BSTJ 40: 1. January 1961: Proving Theorems by Pattern Recognition - II. (Wang, Hao)

19.7 MB·English

(HE BELL SYSTEM
TECHNICAL JOURNAL

Proving Theorems by Pattern
Recognition — II

By HAD WANG.

(aan ose ly, ED

“howe! gwonone concerning the siitin of pacing theorem iy
sackines ave ror ar fromthe sepia’ the axplaaer te eer
ng lois prof procedure for the prot clea egies that en
tne feo minor pecuiar fetus, A foilyextnsie deta af
cin pen ia giv, including w part alin of the (2H(BWN3)
said eee aerate preitive fe the (x9) eee 0?
oiler dead trate of Soe rons. Fe connec eth te (Bh
ean emi combiner prob ie sped tn Seton 1, Sone
{mple mothomatolezenpl aie comand tn Secon VT

sa Noe Ths oe she ar om asf Shi pat

24 The Doin Prom an

Raduaton Prato

With megan to any forsula of ube predicate cle, 6 ae tare
in knowing whotlug ie 4 theorem (che problem of prorabilty) or
caval, bbe et ego bas wy teil al al the pobiesa of
sii. Orginal hi decision problem was civeted to these
Ferme ite prosndn hil appeal to all forzclae of the pre
sate eu Sines it is hos dhl thers ean be ne such eostipeeat
2 ta nun vst nueusieat sooUsAs, saneamT LE

ragedu, the mai prof i ty deviteproosduas eles fr elsser
of onaslae whieh sttis'yeuitahle exnditions

‘The eotrplerrentary problem of reduction is to give alleeive prow
‘lun eich recuee Sromder classes to sarower ones while preserving
provabilty or eatisohilty, Tw this way, a deekion proce for a
alr las eu he made +0 app ton Tage en. Th fry moet monk
fn Geren pean has bea dior tthe scale of fing
ronwdines whch lien al forma of the pte eels to mate
bore of tome epecal ele fo, choso the Sole norm! frm. Tach
sch clas ic alld a recdaction class relative to ati or prowl
ity accaning to wheter satis provabililyb ptrervis by the
tusmsformations (Ret. 2, p. 32). I flows antamaticall thatthe core.
sponding decision problem foreach rrduetnn class ut alate

"he sedeetion clases andl the prosedarca enplayed to issn dem
are boing rancor eth wader esos only of nines ae fe the
Probie of ivarerig pve nis the dwison problems, Mone
Ayes elvan are eetion procedes which mre apolable beth
tiled laine a theton kes aut sng in Dart bere devil
le ears Some vey preliminary reals on (hie mone enor expect
‘the aedtsiem penbem will se deserted in Section V.

For bach the deesion problem and the retin preblen, Chey
sey th “yea a” eto sisal, ste vest of deter
‘mining 2l ede and devising transformation prosedurs whishprecre
sll models. Sach quesioas heve been etadied vo earain extent (Re

1. 20, but ill be disogurded in what flloss

Misrestomary to hutcletiaenaturtion eleser wad desu les
in der of firm iv Ueprote ntl Grr, fey with ll quaatiiees
a the beginning. Brunlines, wilh toga Lo eatisiblity Cor provable
ity) sompunotie (oe abuses) of formate i the prenex somal
foram ate conto. We sal ell ths the eaten prone form.

Tn Brotias ¥, proeeduce wi! give Toe ve aing wy Corals to
‘nite ext of generally eimplerforaulae in the eaten preteen
Such thatthe aignal formula i provable and only if al fora in
the neice act ar, lv this andthe nowt for eoesion we all ly be
concerned wih forme in the extended jresex fora, Fureercre,
‘ev abl ve in Sercion Vy prufaleeson pruvsduce for she quantsee
fr log btn frum the propia tales hy adcing equality,
Fein symbol nici tostnts, Any cereale
«quantifier re tautology aan exten the notion ofa propeitonal
tastolrgy, Wo shall makes eo 02 the Fact shot wo can alvns> decile
whether give for ivy punter tology,
12 A Bri Pormleion ofthe Prone Caleuler

21 Primitive Syoote

stant Vasil 2 v6 of (0 Ininc sb
13 Tidal toneants (x faite or innite set.

£0.18 Peopasitional Bosleun) nperatinns: ~, Sy
‘a. Preiea letters (a inive o inte st),

INIA Bunetion letters Gs (ite ot abate st)

aca Equslty: = (0 opreia)powtiene aymbol

Iai? Quantiestion eyrbals (3, UE).

toca Parentheses,

aa Inductive Dofisition of Toss ond Pormatee

a2 A variable of an ind vidual constent iso term,

taza A tinction eyed ‘ello by a sutable saber of terms ie
ten

V2 A pradicate fllonrsl by a stitable aur of tsa i for
sn (al sn etemie formu; in patieular, sf ay ane teens — (a2)
tiree= 3 30 formal onde atom fore)

Casa Wg, g ace fiwular and a is a vavnble den fale, (ade
mee hie deg 2 he pare formulae

198 Trdtie Datnaion of Those

1.21 A. quantifier inv Lantoogy isa theorem.

Thee Ha digintiny 2 of altoratives ic thneem, ge is one of
the alteriatives anda is votiable, the

fal lf isw rm, Hen the rose of replacing yx by (a)es in is
etheares

{a if isa variable foe in but no few is te other alernis
fund # io does not oe? pa then the reel remhcing et by
(ed in D Sv theorem,

a IP g n= & in teoom,g9 sal

‘he above forsvaaton i seme en’ wth repoce 0 formula in
the extend pve form

1s The Paansntat Phsrom of Laie

‘The main pur of de set few seetons ie 0 slay the desision
prohlest on the tential uation of the Cusdaneatal theorem af
4m mate sera nueunteat ove, senCanre 19HE

lowe, an aproach jvsistod by Skolt ew Horbrand and zecenty

vive ay Church," nud by Klaus! nd Drobon.

sppose Wye sa quantifier roe mats

bas (ey(daieh ese,

kee wen) ~ ME
Lev now Dybe My vos v My and 3, bo MIS, bsing an abbre

‘anion for @F 1, The fundasceneal theorem, whet applied tn 13.1,

statce

sua The following three conditions cv ecuivulnt:
{GY 1.1 ew Haney of the prsinate eles; (6) foe sore Da is
“Aryans fee telly fe! 182 otis

Try isa quantified (aula, en, iy L231, hath ie an the
seul f subetituting distin vanlables for distinet numb fy it ane
“heotems. Fer example, sbopene Ihe mean
tus Moab v Mabe v Med

Wehave: by 12.820),
Mand v Mabe ¥ (6 Mace;

by 12st),

Mout v Mate ¥ LeyiG Stays
Sinilcly,
Maab v (ByhelMays v (spieMays
uyyetAtage v (lane Maye v sit Alaye,
by 1233,
(epee.
by 12.28

covey Mow.

Hee, eon (ipl asin (ave 6) in La
Oe the ther hund, iF uo Dy # quar freo tutulogs then thre
is, foreach Dr , som interpretation of the function and predicate sym
bole on thee [++ 377 stich atisfow SP, By a wel-knosa ang
ment, hore is thea an futerpeetwsior on the dora o all positive in
teger which stises ~D,, ~Ds, ee, simalteneously. Ths, howeves,
rsut Ht unor the interpretation each Sue segnwnt of the inne
tae Mania & tas fe MIB &

ie (ru, Thu shen there eo integer vie 1, auch that for every integer
ty here ea ntoge ei," math ths ~Aaye. Ln other words, 1:2,
ths uogaion of 11, tre ndee the Sstarpretation, Hee, Use nega.
iow of eonon () mplite the negations of editions (a) wd (

tise fake 15 as w rodl of 133, if seme natural $0 rege 9 68
tu indepoudentvargbl, 2me 4 dependent variable and «asa intl
‘avin (he initing ese of x dependent. vazable, function of xem)
fnyuivent), he general prinsp'e of eourieting M, from 131 may

wm sumatiad by saving that eoch initial variable gots» cist
amber, the independent vatiables taking en ell passble yusiliew i=
fegees en value ard the dependent varableealvays taking on nurcbers
ot ned efor.

“the general ve, we must consider w disjawokinn for provability)
cor conjunction (Gor saisisility) of formulae with arbitery string: of
uineem., Then we ean gun eowstrurt the melated. quantifier (co
formulae inthe san way, woth tho uuters tv each clnae proceeding
independently

"Thea, ce wih to study the stihl psn
Aur Ke nm

swe onside acy

130 ak ee WED,

wwhonr ouch ec of the omy nih a 2 O42 02 E hada es
ee |

at ii!) 225 ya = ea) os By

eyclee 0 aM = 24)

ne Santi way of abexining Mi, Ma, ole fr the formula 1.37
gna op neplaning the deperdent suctabice (Uhoze with the loner 3)
sok with a cunetion (snelinus: alle a “Saoles fetion") ofall the
receding fadeperdent varies (these with the Teter 7}, an then
Sropping sll tho charter. Lel the result bo 3#%. Tn parila, the
{iia Gleeson?) voriaier ape nepived by tine ratte whieh
nsy Te viens tevial felis. Suppose eo b= yh +
a yin 87

‘The Skolom functions sre any fuvetions ox,» oy which, taleen

ogee, sai the following eon

bons Ga) ar owe gay gn tied Zot f= eam
woe
6 rue ween sveren neousteat souusat, ranvany LOL

fh) Fa enc gad. 0) ee nly en

{9 Por ony gay toot

te ee BO tf all

‘Hom we en tke Hae sesh son ehh ets te constant
fr che intial (enenlons) varies for an arbitrary’ woustant. when
thon is uo ead initial vueable) wd ie lose with expect to the Solem,
fini Once sch am Cemureable dma ahi, we me
how eiawtale all ie gtuples of members oft deseain. The, Be
fe ly simply the rel obtained fom AP on te independent
‘variables are eplaced ceepetively by members af he sth p-tupe

"Te satieahity problem of 117i thea recluced to Lat of thease
conjunetin’

tas Medeitee

Silly, she sssEabiity problem of 12.0% can be handled hy recing
ach seperately and then taking the conjunction of the « ify
‘conjunctions of the form 13

[Te edstomary tots the peitive integer othe domain, fit eine
‘euumeration of the p-tuples, and spovfy the Skolem fancy
rntiral manser. One false er eeemrion of the pple te Solin
ing

0 Gans = 9) tei o> Dh i er
{a) they are permuterins of cach obhee bu fa, ©, gy) pameen
(yy by) nthe lesengeephie ardor oF

{bras 25 4) = shy = Bas 2a, — 3B oh Ly

aj) reuranged warding tr nomierveaing magibade, prose {5
‘hh simile nner i the Tesiogrphie neers or

9) rmaaleeyag) — mating <= ue Bie < Besar
a) maaan ++ 5 ag) < mast + bh

‘Tho Skolem tetione geo us homo Uae ole through the iat
nite rset 13.9 From leo ight al ng ech tre the mullet

Sed nage Tae the nex eli express oso eval,
"Treks nh o=" tg? TT pt he costal vale y+ +r doa
hae

ML ee dO ooo A oes QE oo Doe gh
‘Barh “ime a unetional eyo get aval, the value sabia
imal Iso corarromoes Ul ann expression,

Th thie way wr aviv uta fran of the fudaanenl (hose of lee
ava ponenwiration of 138

ty novural fu olvorvn that the inite eoxiunckion LA.8 ean be
divided nto eeetione (Ret, p38!

ssa Tho fins seton 5 the oot of Chee M.S in which the ptuplee
Feplacige the epondens veiables ure made up of tategeae sn te act

Pies dilyor eet (1! io, = O; tho Ge Heh ection fe tke mw!
thse Aije ot belonging to che ah sectinn in whieh tho potmples nee

rua) inogees whieh evr in the union of the fst 9 sections.
"This rw on fas bows ssl by Skevem iu explaining stm dein
provers (oe Seton IL iho

st Special Coe of th Decsion Prablem

‘The petal iow deeidab ess a, wi regard Lo catstinnlity
he ftlowing:

T. The manadie cae, The vas ofall formulae whieh contain only
somatic predates sed na Sunatin symbols,

TL Ths BA vais ruse (ie 6 prowabitty cage), The cies of
sil formate in Ue poenes fora with preteen of the form (Bis)
{yest fryhe m2 0, and netio apmbele oe the foes
fy) es GyltFn) = (Ue) fr prowabii

TU, The eajuovticeowtishaitity sas. Every foerala in cho presex
for witha nat whieh 5 conjunetion of lie ferae and thelr
egatins, (guvatentiy, the dis unetive peovabiy’ ease.)

Te The Soom nce very fore ithe poete ferent wo tame

to symbols sl fa asa poi nding with (En) =~ (Boh > ©
fd crery ntomis formula aocusing in the mtr contains either ane
SU the niales joy ss toy ot all the independent vavinbles. [For

paenbliy, a) Gg at the en]

Vo The BALK satihablty one (he ABSA provmility eaze). Boory
formula containing no Raneion aya in he prone for with «prec
Giy't os yehleg tbs = Et

‘VI. Tie evcaans use, Kor satis, over formula whieh cone
tne no fnetion sembols wo yay sig, only a single dyadic pric
fate se, ar he the fer (2) pay 82) Eee)
eS 4,2 cartier

Ii sion tn sae, bv other easse aay be mention:

VIL The Edy mbdetlty cove. Mery formula with the pol
five) and with no Tanetion ssmboa.
SN rme seus even areuntent suuKDAt, saNeaKy IL

NTT. The Suen amram eos, Bar slit
wie Tar equality sn, om fonetion satel wl aio priate
fils, ond has the form (rated Marae, de CeNleDIB aN
MF, ¥ quantile

1 ay tm ots a i ll cu, vl ingle seep of TT,
ho Zonetion ayn are perrsiied. Indeed, vary Title ie Reo abot
the decision poles of frmmlae containing function synbols(oompaa?
Raf 3, pp. A 107), Uniees often alae, we mba alway wenn
theca Finetionsgmbols see.

"a what folios, eaces 1 and VE wil not, be women, So far we the
momaie ease without equality (asubease of Ts concerned, iis ble
Urbain» ison prove Fra ne for ease TT, Senne of the pre
Ter atggete bythe element eee wre oy exouunted Me th
AEA, cas, while other implications of this eae seem to eall for a
‘ser éxito rian witli postr

etal uno saen WETT fom etic sla im he sen ha
taste isan effective procedure Ey whieh every formula, possib® con
‘ining — std fonction symbol, ean be reduoed to one inthe elas with
stisbailty preserved (Hef. 2, p. ). It folows that there exes 10
‘vin preeedaye for this toro It is, Bower, desirable te fn eo
“semideekioa prosedur! forthe elu which ea dosbion procale fr
some subelas of it that i not speed exp in udvunce eis thought
Ghat such semicon procures are n nasal way a extending the
ray of fornia devise by a puaeterve Tite st of gramslaes
Aref desi ie ined sn Seti TV to pot to the rt of thing
‘which ear he dane along chi Hino. Th should eo interes to design emi
Assi: poooedues for case VL, ar wel sf other reduction lasses,

‘The ane VIL iv pechepe the beet Rear arose eae i hae been
‘maationed in varios connections {see eg, lef 11, p_ 57D and Kel. 12,
'p420) ln Seeton 1V procedure mil be given which mey be e decison
procedure for the whole ease but has only en sbown to trmnete for
Serain sposil eases, A proof of Snitenes of the provalure « wanting
Tes thousht that, incomplete ae the vlution i, i uuite samaative
for further sore the deison penbles, Some mer ening com
inaovil pvider ste ugh tela In te oer ot his ene,

An slevial se ceirian proses foe the wuah-ludied ease ¥ ll
be given in Section THY in the equivalent form ly (for etic).

The Soler cas willbe ease in considera dtl in Seevon Tr
‘eng ideas papal 97 Shale! fp. ASR) ard Chr Ip. 261). Rearke
relevant to machine vealizations of che prose wil alo he inside.

“The Skners ease inlies the following spoil eazes
14 ‘Tho AB satiabily esse, Berwnse evry acrmie Jorma bar
to ine some variable and ther ie only une fdepetlen variable
TW, Lor satiety, every formule whore prefix ems with (dp)
ig acd in whien avery aims fornia ening c€ Last exe of
‘he variable y=
Te, Hor sisi, every formal whose prix i
ip) gC + tes =
ond i which ever tome oral comtaina either nll af 2% OF
Bt last one of yo 8
1Va. For aatsiubiis, every formula in the Skolers normal form,
Sieg with prefix fe) => Tenis} +++ {Eyal sch thar every atomic
formula etait low itnc varies,
or the extermive Herature on the devision nile, the grader i
referred tothe bibliographies in Tete 2 and The water haa not been
ble cortuey erwlly rue of che rele itemte, end mot cera
{thet che pronlunes deserved in Sections TT yal ITT nay aot tum eu:
to be inferior to easing ones, Restly, the writer noticed that ides
‘long he nso the sation of the #4 provabilty ease given in Sesion
Boot Part Tan eonraized in Shlem"e wings fea, Ret 4p 235),
‘OF the tv sessing soee Tan TID, some brit eens
sale,

2 Foc Simple Cases

he FA satiny esr TT as greeuble decison proteus wo
Ayrton she ncasnenta Harem of loge (Soe Te 1, p. 13) Th
ews ora to dove @detsion procedure on thr Ins of the fdas
fal Uheccen, Conse

ut ph Cane) 0 (My oo re

‘he en fac ust

My eo MN, kw, or Lhe m = 0.

Th Gat, thi 6 Hing ons of the Cada Mheorem because ro
holes functions are ceo, thk the me coments for the ital
ables ane all we reed foe icing a mode. An tier wos, ei
the nagation of 15:2 -u qmonterinse tautology, and the mtn of
{ibd ina theorem; or 132! has wl, a 1l baw model too, The
perros of the eal sigh is ptr ed, bat. the presence of Fun
reo ny 1.5.1 ld eal the poeta,
wom auvenat, Janus 1991,

‘Tho coujunetive sassabiity cass LLL ye originally solved by Hes
brand (Rel 5, pp. 44-15). Suppose the matsix

at AL Ay BO eves Sy

or, ina diferent noetinn’

uae sey dh BD

Asump fist chat neither equality nor Function symiols oscur. 1? no
‘toicate Jeter oecare both on tho Left side snd on the right edo, then
‘we can Simply chooee to make al predieav occurring on the lett sie
‘ye of allnumbers and these on the right false fra nuzbers and then
‘he itt ronjanetionrorrespotng to the given forme’ fe fr ander
‘ho ineerpotation

‘Whenever Uhre ise ease ne Fe lone de rg hie
sorta she sae prodiate lees, ¢f., i Gabe sd B, 2 Chow, we
Frenpare then and cle whether 54 poss to assign the same inter
ter tduiv anguonte ie some A, and AE, respectively. f the anes ie
$0, the miginal forma cn hav no model, beewuce the intnite con
hot a be away fale. Ifthe answer so Ene every such pa,
he he ginal ferenls hae a ase

“To compare ly aud, we exanite the thee prs of worrsponding
variables. If both yarabls in same pair are distinee dependent sur
ables, thon the two slauses 1, and B, can newer got the same numbers
‘Who shi ue et “or ron oe py wr ems dei the eet
by asking whether there an perkive inet, uch tak afe) = ale,
a) = mi) and e@) — w), whete, For each Yarale in the origina
formal, at i Fanetion giving the number which replaces « in,
This poe ie a ame tn gpuerate seh function fer rach given
fo-matla, When thete axe golitinne for sone pat of laos, the ogi
fees woot.

The formula 5-4 comrains fnetion syonkole but not =, then the
evan fly am ts take sees iat rrr
times. We may hove br tsk whether fogs) = ga), oud os)
tt sw oaltion Tasch enes ee bx eution ety whet fd g
fin the sume Teton, ieenineonerwise we ean eleape give diferent
als bros atl gfQ tne The nee of ayaa ae

‘Wien he unl er aso cera, we have to Ha all the exons
aon yy oy ty if hen sayy a empty og
‘raise TE thor ans ae, er! aly trowel a any ser

tht we oan ln rej! wunibiity 04 the ground of, et avg

tsp «easy B,, saad ap) ~ ip) bas elon in