Stanford Artificial Intelligence Laboratory Memo AIM-298 I Computer Science Department Report ...

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Stanford Artificial Intelligence Laboratory . 1977
Memo AIM-298 I
Computer Science Department
Report No. STAN-CS-77-6 11
T H E L O G I C O F C O M P U T E R P R O G R A M M I NG
bY
ZOHAR MANNA RICHARD WALDINGER
Research sponsored by
eciffO fo lavaNhcraeseR
National Science Foundation
and
decnavdA hcraeseR stcejorPycnegA
RETUPMOC ECNEICSTNEMTRAPED
Stanford University

I
I
Stanford Artificial Intelligence Laboratory August 1977
Memo AIM-298
Computer Science Department
, Report No. STAN-CS-77-611
THE LOGIC OF COMPUTER PROGRAMMING
RAHOZANNAM DRAHCIR REGNIDLAW
laicifitrA ecnegilletnI baL Artificial Intelligence Center
Stanford University IRS lanoitanretnI
Stanford, California olneM ,kraP filaC alnro
Abstract:
Techniques derived from mathematical logic promise to provide an alternative to the
conventional methodology for constructing, debugging, and optimizing computer programs.
,yletamitllL eseht seuqinhcet era dednetni ot dael ot eht noitamotua fo ynam fo eht stecaf foeht
gnimmargorp- .ssecorp
sihT repap sedivorp a deifinu lairotut noitisopxe fo eht lacigol ,seuqinhcet gnitartsulli hcae
htiw .selpmaxe ehT shtgnerts dna snoitatimil fo hcae euqinhcet sa a lacitcarpgnimmargorp
dia era dessessa dna stpmetta ot tnemelpmi eseht sdohtem ni latnemirepxe smetsys era .dessucsid ’
This research was supported in part by the Advanced Research Projects Agency of the
Department of Defense under Contract MDA903-76-C-0206, by the National Science
Fou.ndation under Grant DCRE-03737 A01, by the Office of Naval Research under Contracts
NOOO14-76-C-0687 and ,6180-C-57-41OOON and by a grant from the United States-Israel
Binational Science Foundation (BSF), Jerusalem, Israel.
The views and conclusions contained in this document are those of the authors and should not be
interpreted as necessarily representing the official policies, either expressed or implied, of
Stanford University, Stanford Research Institute, or the VS. Government.
Copyright 0 1977 by Zohar Manna and Richard Waldlnger.

Nlanna & Waldinger ehT cigoL fo retupmoC gnimmargorP
Contents:
I. Introduction ................................................................ 2
.II Partial Correctness ........................................................ 6
.III Termination ..,..,......,.....,.......................,..........,........., 27
.
.VI dednuoF-lleW noitcudnI ,,..,,..,.,...............................,..,*..*. 32
*v Total Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
.IV . . .. .s . m. .a . r. .g . o. .r e. P.v . i. .s . r. . u. .c . e.f s.R o.s . e. .n . t. .c . e..r.roC 04 I
.IIV Program Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... 50
.IIIV mar.gorP Development .................................................... 61
,XI References ................................................................ ‘76
I

Manna & Waldinger ehT cigoL fo retupmoC gnimmargorP
I. Introduction
nI enuJ ,2691 eht tsrif naciremA ecaps eborp ot suneV reniraM( )I tnew ffo esruoc dna dahot
eb deyortsed esuaceb fo na rorre ni eno fo eht ecnadiug smargorp ni sti onboard computer.
enO tnemetats fo eht ,margorp hguoht yllacitcatnys ,tcerroc dah a gninaem rehtegotlatnereffid
from that intended by the programmer. hguohtlA wef sgub evah hcus ralucatceps,stceffe
srorre ni retupmoc smargorp era tneuqerf dna .laitneulfni erehT sah neeb laitnatsbus troffe
yltnecer ot ylppa lacitamehtam rogir ot eht gnimmargorp ssecorp dna ot elbane eht ycaruccafo
eht enihcam ot etasnepmoc rof eht enorp-rorre namuh.dnim
In the late nineteenth and early twentieth century, mathematics underwent a process of
noitazilamrof dna ,noitazitamoixa yllaitrap ni na troffe ot epacse morf sexodarap dna lacigol
srorre deretnuocne yb suoiverp snoitareneg fo .snaicitamehtam A ralimis ssecorp siyawrednu
in the development of a logical theory of programs.’ This theory has already made our
gnidnatsrednu fo smargorp erom esicerp dna yam noos etatilicaf ruo noitcurtsnoc foretupmoc
smargorp sa.llew lacigoL seuqinhcet era gnieb depoleved ot evorp smargorp ,tcerroc ottceted
programming errors, to improve the efficiency of program operation, to extend or modify
gnitsixe ,smargorp dna neve ot tcurtsnoc wen smargorp gniyfsitas a nevig ;noitacificeps ynamfo
these techniques have been implemented in experimental programming systems. In the last
,edaced siht dleif fo hcraeser sah neeb ylemertxe ;evitca ti won sah eht laitnetop ot trexe apeed
ecneulfni no eht yaw retupmoc smargorp era.decudorp
ehT elbaliava seuqinhcet era ydaerla debircsed ni eht ,erutaretil tub eht tnaveler srepap era
derettacs hguorht ynam lacinhcet slanruoj dna ,stroper era nettirw ni a yteirav foelbitapmocni
,snoitaton dna era netfo elbadaernu tuohtiw emos dnuorgkcab ni lacitamehtam .cigol nIsiht
,repap ew tpmetta ot tneserp eht lapicnirp sdohtem nihtiw a deifinu ,krowemarf gniyevnoceht
noitiutni dniheb eht sdohtem yb ,selpmaxe dna gnidiova eht lamrof sutarappa fo eht.snaicigol
To facilitate a comparison between the various techniques, we use a number of different
smhtirogla rof gnimrofrep eht emas :ksat ot etupmoc eht tsetaerg nommoc rosivid fo owt
.sregetni esehT smhtirogla era elpmis hguone ot eb ylidaer ,dootsrednu tub eltbus hguoneot
etartsnomed lacipyt.seitluciffid \
ehT tsetaerg nommoc rosivid fo owt evitagennon sregetni x dna y , detaiverbba sa g&(x ,)ysi
eht tsegral regetni taht sedivid htob x dna 1 . For instance: 9(dcg 12) = 3, g&(12 25) = ,1dna
0(dcg 14) = .41 nehW x dna y era htob orez ereht si on tsetaerg nommoc ,rosividesuaceb
yreve regetni sedivid ;orez no eht rehto ,dnah nehw x ro y si ton ,orez a greatest common
rosivid tsum.tsixe
A evian mhtirogla ot etupmoc eht dcg fo x dna 7 thgim evaheb sa :swollof ekaM stsil folla
the divisors of’x dna& fo lla eht srosivid fo r ; neht ekam a driht tsil fo lla eht srebmuntaht
2
Manna & Waldinger ehT cigoL fo retupmoC gnimmargorP
raeppa ni htob stsil eseht( era eht nommoc srosivid fo x and J ;) finally, find the largest
rebmun ni eht driht tsil siht( si eht tsetaerg nommoc rosivid fo x dna 7 ). ehT cases nihcihw
x ro 7 si orez tsum eb deldnah.yletarapes sihT mhtirogla si drawrofthgiarts tub tneiciffeni
esuaceb ti seriuqer na !visnepxe ,noitarepo gnitupmoc lla eht srosivid fo a nevig ,rebmundna
esuaceb ti tsum rebmemer eerht stsil fo etaidemretni srebmun ot etupmoc a elgnis.rebmun
A erom eltbus tub erom tneiciffe mhtirogla ot etupmoc eht dcg fo owt srebmun nac eb.desived
litnJI eht tsrif rebmun si ,orez taeper eht gniwollof:ssecorp fi eht dnoces rebmun si retaerg
than or equal to the first, ecalper ti yb rieht ecnereffid-- otherwise interchange the two
numbers -- and continue. When the first number becomes zero, the answer is the second
.rebmun sihT rewsna snrut tuo ot eb eht gcd fo eht owt lanigiro .srebmun ehT wenmhtirogla
si erom tneiciffe naht eht evian ,eno esuaceb ti ylno sdeen ot rebmemer owt srebmun ta ynaeno
emit dna ot mrofrep eht elpmis runim operation.
ehT evoba mhtirogla nac eb desserpxe sa a dezilyts:margorp
margorP A eht( evitcartbus :)mhtirogla
input(q) )07
x( )7 + 0x( )07
more: fi x = 0 neht otog enough
fi y 1 x then 7 t x-y esle x( )7 t (r )x
got0 more
enough: .)ituptuo
The notation x( )y t ox( )oy snaem taht eht seulav fo x dna IJ era ylsuoenatlumis tes ot eht
input values 0x dna .y ,suhT eht tnemetats x( )u e (r X) has the effect of interchanging the
values of x dna J . sihT margorp sesuac eht gniwollof ecneuqes fo seulav fo x dna IJ oteb
detareneg ni gnitupmoc eht gcd fo eht tupni seulav 6=0x dna :3-0~
x = 6 dna y = 3,
x = 3 and y = 6,
x = 3 dna y = 3,
x = 3 dna r = 0,
x = 0 and IJ = 3.
,suhT eht tuptuo fo eht margorp si.3
hguohtlA eht reilrae evian mhtirogla saw ylsuoivbo ,tcerroc esuaceb ti ylesolc dewollof eht
noitinifed fo gcd, ti si yb on snaem tnedive taht margorP A setupmoc eht gcd function. First
fo ,lla ti si ton raelc taht nehw x semoceb ,orez eht eulav fo y lliw eb eht 4tg fo eht;stupni
taht siht si os sdneped no seitreporp fo eht gcd .noitcnuf ,eromrehtruF ti si ton suoivbo taht x
3
Manna & Waldinger ehT cigoL fo retupmoC gnimmargorP
I
lliw reve emoceb ;orez ew thgim yfdetaeper etucexe eht If-then-else statement forever. For
,ecnatsni redisnoc eht margorp A ’ deniatbo morf A yb gnicalper ehtlanoitidnoc
fi y 1 x then y c y-x else x( y) e y( x)
yb
fi y i’x nehi y t x-9 else x t x-y.
sihT margorp ylesolc selbmeser margorP ,A dna ti yllautca seod etupmoc eht gcd fo stistupni
nehw ti sneppah ot ecudorp na .tuptuo ,revewoH ti lliw nur reverof dna reven ecudorpna
tuptuo rof ynam elbissop tupni ;seulav rof ,ecnatsni fi x0*0 dna ,O=oy ro fi *0x 0 dna .~x=~IJ
,suhT fi =oy=ox ,e 3 cg enn sveieei uwhusfqotlsoel aeslfvcoocfus x dna g : segreme
x = 3 dna y = 3,
x = 3 dna y = 0,
x = 3 and y = 0,
x - 3 dna y = 0,. . . .
esehT smargorp era sa elpmis sa yna ew era ylekil ot ,retnuocne dna tey rieht ssentcerroc siton
yletaidemmi .raelc tI si ton ,gnisirprus ,erofereht taht sgub rucco ni egral erawtfos.smetsys
hguohtlA smargorp yam eb detcejbus ot evisnetxe ,gnitset eltbus sgub yltneuqerf evivruseht
gnitset.ssecorp nA evitanretla hcaorppa si ot evorp yllacitamehtam taht sgub tonnacylbissop
rucco- ni eht .margorp hguohtlA erom tluciffid ot ylppa naht ,gnitset hcus lacitamehtamsfoorp
tpmetta ot trapmi etulosba ytniatrec taht eht margorp ,si ,deedni.tcerroc
Techniques derived from mathematical logic have been applied to many aspects of the
gnimmargorp ,ssecorp -:gnidulcni ’
0 :ssentcerroc gnivorp taht a nevig margorp secudorp eht dednetni.stluser
l termination: gn itvaohrtp y a lm.nlap earolvugtlitosignrwepve
l transformation: gnignahc a nevig margorp otni na tnelaviuqe ,eno netfo ot evorpmisti
ycneiciffe (optimization).
@ development: gnitcurts naoc margorp ot teem a.no inteavciigficeps
esehT seuqinhcet era dednetni ot eb deilppa yb eht ,remmargorp yllausu htiw emos eergedfo
computer .ecnatissa emoS fo eht seuqinhcet era ylriaf llew dootsrednu dna era ydaerlagnieb
incorporated into experimental programming systems. Others are just beginning to be
detalumrof dna era ylekilnu ot eb fo lacitcarp eulav rof emos.emit
1
4
Manna & Waldinger ehT cigoL fo retupmoC gnimmargorP
ruO noitisopxe si dedivid neewteb a cisab ,txet nevig ni na yranidro epyt tnof
and secondary notes interspersed throughout the text in a smaller font. The
basic text presents the principal logical techniques as they would be applied
by hand; the secondary notes discuss subsidiary topics, report on
implementation efforts, and include bibliographical remarks. Only, a few
references are given for each topic, even though we are likely to lose some
good friends in this way. ehT ytsah redaer yam piks lla eht yradnocesseton
without loss of continuity.
In the following ,s e ehglwcal upinohwotcae fo ;scipot we nigeb htiw ,ssentcerroc the
detagitsevni and best understood of them .lla