi nt endedt oi ntrodunudecrger audate Computing Sciencset udents
tof ormarle asonianbgo uti mperatiprvoegr ams. Oursp ecfiicg oasli ncltuhdee
following:
1.im prove intuitainodna biliyt ini mperatvie rpogramming, through under-
standgi tnhes emanictso fp rogramsa nds eieng numerouesx amples;
2.t eahcohw top rovper opertoifep sr ograms;
3.d evelop ther elevamnatt hematibcacaklgr ound;
4.p resetnhte O BJs3y staendmu sei tf ora lplr oofasn;d
5.sh ow thatth icsa nb ed onei n waa yt hait sco mplretiegloyry toen uostt,o o
difficulotrt ooa bstract, byu sing equationloagilc , whiicsshim plyt he olgic
ofs ubstituetqiuanlsg f oreq uals.
Thaelg ebarics emantiofci sm perativep rogriasme sscrd ibed bys peciaf cyliansgs
ofab stramcatc hinaensd g ivinge quatoniala xiowmhsi csphec fiyt hee ffecotf p ro
gramso ns uchm achineTsh.e p rogrammilnangg uage featurterse atiendt heb ook
area:s signmesenqtue,n tialc omposition, conditpriocoednurael d,efi nwihile-loop,
tiona ndp rocedure callT.h ee quatioanxaiolm s whicdhe scritbhees emantics of
thesfee aturaerseu sedt op rovteh ec orrecotfpn reosgsra mTsh.is w ork shothwast
imperatpirvoeg racmasn b es eeans f ullfyo .,.malmiastheedma ticale ntitiaebso,u t
whicthh eoremsc anb ep rovejdu, sas ti na nyo ther branocfmh a thematics.
proofs
Thef undamentali deoaf p rogracmo rrectness goesb ack tow orko fv on
Neumann andT urnigi nt he1 94[0s25 76,] a:n i nva.,.iafnorta ni teraitsia on
property rtehmaati ntrsu oef t he stateeac h timet hel oopc odiese xecuteOdu.r
approach differfsr omo thedre velopmeonftt sh iidesa ino urc hoicoeffi rsotr der
equatiolnoalgas iacf o undatoin andi no ursy stematicu soefa n i mplementedf ormal
notatitoonp rovidec omputesrpu portf orp roofIst.a lsdoi ffeirnts h atw e define
these manticso fp rogramussin g an equatinoals pecificaftoirao nc laosfas b stract
machinfeors s torage, bys pecifyingt heeff ecotfps r ogramosn t he statoefst hese
machineAsn.a dvantage oft hisa pproaicshth ati ta dmitass modelsa nyd esired
organisaitoonfm emory,f oerx amplinev,o lvingc achsea nd/ord isTchsii.sas c hieved
bya xiomatising thper operties thaatn ys uitabslteo ramgues th ave.
Equationloaglci hass omea dvantgaeso veort herm,or e complelxogi cs:
1.i ti sve rsyi mpl-et helo gico fs ubsttuitign equalfosr eq uals;
2.m anypr oblems associatweidt heq uationall ogaircdee c idable thata ren ot
decidable inmo rec ompleloxg ics;
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