Semantic Interpretation as Computation in Nonmonotonic Logic: The
Studies on the logic of automatic computation ...
134
Transcript of Studies on the logic of automatic computation ...
STUDIES on 'iim wcnic or Ayro'MTXc co!]?uvat5:o:j
( Incs»«iisnt»l Data Assivailatior. in
Man - Ccsputer Syst-sss)
by Licnslic A. Lcnbairdi
STUDIES OM THE L^GIC OF Ain'OHATIC CC.f?JJTA7iOH
(Ir.ci'o:T:£ntai Dsta Af-siailation in Men - Cat-putcr S73te?ns)
by Licneilo Ao Lorafaardi
Usssschiisatts Instituta of TechnoJ.ogy
SUilHARY
T1t3 nain pjs-5b.l<*a of siodsm ccraputaticn thttcry and ncthod-
:.do3y 5?i3ffii fro:i tho fact thst conventicnai digii-.ai ccupytei'Sj
developed foilowins ths ciESsical idaat? of Turisig sr.d von JJeisaann^
f.-:;il to r,S3t raeny r-cqaiiGinsnts ea ocsipor-erits ©r tei-^inaJ^; of
cc-np.le.'i iT5r.n=corsputir inforinatioii 3y3tcra textures o 'fheir main
litiitatiors in such context is sc^stiKes idaatixisd as thair
eKCOtedingiy hi^h nesds . rsgai'-ding the spsoificity of both tha algo-
i'xtliT-iS that thsy csn accept for Q;-:ae5.stlon,and their dsts^ irhich
i?.3f:33 t'fis^ not p>'s.r;:ariiy 3^?it^;^i2 as orpcitis for iiicr-csjental dcta
sssi/iTsilation •j:hro»:i5h adaptivaly gs-owing and ineraaentally GCKlifi-
able aigorithsjso Accordingly ^ basic r^sear-ch is boing carried
ouv en designing n-w fouKdatlons for> the? icgic of a«tc;.iati.c
dij;ital cca-vjtationo
'"his pspar- corjsists of two pr-sliaitiary probes into ths idsd
of ths "xnc5r's;a£n1:ai dsclarativs^ CGtsput£r'"i:H bciids for a poijsibia
solution to this probi'Sajo l^ha first pes't is davotod to d^jvcdoping
ths synt^^: cf a prcgrsssaing lar.guaga for such ccraputor baaed on
3 rav5.sion of Karl Kongsp's notation* llsough it haa b^-^fin r^c^ntly
dc-^ci-Jcd to discss-d this nctatiori snd rt-plsfie it by a Giibstsr.ticilly
tss;? cr.a.. to t.'hid-s tha hansn ifs ko^'S irijTjsdist-aiy rsspotisivaj still
this f£i^.t pai't uhould ha ccnsidarxid asj an car'iy version cf the
fi5-3t chsptGr- cf the bosk davotsd to thsi dsvaXop-riSr.t of e> r.-xj
craipi.tc.vj.on philosophy, th£;t this cuthoi" ia writing c Tr.s ascend
par-'w ( ccs-sraspcidinw to the accorsd cheptss' of th-s book) pracGnts
e2 nsw kind of n:3soi''y or-^aniaation bssad on t:io idaas of Mswclit
Shcvj and Siasn^ but «.:h&k--9 such idsas sr-a revised in a way to
en.sblo tha ccnputcz" to acan ssyirbolic eKprassicns isibaddad in
iistn irl.^ht to left (Io'S^b --'^ t"s cj'tiimdad L>i'-a;3i6'.-7ic2 s-anr^a)^
ifhich wcrJ.d not ba siuy if tha li-LV, LIS? or FL?L .list-atr-uctui-ad
I'saory ori^enisaticn «as^a odop-tcdo
Tnare sr-s hfer-a no hiuts to the othax' cheptera, clevotsd to
th« dsasign of tha control unit of the ticn-j ccsnp-ater a-ad racm'aivs
fitnations of s;^':3i3olic sripraasicns^ its inr-jut-oiitput sy:3t€rSj
algov-ithna to cs-Ci'dinate tha infcsviation ficy^ randccs scvC-saaiiig
end infoi-^iiticn si'atris^'alj, and idaritifictsticn of pji?2:i;3-i:ar3 to
dcacilba cs-apiit*? syatess qua'iititatirely j, z'-ss^scti^-^aiy.,
Jar.viSEV 30. iS53
Ttils c.^i-^^r.f'-r Ih aevcteo. to ivxrcKincing, « prlfsttive alphabet
..r Cir^,'-.-., : : '••soiiaiiing rules by s*s<ins oi vraicfa marks can be
ijfioffBbieci to forffi tacse r-'^rr IcxLar aggjeg^ces -» the i'orT;d -•
'iti ' !be definition cf the functions that
ths v;;'.:.Ti;v:cT cv^^ip'^toi -'U! r.c;!nputt, studying prciperti.&3 cf «uch
fora-sj;, such 3S the rv^jsTb-. .1' * ^- •"f ^ht'ir cononicsl dsco'ccositioa
•:7.rhT-> «"'r'e sp'-:=~e :.. u'3«ing the detgchaent
c .forn3 up.y oro.e'ed co-iple cf fcras iat-o a
:-(ra one iietacliS'Sut is the basic opsx'ation of this theory,
xixe aecantict belti^ aefiae--] Ic tenss of reccx'sive detscanjeatSo
Ito Diost important prop-iJ-v -!>^ -j< <^o^)i'^-^ v -^ v, sjhicf' ir^ proved
Mere in tbe 8?~-'>t. generaH
,
!.;- a]-i;b^!.\-'- pnesent.ed in thj.s chapter is restric'ced to a p.re-
'7-.-= •:.-. > of sarl-ce vb-it^i.ls aecess&ry Toi introducing foriiiis-
• jc/; ryl&6 gad eiesjeiitary ev.*2luation ffi-:?tccxi6, Lnter on the
be 3Xt«ndsd in order to aliou i'or mcjxe cnrcpact
- of fuiict-Jons and raore gereral evaiastiori asethods^
C^tistfiDii; 2«Y bs c-r fw .^
finite 3*j.3uence^ of de.'
fb-ev =-'v ;; •. vhey ^5:« t&lsien
r^cttsi'^t of Vln3t«? ,
dlglt«j, at Ieai*fc. c
, rntlot.\£.i, (r«pr«S'--;nt.«^.l :;«
, Alpha -riijastlc. cor!fc:'?.<«nt«
" >' arid dftcloisl
:yi: a welt sstsbll^tvi':
a«fi them ?\i.rt.Uf^' !* herx
Thcra air* six ^^perstiv? r^sa:-
^"^ cod Yj ^Cii lei vv( •
iv -her 1nt r ctji ucc ^
I^iBtly, both ccJi'^.^.v • -;• .:v'^'
^r* r<-pr';!.»^f;"-Trd by •« 3 i'f >* re- ,-:
T-'i'jh.le letter,
. ; ract function
;rated dxirlng computation.
'i.u'iC'i '.).'} '5,'.
Sf.ci 1 on 1
.
2 Forms
Any oxiSeTed finite ae^ieiiCe of n;'.3ijt3 vxth-suiv occuri"i»nces of th^j
cu.lca Aj^ csJ.JE-d an f^g^n^^a,.
A pni-ticulsr kind of fsggxegates, referred to as ?£!:«£> pi'-y 3
ciore ij!ip'...>i-t3n1" role 'and neea a thoTX-ugij di^fiisslon , The first
jnd sirapi "'", se covered b.y cba fOii<?wing;
De.t1.at n to . . . ._ -_ ., . -. . Ive mi3rko >jrci forms
It ohoai-d ts no^ice.d that punctusr 1 on Esrkr, are cot fon:--.
Ac'Ciiing ;:^^ rhl.s first, definitaouj 2.-5,. , £ ^^^^ ^^'J-^ sre
f on-D?3 . An eggregete^ irfilctJ cjsri be pr&"/ed t.o be a x'or72 on the bas.l3
of dsflnitiou 1 alone via I m refsixad to qb atogjir: I'oi-e o.r as
If s ead b era two 'iggjegytes, the aggregate 3,0
V1.11 be csiJ.ed cooc-' -'" - "^"r r " ' :': rXy concstenar.ion
ci'.'^fj.niblon,-? .1 one! 2 it can be pro-vsd thst the si^^rKt-i^rto.:
.' vhieh csn be proved to te
r:-.'ii c-,-i;y !;• i and 2 B finite nuEJbf-.r of
For ex-affifie.,
v-3.Al<?aA. 1,2) {21..}
Lei- vjs CC!? assoei!?»;e to e-3eh forci su xaxegar. cylleii l^vel, ea
The l-jvel ci y/, atom Is
The levei of ? form o_ ot^slned by concats'ist-Jon r:f .
leveis of 3, 'iiid a^,
•TJis jx-vei of ri form o., cbt-3lnfcd by eccicslng in. p-f-.reTi'
-ibessB -ibe fo^'m s_, of TLf-v>-'J. .i is j ^ 1
is a forsi of level. 0,
(f), SUIL [2), 3au, -2
;.". Ci fora b? level .i^ and
((1), SUM, 3.1^^ ili^.t 2*TJ)
lt-3 being; •q forts i'cll'jvs ft ora a fla.lte n'Knber 2L.j£^^'^il^'\ JiL
Fox- exsapia^ neither )x/ noz ff nor O sre foxas, because one
.jjiLy iirc.'e that no finite seiiucncs o.f instances of defin-
It-ious 1. ?.f 3 could prove ^hoa to be ferns
-
Definiition h seys that, for e?.c.h aggregate vblcb. is a fonc,
there icirst be at. Ic-sst. oae pioof, that vlXl be called structurBl
r;xoof, of its b2jng eo,- con&letlng of a fjcite seouence of
lastencea of defiDJ'T-i.or.? 1. 2 and 3., such "^^qX each .lastance
ot definirioa 1 is alvaya spijllrsd lo a const-ant or sn cpsra-
tive isark -and each jnslnnce oi" defxaitioa 2 aud 5 is applied
to the con.catenatiCQ ' r (snclosure la perentheses, respectively,
of -iggregates vhicb bsvo baen proved to be forras sa result of
Instances of defiaitiOM \., 2. ox 3 piecedung it jn such proof
We Kholl alvvsyo RoSusk* that the last i;tatQir«nt of any strv.c-
>:u.tal prc'Of of an e^oresaicr. s Si;sites e being -i fOT?ao
Consider subseciu&ncga of st^tsnents of s atructural proof of
e conalstliis of etetessents each of which, excluding the first,
sppllao to at leeist one ag.sregat« which is s-tated t-o be '* forin
by the precsdins statssaent of the subsequiTice . ecd vhose last
st^jteaent is the Isst- c.-r^e of vUvi gtructtirsl proof- If q state-
seat doss not belong to soy such ^lubBe-.iuenee, it. is evidently
Irxelsvt.T-t to th« proof o Ws shall ^ilvays assusr^ that etr-uctxirsl
pxoof?n do not cents iix such irrelevant Btateaeats <,
The p.uniber of occvi.rrsnces of icar-ks in sn sjigregate e is denoted L(e).
An aggregste coastetlag of the la^L c occurreaci?3 of inarks of d forw
hs-n'og at ie'ast n ocouxxenceg of L'^arks i?- referred to ^j3 part^lal fora.
^S??iJ:L.-±^l-lii!£li[!^. -t^'S:. l:SMk..B!^'. -,,-;.l
%^rj:penj5ud cloned var't-u^h ^ia-ii Is; :t ''1.2 '
T-^oof! Or)vi..vui?v !5y cle.fla.5t.ion 's aect-i.oa lo2. In fact, the
j-,iy provision i'or ixitx-c«3uclxig part-ntnese^ in the six-uct-mai
proof of a form is given by tief init-iori 3.> section l.-S,. vhich
T'?:qui.res. intrc-duct^on f:S per^inth^sea in couplss of ojs cp^n oae
aad 5 closed v":.r.?;.
I>^t nt'sv e ce an aggregate^ an^. let s ba an. crciirr;^nre of e
.ii'jrk different, frcai ?j parentbt'sls In C: i^et 0.«a>^-) tirid
C i'£, ft ) denote the taf-ai number of occxiX'r?i>C'5s of op^a and
.•;.lcsHsd j>areE>t'a«Be3., r«sp'?;ctiveiy^ t.-r; the left d-f a in e -so?/
o/£,e') j3na C (£,e) t.Jie total number of oceia rerice* of cj-ea
an3. <:].c»ed par«nUbs£!»?:Sj rfe^pectivsly, fo tbe rlgiit- of • ^r-•
>.vi- us prove tho folIov-:(..ig
Theor^a 2; If £ is 2 fonti, then, for i^ll 9^.
Proof ; by Itiductici-i vtt-b reapect to the level, n of V-
>-'trst of a.ll, (1) i& true for a = C^ l^ec6u;3e In tibilf
the forirat (l.) of sscticn 1^2^ vtiere sli t^, «re atui-.>, ^^..^
there s>rt no onciii'rscces cf p.t:f-^nt?;;-3??.; .Assunje row •b*'t. r,hi?
v; •'/•;. :.•. be«n proved fO' -:uci\ that
-.;: UJ .: coriSidery iu a atnictux^l proof of s^ the first luBtaoce
if definltioa 2 or .3. B-fction "i 2, a-uyi-nr- fhci
say e-, cccitaiaiag a 3s a i'orsi of l-svei n
ccn><j?;: -•: t:!:»« eacl.osirrf In psrv::m,.h'r^ of ievei
ix-l, tor b'iC.suGe «?= it* tiie re£«ui f- of corjc-j-s^eTiciv:;:; .-'.ffis
e."* sad ssj-, one of wiilch^ B&y eji,^ has xc-vel ri vljiie .rn; r,?:!-;,"^-,
oay e,, -ncn-taioa a end has leveK n. The rt-irtfiij nder ox" cae
proof cennot- conrsin In-stCicces of definition 3> sectiori 1,2.
ep'plied to cont3lnJx-g s Bi^ partj, beceuse other-
vl3e e vould be or a le-"el great - Consequently. «
he3 the st.ro.cture
% % -• :.
-"^
.ji
hs^pit- The eorvtr-*but; -. -
?;b ussntiei iishe:> by ieu-a^
'y in tii3 second ease coatrihaze one
ootu ueciuo aaty ia the second case.
'oilows by subtrecTJ pp?vicat.lr.n
'•' ' '-^' }:i the f'.r<r.l cft^'fi there sre tvo tilation:^
and ths conclueion fo.Liovs s.1aiiilarly . 'i^s-? r.heorsrn is thne. pravord..
Taeoreta '^h)% If e irJ a fono. thsa^ aOX sli q
Prccf . AgaJi (Bts) 1* %r\ie if the ls\-»3. n cl' a is ('.- Let us
rspeax t-bs iiyport-eocs a>id r*?3;5orijns of ihe proof of tJu;,>rt;in 2
viitxi the point of oeflfilng the forms ('?.), (3) sna (U).
;^ -'•--'h eased
by hypothesis. The coutributicn. of e| 3i:d of all s, dU.Terent
frcic eJi' to both s^:uit)erB of iiia ) vanishes by learoa i^ whil^,
la the s^jcorai case, xhss occurx-eoce of »3n jpen perenthssis Ixraitiaj^
e.^'' to the le.tt- contributes e unit to the left aaaiber of (8a) and
none ix) ;he ri<nt oce . Frr:aE t-his poxnt^ the proof folioys like
Anaic-gc^asiy Of^e c-jri prove
llieyr&ffi 3bi ; X-f e is 9 form, th^n^ for sil s.
Lot s snd b be aggre&3te«M. snd let a he another •itn^eQate con-
aisTiog c'f s sequence of coa6ecut.iv>5 occux-Tencea of oiarkb in e
.
:ihea the segregate obtained by replacitsg b for a in a_ vili be
called j?ubsr it;..^ -> of b f r a in e ^ik2 deDctad Sfb^^e)
an 2':oTi! isi e, then S(b, -j^,£) X£ a_ -fcr-gio
Proof- Lsi: c. bs t.be -'?!.om occurrrng In a, Consider^ in <any one
3truciiu-^l pxr.cf of e, the inatsace of definition .1, section
1.2, stating thQt c is 2 fovK, and inseiiv risiit aftsr the above
occurrence s e.t.ructur'al proof of bo Then replace gi?3phlcally
b for e in all ststec-ents of the struct'iral proof foLlowia'?;
such insertion, acd th^.n reraove ai.l iTTeievj^nt ststacsats frosi
the result, ii' there are scs-st. Tua resulting sequeric-e of state-
racnta is o s-fue*ufai proof of S(bjev«) vtvioh is therefore s
fOXIE.
'Ti-'e re-'SBcn for keeping tb/i stateffifnt. that, c is a fora instj^ad
of 3irup.ly replectng it with a etructural prnof of b 15 that
there taight be occxtTi-^nces of c In s other than Q_^
So for the concflrpf, of level of ^j form has besn envJoaged as
dependent of rhe proof tbst ona can gi'ro taat an aggregate is
3uch.. 'Tals proof is not necesHarlly xsnliiue, ana It l& easy
to give ex5n4>ls>s wcicre more than one proof in possible. Cense-
quent'.Ly, the level is aot a priori unique^ but I'C ts uecessaAy
for fuxther developasnt to give 3 -prooi of its imxqueaess.
Before doing this, hovevar, It 1-3 handy to introduce s view
concept r ^re yhall c.'^ll ieft__5egr^ snd £ifM,££H:b.' respectively,
of ths occucrence of « tmiK a different frcsn p5renth«8ls la an
1.0-
^.,.^1'£) "-" ^.(£'£.) ' ^^ «£'£) ^9^)
By IheoretD 5, r'eixher dept-ia Is ever nfgstlve wbea t is e lofo).
Tfaeorga J^^__f'^f , f^>'^ ^'^.,„'r^p^:^^?-o^ occurrences cf atoirs In s
".• _ e ii, the level of e
Tiie vasia con&eqvis-ace of this fnoorea is rh« fact, that xt px'OYides
a gu-ar^ntea cf the liijiqueaess of Iks level of ^ny i-^-^r y
Proof: Kotjce firfot that- esch cccurreucs of definltiOii 1,.
section U. yields 6 for^ of level 0, yliicb tbexefore has no
occui-rene re cf piirsntheaes sad whose only atom has consequently
depth 0. E;>ch occurreace of definition 2^ section 1^2, jSeids
fcrn's -i/iere i-.iie issxlsurn level of occurr-ences of stems is tbs
k;^x:5du'3 of the oa^s of tb-e fonus vblch sre conc-Jte-nated ; in
fact:, by leQ22« I applied to Bny tvo foi^& e. aod Sp, the coacat-
en.^t.^OQ of e^ and e^ a'?es r.ct, yield any chenge in the de.i.-t.h of
ezry ;'acva'rence of atoms in either e, oi- e^ . Each occuireace of
definitiorLH; section 1^2, Increaees by one the depth of all
iff :ctiici. occurr.=:,oces cf atoois. Ihen all tiirae definite one ;yieid
thi fiarse gcieratloa 01 varisticn hoth of cKxlmum depth end level,
Ttis applies In particular to any sequence of occvlxtsqcc-s of
(! :finiticue 1. 'co:'. "^^ section 1 = 2, cou?Mr.utla^ i; stnactural
:'.roof of ej the theoret:i =
-11-
Xf a denotes av. fiS;;rogatc. let a~" dcniot.o the inverae agsJ^gatc
in 'i^Iic reverse oriSer ana vhaxx-; open pareiit>?,ese3 ere repieced by
closed ones in hII of tbaii* ccciirvances, end vies versa » Then
there s\ibRi3t.'5 the following
Prcyo£'- la fact, n structural procf i
cy oiiiipl;;,- invertilag tiis order of all couples of for^s vhich
are concotenated by occurrerices of defiuition 2_j section 1„2„,
in the Gtvniotural proof of a.
.ei. e be -a fora of ie'w;i.v n .lad ,l^': uo define th^ii,^'? C,v
' ' .':.
'^'stioa^ ci»l element
:.-r.~; r'''<- 7.: <5 pcslti^-e jnt'cgeTj eaLled nuxaber of canonical ';J«-rfao.t.3
'•l e, TttC" cscoaicsl dvicoajscsitica of e_ Is bssed on. in'-: oe.ciirr&ucea
of coij-raas of aeptb m £.. TI t,here sre no such occ--. *>:?vi
M(e) = i and the eaaonlcal decowposit ion of e Is e, ^ _ id
-o be unaecomposable c If (here ere e-J such c--53!K'iS. sben M(e) = m,
. srid e,.,- , are tije aggvcjTat-es v.rireaces
. .;-:; ;;-j pxec2ding tae first of ^;nces or ioxluving tba
•#;.il.e e,, (i—.' '' is tbe
^har? 'S^^fir'f*'^ Is obvious
;
coapoii.entg of g-
Fxoof ; Let lis prove This -fbeoreia for the eanonlcsX eosrporsenb e
- £4 ^<="^--
i
nicturaX pruc-f or e^
of Soch . ,'roof atatlng that an -^ggr-egflte &^ contyinias
e, :r. .i j'o:-.;;;,, be an occurrence of --a rtomi^i of deptb
oit;b.er to. the; r—- "igh^ of e^ In e, «c<5 let u--3 call it
c„ Let ijs 3bf..''' _ ;r. are identicel, vhlch vaalA prove
tb>r: Mi>'5C.r<im. In f'e-ct^ suppose they vers not: In this case t cencot
be <5ri j'n&t-a'.jce of ^iHfivAtiav: 3^ section i-.2. becaii-AJe? if it veie-
c
could CO ioiiger hii'/e deprij in e,-, 'JtieT-efors-, t J.s aa occxirr«-nce of
definition 2, statias- that e* Is s forns beceuse it -Is t-he concstona"
coimB liD-lxmc; such concatenrit.ioa. which aust be in e,, ocbsrwise
altber e^ or ej? woii/id coat-'Jln e..> aad S. '.fOu.Vd not be the firGv.
atetOKant yle-ldiiig a foxTO cental us ng «. FurtberBOr'j c*' KXist have
depth in &>'.. I.st vij:; shew that c^- has slso depth in e. In f3ct.>
each ei^staimnt of the sti^ictur'ai proof of erctslng the depth of c*
vould also raiee ^se one of £j sad c-f-' and. g lotist confieiiuently ii^ve
v'as s»ae depca i:i e.. vtiich Is 0.. Eort e. e.>5rmot contcln coansae of
dsptii in e., -?mich ends the proof of the theoieiCc
SeclvvaJ. .;> per? ^sl,..;^:
Let us call gBri'^^bXe Ic . . ^ : js of Vevel 1 cor-siisting of
the enclosure xa psjentijefje-s of t-fae conestaxation of -Cie
varisbi? raark x 3ud ^ positive iritf^ger, called subscript,^
For ex«cpi.e» the ioxm
'
(x,8) (1)
cyntux before iatroduciag ar •: r.l eic-s?eat3c Conssoiuently^
It- .13 .Ujpoo3lW.e 3o fsir •>'• :;>tiin5 to sny nacatiou,,
^<>:;::'''or^ ;:-.voc>th acqua:.
:
Uuig^iSge of fche 3bstr9c!^
f a.^o-ata.T shcnjild ^-t^ jji^iCiYed bj* re^diiig '"•he vai'icblr? letter
?ar ex3mpley
;_ Xi4isr-3 nr! ^tsri-sble lettfri- (x,r) ever c-ccurs M^Jesii; :
t suca thac 0< t '^r. rher ' -
: th- VJ l--i- ^^'-" U,£, .
^:y...
--' 3C-quei j1 : .rs cleaj; thst the -as? of ccry:^
:. a protectica .^geinst iu:controlled groifing of the
eubscr: . -roile Jshters while cotnputatiCns progicese .
^ •;^ and 'let v be sjn occurreaca of ^
verl2bie lett«r In e.. We have the foiiowing
Tb.eore;Ti 1- S(b;.v,e) '
Frcr-f- Tn. fact ''he gtructursj. pr-xif of e rjMtzt corjt-!>la the
Etatercf^n*. tbat the variable l&'tter occxvrring la v is a form;
1-hjs ca-i be pr':r^'od vrfth 3 technique 8t taE. sLrailtir to fcti^ one
U£.t.d tc px'ove The -nDnicsi (3<rra'T5)oaatij.oa rJtieorc:% section I.,U .,
thai-; 3b, by considering tlw f.1.rftt noti-irxei^^vanr. sr^steecent t
stating that, a form e'^ contsicius v i& a form.; c_ cannot obviously
be itj i3i>tsac£- of definition 1, section i,?,, aad if s'' difi'ers
frova the forra occurring in v^ it could r-or. fce sii iriBtanc* of dsfin-
itton 2, {sanre BCctJon, because If it vers orje cf the tvo ccacat-
enatsd fortarii vcuid have an occurrence an ar.om or coj^ia of ni»i.y4-
'ive dejri:.h.- Nox' can t- b;ii an Instsince of deflaitton 3y 'iaiae
section, becs^oss if 11. K^re it vould sncloec %.v. parc-ntnesen sn
-SS^sgste already coatsining v. jfYous this pzur/c, tie pj-oof follcws
like ihe on-': of theorem ••-, sccrjorj 1.'...
Tola proof csn Iniffir-dlately be extended to the csee viaere v
J 3 an c-cci-u-reace of any -ondeccrcrjcsable foma.. In contrasts if
such TcTTs. has several com;poa'*ats< this proof -fould no longer hold»
The th^yort-ia „'Oulcl ttiil be tree but should, be pi'tivcid in .;5 differ-
If £;_ is a fora, the largest of fill vslusa of the subset ic-t.s of
tiie v-3riable 3ett..?.r^> of s wrll he denoted K(£)>
Lf^t2.
'^''^j^ ^^^ foniiSj and suppose that,
^V- K(h) '2^
Suppose favthyr that, b ha& ?i to-^vTl of k, (ic'-i^O), occurrences of
righf"- by denot-sng th<?ro v., .1 (x, 1 ) d-znare
Mie variable letter ac > I<--: •,;.3 couei.der a new foro
obteliJied fro-?' o by repi:.. .. .,- ... v^, C:r:^r^k), the nev v^riabl*;
letter (£ji, - >i(B,) ^- I^C^.)) 1*' h^^^.sJ- "- otherviee tbs i,,-^-^
^ -^vorsicsi ccsKJOnsnt ?*, of e. This i^*v> form is called partly!
GetachrK'at or t, from a snd ciet\ot:,rd
I?/ Theorens i. tiie purtta'i. derachraent of t-«o fores 1* a tosiM.
.?or e.f.^.sxple. J.et a c$ ((x, 2)., iic, l)'i and 1^-t b be (x,3)i
2. U,2)); t'aea H(a) la ;(x.l)* 2. (s.,2V); t'aea H(a) la 2, HU) is 1, «md(b | a) u
cf/aipcfct-. i'cy:;i i^i 3l::?o ;i c-;:.>a:^-act fore, t.bat i>J. tK!rt 1 -si ; deta cXi 'j-^^jnt
Ari<:'%h;^i- ?,'i Dperty cr f^rfrtiai dc:tac:a::;-:.:it I'i itist there <;flti be
t^?o occurrences of e sii^gl<» ysrieble letter in (b i, ,^)i''^ they
are both the x-'ssult of repl^clcxg tvo occtirrencHS of a t-irigle
-.srl^bie letter -wblcb orlglaaUy vere contained eitaer both in
J or bach in b. In other terras, part la1 j_gt5(^''^"^" operata<:a
T.'ie.Gsrves the rtcrcattca^l aistduction cf vGilBbie letters.
ijst un proye the foll'^rinp-
foina '^a-:a *hBi ficn p-^ tiJteir^s (c J ia I bU
W.3.)'iM\>} - >i^b.! (3)
(6!
Prciof ' If c ioes not r:<:jni.^lD oc':urT£ticafi of V'^ria^\e li^i:•.l:<^^^
the vbeoxeffi is tru.e beca'iae, for cil foras d^, ''c'^" -£_*
0-;i';'---jse.. the tvo srembers of (6) are both cVr-ained Trcjia £ t«y
merea^/ yeplfjicl;*^; Al ocoitrenctt' of variAblir .lettei^Sj, and t.h»
tbeori^K i-ihc-iild be proY^id by shovjcLg that sucn repUcesx^nte are
Jtleo'ic--,! iti both c--ia?ts.. Let v be such >.iu oc-c^rrefii*©, aay of
'th^ vartsfcie letter (x..'
'' "" - .','- ':,. left ceciber cf (6),
V is replac!?:d by the i-' ^' ' /,. ybich is c-btaS.nea
:-'j' tae j^'-tA c'vnonlca.i corrfionear, b^ of b by rep'Uclne e-ach
occvijrrence C4 • .letter (x,^). euch th^t 4-='-^'''^''.« ^'?
''5.> A •
-^(S.) + ^(s.i)> sad e^ca occurrence of arxy other 'fz/'-rt-V >.-
l6-l";£r C^.,l>.)^-ly t-be ^.-th c<.) j.oGic.el coaipon?j;at aij., of «_. Ix: co;?,-
puttfiif: tbe right i^^sb^ar of (6), v Is first; .replaced by h , auj
then each occurrenc-i: c-f ^ T-xn.cticn let,''^er (x^ k) yithlii tiiis
retiiBC8ES:nt is T«:pl6c-ed by (x, k - .Mf«) -- Ms)) or by *i, , ftepend-
in^ on "f'o.othiT »:yr not, k.> M(6) , Xq fchl.3 c^sse the resulted -are obviousl;
^r-r-'-yh'i because tbey coise frca the sense actirai tflken on tiie efiuie
0.8-
data, ^^jhicb are siirrpiy different3.y located %jhen the ectioa la
taken,
Consiclcr ncv the cr^se i,>K(b). In this case, in oi-:ier to coispute
ths lerb Bcncsr of (6).v ±s replaced by (x, i-M(b|a) + H(b ja)),
that is by (x, x~M(q) -s- E(b) i- H(a)-K(r.)), vhile in order to com-
pute the i-ight maabex of (5) it is firct replaced by (x, i-M(b) +
n(b)), to compute(a| b)j then, becaiice of ('4-), it is finally re-
placed by (x, i -H(b) + E(b)-M(a) + H(a))o These re.plscements are
identical, and the theorea is thus proved.
^co-s of da!:o . ,: de^so-b-
Tti<? ahilit-y to de-tvii;: .-^ic n--2tu.rft of t,tj!^ abrftrKci
llie rs,.s\-^ac.e of CiSc^icnttient is yi*? uitJ by ,5 propertY
• ;pi>t.t;«nce goes iVir beyond the i-r^fj? • -i t-t^j ot-j>er
preserved "c
/ th'3 defiiiifcion of j-.ts detachEsnt opGrarion.
For ijjcat pj^scticsl purpocjs^ in this particulor theory, detachtrien'c
ir, iisefiii •vfiien (?),, iiectlon i^J.ia sati^sfied,, Gr?(; its sssoclativity
iB utilized oaly under the hypotheses of taeorerii 2, section lo,
yhich gJ.co allo-r foi' a trivial proof. Hcr^'everj for the scks of ecu:-
pletensss, thic- section i;; devoted to defirdn^ r-h-is osslc detsch-
t'iGnt op€r.3tion and t6 proving ita associ.:5tivity in tlis cost general
C5ce, that is^ to proving the rollo-;d.rj.g
Theorem 1 (/issocistivit.y Theoi-eB/ If a, o aad c sre foriss, theii
(c \(£;n)) =. ((cl b) '8) (1)
Proof: Let r "dc g uon-ncgstive Integer;, and let QX^^) denote the
iafiGiLbe eequeace of vsriJ^ble ler,tsr3j separated by ccnaes
(x,t;rl), (x;t;!-3), (x,n3)>-».".'.»^^'.-^ (-)
Lefc £ be e fors, end let e' denote -the sequence of forEG, separ-
ated l)-j corsias, vnoss firt>t M(e) e3.eEf;uts &re rha cononical ec/ii-
pcnauts of e, aad the fQllcifing Stva the elenronts of the saqusucs
Q(H(e)). ?or i''*M(s), coGse«uent-JS.yj,tha i"th eleu--eut. of e,^ is
^•^^S^5.''"'^"^^-£^"*'i)'' Sequences ottsined fro.-a aCTwc like e- vlll be
called infiaite forcus and e'' Yrill bi; called the iafiuits extsusjjon
yne forc;'ili25tion oC this concept csn ijr:sediately bo obtained^
for exr^cple., by imljGdding iiafiaite foiT^- 3.nto t:ue class of infinite
setii.-cricGS of cox-ks obtnincd by replscj.ng "f.ijiite oi- itifinite" for
"finite" in definition h, eectlcn i.2» 'j-i;e concept of csrionicci
ulstcaat. of Lle.r) ana. Ri£, r) can l>-; ?iiT)ediat«JLy e>rf-,end-?;'j to
ififial-)-,e fomsc
I»et new g be 3 for;s cz'' aa Inrjnito fona ajicl 3.et 3(/.;) l/^ the SEaliesi
non-nesative Jnfcegor i. such that.
R{g, 1) =: Q(I(£,1)
)
V7e will feen call cont-ractloa of z 'Sncl aenote ^ tb.a fcra
If tvo foruS a Gti;! b bave iuGntloal Infinite exteuslon::^, they have
also i'-ieni ical contract.I.cns sud vi.ll ba called glrnllar, a-iie slsi-
?v.firit.y relstlop. of fcrrrs, denoted e/^-'i?, is reflexive, eorrraarativs
and transit lip's, ddcI cJas£es of eqvii.vsieiice binder sucli relr^tion
can be .represented by elti\..::r the comaon Infinite oxtenstcn or t-b.tE
cciKKon coatraetion of theii' ECHb^i^s. The conxponen.t£ of a term
H which are not also coioponents os!;i^
vriil be called trailed
£.'iri?!S£-%,5}'l''^'^'
5}S?X_£!^^?5SS:'^ °^ S° ^'^^ simili:;- fores can on'iy
differ by the number of trailed corrponentf?..,
If B «od b sre covitracted forsy^; t.h;'2t is, for-if.s Mltbout trailed
co!::poneats, then (bja) Is also coatractadc
If n, b .lud c are foxtis Eucii that »3y^b, th»n
anol
Kow let c and d b-:- infinj.ts foria. Tee per'^xai dctacbrjent
(c id;•-" Ip—
;•-?; tu;iax:td like .t..i ijecT-lon .!..> t'ct rori-na, v.' .ii roe C"ily c:,jiei-
ence that the litaitatlon (2), oecxion l.;?, doee not apply in tliizi
-23-
cjiiO. Vax-llal. detGeiuiieji^ of ini'lnitK forcis :is nor. only
GlvmyG defined; but alco evj.deatly SGSCciative, as .van be proved
by repaatiag tea first mii.' of Theorem 2, aeotion 1 Fiirther-
more, for ir." ^vo forrss a "nd b,
Ttiea, for say t'arc~.Q : ^r - -.; £. b and £
Eenc-s, by contrscting,
He sm-al- aov prove that the two njembsrs of (5) have the sa-ne
D.i«.ubsr or trailed coeDroonentSy tiiat is, tbot thsy sm eqxial.
In order to do so, .let vs denote a s eoiistant viaicb never ocovcs
:l'-i 3, b, (,'v c. arid 2.et us call s?"-, V- and C" the nc^r form? ob-
XBinQd by substituting for each trailed coErpcnent, sey (x^i), of
a, b or ^, respe'Ct.l-<relirj the forni
(a, (?.,i)). (6)
Tac: fCiT^ £--^, bi-- end c*- are ccntr.Tcted, and;, bj rapJ^clD^j theni
for s, b and c, respectively ^In !>), sine-?? ibe detnil3ffient of
contracted forsES .la contracted^ ve obtain
(^ \(^ I e^) )=:((£*[ b<*)\ c;^) (?)
Let us now replace i^^jj^ ror all corresponding foraii? of the typ'?
\6\ coAtaiued in %7), Because of the waya traa ciiosea,, this
3T"plf>cen:<:rnt trciisforss the lei^ and rj.j^t meraber of (7) into
(a^ (feta)) ^^ (Cslb) je), respectivalyj hsnce (i)^ and the theorea
is conrp3.etGly proved
«
.24-
Keffisrk'^^TIie proof of the fisscci-stivlty theorem given
in tha.s section^ vhich is bosed cii sa isoiaorphlsm,, Is
sl!x;ple but hea the dlBacrrantaz^ oi* isjpi.i.cltly re<^uiring
tho por.tuis^-e of irbe exir;tsnca cf a space- of setrs of
denufoerabxy !:;ODy markS; that is., r»e-s of pc^e; f'-^n' This
i postulate cauuot be represented in any automaton, sr^d
i
consequerniy one; co-.jld 3 priori think tbst. assooiativSty
hoicts only provided that the flsnu"icrabls space of forms
is iisbedded j.a an appropriate Gp;5ce of p:>wsr j\ ^ . In
order to sb.o':o that asBOciativity holds infiepeaaearly of
."L^uch riD-beddin^s one should prove it independeritiy of tho
abova postulate- v:iiat is. one shou).d give a proof in the
fanite of asaociativity.. Hove>'er, this logical poiat not
fe,llin/5 withia the ccope of this ^•rork, it •ifil]. not here
be the object of further elaboration.
Rarasrk 2i Let.A be the erspty e:;q>rsssioa •., which can be
represontod, for exiicple. by (x.1), vhtch in sirailar to ix.]
Tiisn for aliforaiB a
and
Ir. other vordo, if vs consider ths; faasily of ail oinsaes of
equivalence by slBllarity of forms, theri; under the biriary
operation Induced by detaehfiient, this fataJi.ly JQ & Gemj.sroup
having the class of eciuivalence offt as ualt elet:eaf . both
left end right. See Clifford and Preston, j
Let UG nssccinte to each forci e_ sn inocgei- i2£!££S£k2^1 ^^£)
cuch tbsv
n(a)2i K(a) (1)
Faenovsr a foi^ n ic cevly iritrocaicod -.fitbcivo specif^jring 5.tG
parar^cter, this one ir> tnlzezi to do £(2)
P.»EiL-;iinG the notaticn unsd in r;£ctio.a I.5 to dsfino partial
c-et2ch~cnt;, lot i-is consldar the nc? fora ootainsd ^roa 1> by
xeplnrSnr; for }r_„. (l- r^k), tl^.e n-bv variable Isttsr
(:£,i^ - H(s) .;. K(a)) if
4^M(o), (2)
or else the 1 -th canonical ccnTPon^rit a^ of c;. ^irhis ne-i? for^n
diffara frca (b ^ a) only by the variabla letters sr-tififj^lGg (2)^
-ijhlch in thio c2se have o siiuscript -t.iaich exceeds the one tbsy
vov'-lc brrvc in (b 5 _.£) by 7l(3)-n(s}.
ruis u£-.r fcrtTi ii: called ^arcpotria r'a.yticj.^^^d^
frc-a 3 H-ith paracstor H(o) cad denotea
b-^
f^ud Hjb f.^ G jlR defined as
K(b) ^ M(g) .. %) (3).
i=he.csGO.nio.tivity of partilal dctach'^ont of for-as ca;i bs eicfcecdsd
to parfiusstric pr^rtial dGtach'.r.'5at cf forms- th-as obt3inj.as xhc
H(b) ^ M(£)
11(c) ?^. :.:(b)
Proof: Can bo cbtaine<3 by grrciixloGlly replaclQ;^ .fcru* ccch cccur-
i^ence of K aa occvrrcncG of H in tbeoresi! 2;, section 1.5-
An undocoiqposa&le for;n of lovel 1 .io called aortal. The fi'i'st
occia'A-eace a, of- a ir:3.r;;i ia a nors-al i'ora o K.ust 'cg the one of
an cpsn porant-hosis- otheryloe 8^=, vaich Kunt exist b-accusQ e^
hns ?LSvoi ^1, ^.-cvdc- aave to bo ths oscun'encs of depth of a
oc/cr.:", vxx\ a v.-ould not bs unaocOLqiosoble . Sirnllsrly^ the 3^Gt
cccuiTcncs of Q E'nrk in £ tnu^st be the cxje of a clcstd pai-cnthssrls.
There casiaot be in a occu.n'cnces of ijjarlcs at nsptn CK buccuse
thii! woiud 1-T32distely yiel.ci t'-.n cocv.ri'snca ot dapth of copsas.
Consider a structural pi-oof v.. <.-. ..w.^v.^l fos-m s. Its lest st-sts-
r;:;jnt, statins fc'^^'t Q ^s a foi-m; csnnot be aa ccci!i-r.':uce of c-J.thsr
definition 1 or 2^ cecticn 1.2, bscaxjse in the first esse £ vould
have level and in the soconcl one it vrcu3.cx ba decoirposable.
O.'herefora tbifs stotSEont caist be an occuii'f^nce of dcficiiticn 3,
section 1.2, atatins that a is a form bscanse it results froc the
enclcr-iji-e in pax'sathesec of anorthsr fora e*'.Cont.cq.uontly, if one
^21-
i!?fficve£? tho iirlticil open and the I'in.nl clC'sed p«irent;hcr7es fron:
a rtorcTil fo)-tn a he vi3..L x vvd anat.her form _e*. !i'^hlB parentheses
romovsl opersitica is called pee}.liv|^ and a"-* ic called Interior
of a .
The r.oneopt of pariaeti'lc parti-ai aet,;jchr.-«r t needs G-Jnif: JusTi-
f-icatloa-; Let fi4."Y j i=1^2. oK^ be a oequence c;f foncs. and
let e bo ?3nother form' ouch it M(e ) = M i^ad such iiiat ail can"*n-Lc;Dl
c.c?spoaeiati'. of e are nontiai. ConaidKr the forte
whei'e £. clenotes the? -Jiiterlor cf the 1-th eanopical corapoaent of
e. In geaetal, a. msiy be n^rt conjpsct, snd It csa happen, for up
to all va.lvi.e3 of _i but one, that Il{«. )^- ;.I(e)v If thic happens
and Hlg. ).''M(e, ), then tha 5-th coaponenv; of ths foivr; i 5^ contaioB
c>c<-arTC!nces of e variable letter (x, H(e )il) wh.ich is generst«G
by cb.sngin^ the subscript of the variabie letter (x, M(e^)-i-l)
occui-rlna: in g^. , Let nov e, be such that R{^.) - HCe), aaS assuroe
thr/t e i3 als?o compact.. 'Ihe J-th concponeat of (j) st.lll contains
oc-cirrreiicc;; of the variable lsttc^' (x, H{e^. )(-i); whxch, hovever-;,
has a ccmple-cely differenl origin, bcrv^ausc such occvxrenzes pre-
fxiflted ia e^. Iliia e',casple chtXiis that i'orms g., are detached
from the Interior of the correspoucling cocrponentfi of unother fora £,
then the globcl notatjonsl distiaction of vsri-sble iettero c.n be no
longer preser-zed. Hcvevcr, If (5) i« replaced by
/ |H{j;) V / JH(e.) r / |H(e) s
• 19^: Is .5 c-squec'
^'^T Out of -i :; -Mti;
lisv int7"v>dur.ecl bv ]^ev,'eiJ. end
-3-
(the free: storagf; list. ) as soon cs their rv; nteatr, becrcroe Irt'elc-
vanl- to further proceuslng !'5ainf:enanra c.r t.his po«I is carried
cut sutotj^vicaily by the ahstrsct rcmpufcer. The perceritsge «f
iL'emory space devoted to orgsniaationa i overhead is thus constanii
with respect to she Ifcngth of the eggregat.e^ stored there. A3 it
will be seen in the sequel, slso the percenvage of time devored
tc overhecd cperatioife, sur.h as addiessiog. Is alsc? constent vlth
respect to the length of the aggregates operated upon. This con-
st-ant; ratio of overhsad space and tittse is an Iniportaat pt?cuilar
featuf? of thio Zits^-roct computer, and tr^xiy aspects of its design
have been. devJsed in order to provide it -with this featvire. The
study of coaputation ^chesei; vhex'o the ratio betveen average
organisationel. overheEU in -"p-^.ce or lima snd "ot3?i. space or time,
rsspectively^ increasei* and tends to I when t:he iengch o.r the
comp3.exity of the inforc!8tion to be processed iticreoses, rady have
3 cGi-toin tGetbeffistical interest but. cannot ....'i:.!?ibly give good
;5ndlcstioas for the euveiaceEent, of the coisrputation ''"-' thodology.
In fsot:, Khlie a coastsnt ovarhead I'atio^ even very high, cea
potent-lally be reduced by Gkillfui tailoring, this is not the
esse for overhead x*ar,tos not bounded abo'/t:-' by a nu>uber <^» lo
This iG^ in ossenoe, the engie fro^a which the desigi'; of the memory
orgeiiizction of this abstract computer should be viewed.
For '.
>•«! ^ oc
re;.oec; i '••«!=>•, Let; S denote this n, -£p secUon of 3. Tne
^- )"£;> I:-L'iS.^^£lLii^ ^ ^^' ^ il^-t consls-f-lng '>f aXI voi-d? of 3 no;
oaloiiprirv;; to '?' vho-re. if n. - 1. the contents of the liglit
fleifi of the {n.-l)- zepla-^'d. hj tn?" eddress of tnp.
(n^-rl)-th woifi or by e 0, deper letber or ; . .:<
The Ini-ttal ac' •^A:>
£-i'"IJr^'-i^^^-' J^-'-cr of ^
or the cnc ^ .-iv-iii v ••:= -^r :ng on v -if.- .
n. >• I, vl- pqI 5cldre;i. -, -s of S o?--i.
;, n -i^'rh word. deper.dlnfi; on vtjetber c-r no rv.
If bnfh n 1 one. u, .13 . h- - '' vi]/' ce coiiveied c:' s^y ^i<i
fhs n, -n._, .
z-idr esses of t.>:- -. • v. -na u, ,-
referring
those cases wlisia v.. oiid '/^ «r« iv.
xae uouCl.;- ., , gdaresa section anr
t^'!nft I'rcta ttae n;^'*ih wx" •. .ha
-'•nc •* '•. =- ".' invc'i'-ed
Ce.r-.i.:.;i 1A£ t-?; :>i2ich ones ej:>ictly v^-111 be diBc.u.r,r,ed. la tlie sequel)
need s perwHrient r«feren- 'r--- -- - • -- - -- -^;
proceisseci by Ti;e abfaira
List S, a word cabled eoni.rol .y^jrci v'll'i bf=; se*?. Igrjc-d tc .iv, , liiis
word -vj 11 ^lo;^ h^nve chrae fl^lci^, c-f vbich ^i. .'^:....-,
%he flrs-fe address of tiae life" .rai vord
1.S displayed c-c a . : • >• •
>;f nhe li:^-i '•: .-i:;.',
•;
'i-- :! •>:' d l.n odd.T' '
i03
atains s
Section S.g f-'MP;'-<^iS^^,._l^.-L!L-
Let S be o llsir, « -'^n 3c;w,--eg?.te ar.o r; s positive iareger, ?aica that
: n + K(3) - ] i!
Let. 3: , • i^si, 2„, . ,1,'. a_)) denote vhe J.~th occurrence of a rjark in £,
Tf for ail i iihe nsi-k occi ',
.1<^' located In r-htr centre
fi.sjlrl of the \n + L(s) - ^' ^i of ". h^';' ri.f: o~;-;-e-c;te £
Ir. S8i.d to 'oi? r>gllcc3tc^'^. cf
the n-'J'i^ ii-ora of 3, •. .• r csjd
t-p bs v-ad(5res3 aJlcrexed jip*eg«tes ara fj.t.i.ocar.fid
In listo in t-he rever-se ord?r, j,:i tD.e :;'.x.;:: ::.-3t inp • vr-
reacs of e •''.•
•^- • - -^ - •-^'^rd. etc. '
' -
it \a tak-u ^^^e -, v^.-
der*nir:;x? -.?:' ': .-currcinc:^ • vhe esse
of occiirr^ness of pgren fixes es.
«;cteasicn shouJ 1 • .
;..: r>r cioseci c-;/c. ',:;;n;'r -r
rlr^ht- or left., xer."r..'->r'v.lvp!i y. of q,. In ;?
.
or ir f^^^ a), xf Ui, 3n deFtii is -l or +1^ de-
peadlng oa -^aeoVier it is o closed or open psrenthee
-
£.. '.n .-ft de-pth is asfiiy^a i: 1"^ ;-j^_,..=2;-:I.
If It >_ , ..'v.. ... '^'^(£~j i) "^s either r^ (ij.i'S.' '^'^
LV^ (ja^_, ,a), depend li^^ ,-r or not a^ , v?r th-" '^rcvTTi^nco cf s
c-losed pareniibecis. Conversely, D.(e. /,,,8) :
0. , .. .'.a tie, occurrence of a ci.c2t.-ci o.^ open p??reuUr?su>, re'ip'^ct.U'e .y,
ac /I D ,', 3 , 3 ) . for i-i. L< 8 ) , ,t 5 D^^.s ^^ .,
, e^)
-
^ or D^;e^ .y £ '.> <5^?2tidlne
'.'f a •>!> a form, left and right deptbi- thus deflnr.'d *»:;? ajvays r'.:3aal
and tie-y^r aegata-v?-
After reQiadiDg tbat t'be rlg,ht. depth of ccc vi-Tiences Ci mark£ io
porc-iel forsiS can never be negative, let <_. he sa oci-urrence of i it:i-t.
deptb d_ of aa cpen parenthesis in ••'••' -••''' •"• ^-^^i^; a. Then there 'rr?'
be cccurrsac«s of depr.h -^ rf c3'"S'. cuasas t.--; the
right of £ in d, sad i ,
• igbtrsost .
Let. us extend this concept cf laot-e to v.he esse vhere a. is an crifur-
rence at dapth d of a ccHUca In a. If chere ^re in a and to r,he .:-!<:::
che lefj.n^'.^t cf theoi is zhe fsate of a „ OCiienj.ise, ^., Is scsii '
uciKL'ited In ;3 . If a is the OKate of .3., then a, vi'i I be c^jlle'l t;
aatl^ateaf <.
ft' £ contaHas pcciinrencee of coocj^js of dsp'M 0.
iefv,moK( . 3 is cai.lect the Inl t l.a 1 icti fce ox s, Otherrfitie,
£ Is SDld to be la^
.cence cf dflplii d cf :i --rv.-o -. in a, thea, if'^LOtie exit. .,
in £ to -the 3,'3ft of p.. occuxi :. . .. .- ..psn parentheses at depth d,
then right-ssost of them 1g l.he cTjste of 3 Notice that the snticnot^r
o-;- l/uc Evate cf 2 .,.le not necessarily a. .
For ey^mpie, in t.he form: the roats 'M
.he rcate of ^-S-*^
/ 'he "ccurror.-:"''; ''U psr»^n-fh.
cotr.rau. the icate of the second occur rc-^rice of a ccsKaa is i:he occur-
rence cf the closf^id pa rant ties " *, valle the first occurrence of a
co-irra is iinfflu'ed. Tac initi^i mate of tblo form is i/ae first occux--
xeiice oi a ccmts. The fintimiste of the cccu/rence of the ciOBed
parHnthrjais is -cas oecond oce.m'reuc-e cf a cornffia, wjile it.£. ayte li-
the oc'urrcncs of ta.? open porentheaifi,
Sy ffieans of en arGU!;;f.vat similar t,D tho' one used. to prove the canon-
icjiLl decimposition tb.eorera one can easily prove that, the aggregate
consistiog of a3.1 ozc\o:retiC:ei>' of marks ia a foiTi i.ncliided between.
p.n occurr'enr:e of g mark «r.d its r^vte. esciusivc of ext-reicea, or
piecetiins -jn xizisoted occuri'ence of e corisa, sre forais.
a-^ilooai'ed
L-:^ ^b? i^, in the iiat S vllb connrol
V'Crd Gad c-Uui.HOae tii,
of " - ;;3r-?r!Wiesj.G, the (n-.'
^:;i:' :rA ccc\irr'.n:; • n ^ . .
.
ii^ fU';'- fi?ld t))/? sddr^:- - : ';ronds tc
:t.c- ; is fcfit-jd, or if a: ^^ xurni^t^d. the address
•.' c-:e ;n- : --.r; vord of S or cne one of the control vcrd of B,
depcnrling an v.acrhsr or r . Lia)-I.^ aJ-.S),, suppose .further
>/nGt the ?';;>'. field of tr.c .,.-:;,.». . . . jrd of S ccntfiios tiae addresscarrying
rif the word of S the Initial r^^t? '^f ?:, if it exist,
o:'. if a is inli..i.elly uoEiQtcd, !:ae sddreGU cf ' -^h word of S
if n ' -.1 the c;orrtrol word j;.\£)-i^N;S),
sup;/ • yoovo! address ia conta?^ned ia a LJp'.icxa'L vord assoi*.-
>
'.e dofti the eavae vpt^rn.-
' ling » nisLed i^^ccurr^^nce *?, •.;
iuc Lu^', .. '9.. f5x-t.rfn*: <?
J of tt'> • waere fi'j-av:;
tilt, iaroi . ''tr car-ar-!- :'
, Uxe corn .
.
;:rejice of a caarK after tae other. While t;;-;-
Me occurrence of a ^rk throuj^h the
. . proportioutil tc the d-^pth of euch
-.1 the leri-(.'': .{' th^' r.j^ai Jri^'Viivefl.
.• 1. c.•'' -ailaed by •
of tiie i........ .. .„.. , tt fietsfti.
Y £\\*.+ f;blp l'(ir i-i fiximnr. a«cnje«t^i
i. >6tCHi thrRUdli'iR Is? quite r.;f-
'it-ytrfict touiputer all-, w-,
L,ii5 directly ^nd thus ler-
uf logisal tJtire^d,i Lig, the
ii --^ing Xu the ahftrsct
'i;<aTe»K«ites on which
rc'-r.fl dettcrisents el'ficieatly . xv_c
r3T i of ovej-heati ocerailoa:J due „
. J ;ter Is uidetiendent: of the & i
.iple< for; iy as «lap.
alng, whtlP! beyond 3:.y
wou'.d Itapiy a ratio -t
-pfv^'e" titrorcly l.n<. <•
. K-h a sovj
,. v.iuptious ox" the dej.:iirri of tbe abBtrfica ^iooputtr.
-^ also be noticed that., while pbyslcei threading moves
'.£ itDbedded In 11 sts^ lop
, mx-,u i.ac uiiiy exception of when occaxVoatS-^ csl c.lu'
'- ';'.?red. ''As previoaaly said, wordb ol mesory
iree fleldc, :>f vhlcb th« central otr-i >ibr;'
'
it i;i..Cl
.tea, the cidrccs of the
ii-al list;
whic±i ua3
subilut or S conststine?
;?:. r.)p uf tr.; icg.l':;-3i libt.
,
:£ bot?:i defined to consist
rse pny^lcal threadlag, is that
j'der, and toa end
i9-
jfcr phjrei'cgX threaa.ing ic the beginning for Icgioal thresdiogc
jThe units ojT lufoxuv^.ticxi of vhich the abotract ccitrputer oper-
jstea are towjS^A t-uelr canonical coGiponontp,, not Earks. and vh^n
cancnicei ccirjfionenos of a fcna £ icdbedded in a l.;'/;1cyp lii^: ;"• .irv?
to b=> rrparsted upon the i-th of thea ia found "^^ imasdlately,
£».lnri? its .rjgh-ctiicst occirrenca cf y ir.^ri: is tr^, ...-"^.-e,
Th^ addressing
£:yst'?in of tbft abstr'^ct oomputer Is based on iogicni thi'e'uiJ.ng,
and '-'113 should t.bini; of a logical list _a in -cercis Df a list or
lists of lists, etc. linked by the logical tJireediag rather thea
in terias of a list of vx)rds linked by physicel tnresdins^ T"ce
i-.ole purpose of -0^:701091 threading is to achieve maxitBum utiii-
^aticn of pbysicsl storage, and the reason why it goas bacln^rd,?
jvith rep.peoT. to logical threeding caiy dep??ads on the s'xie of
j"peratlon of the absiract computer vhich, ss it will be explained
jin chapter 3, must read the fonas backvards in order to e-i/gluate
I
them.
Iji the metalsnguage the conf^nts of the top pcintsr of logical
lists vill altrays be displayed on a Una at the end, rseparafced
from the conicats of the 'rords of the list by 3 facriaontaX
ruieo
In the sei} isl^ s ll8t oxhei' \hs.i::L a logicel list vill be rcferre
to as- physical XlBt, Physical lists are not suii.able to carry
3truc!;ui-ed inforxstion such ae partial fomis^ and th-sir main usage-
is connected with storing and sddresGing in sequence marks, tkinly
constants ^
20-
Under the -aijtsuEptict" tbfil, ail of storage available for ^ilo-
ca-ttnp; nssrks is origlualLy l:i the free atcrRge ilst, iio proTjleras
bb-rraJd ari^e 3is far ?ii? the n;s;intenanoe of pbysica', threadjag is
concerned.. In fgc-tv adclltiona, secliioas and re3:inDder? de-scribed
abcvs are sii operstions waicii pres-rirve physical vhyeadini-;. But
this is not the cex-fe cf loglcsi tareadlrjgj, ualetiS somecaiag is
dons .Mouut it. Ihe scherc*5 dl scusoed lu ttiis section 3j.1gws to
so3.ve. ta-is problem by aaeo'i'jeting to e-JCh icgicai Xiivc S enocber
lis?^ U, rsl.L'?a sequenctxig I r^-c- of S. surh thv^t the Itxgical iteia?
slJxvcated in It axQ nor aggregace^ but. .stsply iriarks, &-.• that;
phj?sic;al thrS'Stiing c^n bs expeptlon-aily used sJjso for adcire.ssing
pu.rpo3e8. An glgoritbin to produce the "U'Ricsl t.areadlng of p-^r-
tiJ&l form.';; Utbedded Jin logipai Hot .is Thus defa ri.ed :in ts-m of
the allocation of Eai-ks into a pbyalcQJ. list..
'ibs cperailcQ of the simple pbysi-t-ai ij.st referred to ee L? in
t'uis sectiou is at sil similar co the one of 6evlc.es., BOG3etin?.^p
celled puah dovn :Uats or :?Taclts^ vbicb era preRent- In VTograa'2>ed
forra^; %n a;jst prcgraa corapiier-3 for eoa.ven-sion.al con5pu-t;er3 and
In wired foi^ra in certaia semi. -conventional compuTtjrs.. Such Ii8*:8
are lists of Erarks, not of struotured ijifortcat 3 •?n, and thsre-
fore c-an be phya.lcal.ly addressed . On the other bend, the IqqiqhI
lists used by thia ahctmct computer are a taore general concept,
for waich there is tio intiii.tive equi vs.'.ent in ocnveiii Icnal o.r
£>enil-conveni;ic-a«3l mach.tnes ..
Let a be a ptjrclal form cmd a^ t;he i-th ocemveace froai the
right of a r^^Tk ma, Tae fir^t chins to do is t.^ devAse n .im^i^fcr '
''^'
aigorith^. assccioting to each £^ its rigat deptia. For tbis
purpose, one u^xl^ze^ o sequsnre b^, f j=o„l. ... t,(a)) of nunbr^v;.
and 5 sequence c of trutii values defined as foilcws:
/^i-j ^ ^* v-iienerer a^ la the occurrence =f a
Iclosed VGi'^vt'iiisis
k-i " '•• ''^^"'^ver £,__ :-ia& the value fL-J
b^_^ ctherTrlsc;
•^fblle
^^ _C% wlienevsr 3^ is vae o-cur-renee of an open Daren^h»«i
\ / otasrvise
I^^^ purpose of the definitions is to be able, for ail i, to con)pute
^he integer D^(aj,a), vhich equals b or the basi^ of Infox^ation"
inln-s oo a^ and a si one
i.'- i:/; consider the prcbJem of proceeding fron: tne parciai fora a
to seneratlns a losicsl list S with top poirrters.,and control vcrd
L, such that H(S) . !.(.), in vta.h a it ^-inibeddod. The b^sic tool
for this operation is s list U, vhich initially con^iste of Jua,
one vcrd, containing In Ite centre field the address of ^.^., The
list S is Inin.^Uy e Q-Ust, conssquenoly botlJ'^,^ and
"^-'^"^_o^
address of ^-^.Tiie algorithm consists of L{a) steps each
of whlcb stflrla with the ccmput-anion of b^ ':iad 2, ^^'^ '^^^ addition
to rue. '^nd cf S of th^? xaarli m occurri'ng la Sj-. tiii« inrpji<r:8 the
replacejiJ-rni; c.f,^£^ by 1:he p.r'eviuus coatento -^f the left fJe.ld of
the ccnlro? wortl cf tne free K'tora^ llan,
La? u d'-^note the .1-tti word of IJ, Tben. if o, is « oloaed pargnrb&ess,—J
i- _i
nc--j w.^rdr, cfarry.lng n; , .la its centre field and the address «^ Sc3« of the
i~lh vord of S in its rlfjht field Is 3dd«d to t,be h'Sgiuniug of Uo
If ra. is a camni^, <5^ u le placed int.-- the rigbi: field ct thf! _{-tb
vord cf S, u-ho~e iddrfcs iScr(^£o* ; tbr^n, i.ft^ u^ is; a coosna. 'Ji u,
is replaced hyji^ i<?* .- wbil'*. i*"??H.s
^'^ ^ clostvd par-cthpe-is, s
r.evr v?ora, catryin^ ^ ,i.n its ceatvp field and^^So^- iu Its rlgiit f1.#]d
la added to tiie b«gina.Lr.tg of U If m. ts an cpen parer;i:he£i.s,;/i u,, is pis ceo
Into t.he right fleid of the i--tb word t.-f S; then. If ^ is the
&ii3ile3r. pOis.ltlve iategei" such "h-st'/s, a^
is 3 ciused parenthesis,
c^M<^ is placed inxo the right fieid of the vorri of U tiavirifj 6b u
,
liS addre^^s; flnall^v,, the fltsf j vfordc- of U are eraof^d and gtveri besok
t.o free ttoragev Xa all other esses no f^arther miction -Is takeuc At
the end cjf the performsnce cf the s-Igorit he, U cc-'nt-alne as oany vords
ss the difference between the nu!i;ber of occurs:eticeB of open stid closed
Consider now the problem of proceeding froia th*». above logi'?«l list
vlvh a^ i-iiab^dcied In. it to another logical list S ' rar^u-vting froni adding
V p^ivtiai express,! nn a* to the end of S,- T'hl& operation csn be
•?-3-
eaaily performed px-ov.ic?.eu tbac the statuo of U at the end of
the generation of 3, the iatec^r ^t ;" .^ viitch is the .length of
U alnvoj one, and ubs tra^h value t.^
. which id $ iff the
nu ,• •, ;iB a oJosed caren'Vhesis, sre avaiiahle™ In f-act, S' can be
ganerctsd Just by continuing xhe operatioa of forcing S 'fceyoad the
LU)-'t-h Btsp, C'jiisideririg a" a -r, sxtGnsion of £>
This proceQur'2 alicws the relevance to the ooexatians of any [ioglcei
liitfc S cf three items, nam'sly the ilst U tbe registers b find t
vhich ot, e9-:t step conteln the. right depth h. and the constant t.,
respect iv.?3y. Ll8t U Is called as6ccfer<edl3st of S, vhile the
coaipoLmd of b and t. vhlch, for ail pxir-pcses; cen bs pieced In the
8 rise vord of it^.-iOry, is called che Rtsaociated depth regi ster of S
and denoned 3^,
The above bestc prcced'jxe csn ba trivially ext-^eadcd to n-iiBbedding
of par: iel .fores a into logicel .lists S such that K^S) n + L(a) - 1.
Fx'om this point on, if s capital Roasn character vrith cr vlthout
£ur.H2rscript;g danotet:- a liKt, the corresponding iov'sr case charocter
vita the same axiperscripte end an. iateg«r sixbscript i vill be denoted
the l-th word'Orf thao list, if i 0, cr its control word if :l^-
The Byir^bolic subscript je \?sed to identify the top pointer. For
example. If S is a list, then s 1'3 its control vncd, s^. its- fifth vorri~Q -5
and c., ^ its top pointer..
• - Mat log '.'-f Crtidltloriti .1.j^'grk'r
T^.ert' wtii be ot*^ci for m addUlcn to tirie U'^c:'.'n thr^oci J.r.^-
wi whic:b syitT-ic i,-hA typjfcsi vriy c-f sdrJretis.lruj, of the abstract
cr.rjxpurtr It; beaed .. Tiiis addition is irrtrojuced iti -.-^rder W aii.vsv
for tiiii^isr ov»»rasl pf'fi::leacy - ^'l c.f nco-J.nhereot, uaootu-
put.!(i/lL,U,y prc-isXeraaj, as will - •-l.y exyi'.aiaeci. ia cii»=piier 3'
So f^r, aot beio^ yet ailc.-v-d • '--'^r:iat/s to pcT.st.lcs.w, it,
ie reth-^r awkwvjrci to preswr.f. witb- stior. fhis •sdalticcy
con*xs"Cing of tJii^; b-gc'cy-arj Ic-glc-s.'! rti'-jf/ing cf 1-ti^ ocyurrt!n<;«!S .'f
the roiidltu-nsi- aarli^ .j"''"^ acvi - - -e^;«ry v.;.
do Sfj .In »5i-dff r.o corufiiete. Ui-- j.,.,-.
Infers-'?*? i^•t.' .Jfi cw^ac'-rv.i
Let a tw? 3 pan 13.1 r:;-rau h.. \i-i.-^ , ; i ' i^^. >. v U'S i-rp .-ccurr-^tice
from the right • f <> a:--rk':c;; 4 !r. 'vht; cmtk occ^j,rv:;j <r -; . ^nd ••
th« rj^^lt d«?p^ . u ttift 75s?r'_ . ,
tb5V ^ "ororsa wccur?? free in- «s . .If tn^ *.£< a coinjiis au-j ttit-fce.r .'^i or^
a . Ito n-:.t, 'c' s^ Ir the'
i>- -urieticvii^ .if 5','f however, ar? of l.tttle cr no relc.-rsnce to
-soraputatioxi) . For examjvle, t&e oete of the occiirre-'ce of ^'f
in th«? fQ.r^
ic 1:tts second ocf.utren'. -' .• i/.-Jij the iei'^
a Qf ^mbe
be icc^d
two rx'^^i
:>ntrol v re
the i-\h -'-.tii^ li :.., iv tie
o
^atJlng ojj'
cont?.1j23 In its right field the addiess oi cnw i,-vLi vvrO. of th£
list S, vhile the first one contains in Its right field the
x-ight depth B„ o^" £ and in its cent::? field n . « If a, is a
conra, then, if octhC' -'i^ j- b, and()',r,"r-
h., . no action is t'^ken.
If^^i-'- - b. . V. = -^y nvA'C'^l -'iTxi "t-bie meana that a. is the mate
Of the oceui'j.'.sricr; of iSf"., lihose ao.dress 5,sl\ jI . and the action to
be t3kon ccnsists of placing the address cf the i-th word of S into
the left field of tho '.rord cvf S vhose iiddre^o i3('^,u^j and taeu
r-emovin^ tba first tvro vords of U' (that is, rcpl.ncins U by the
2-rnE:;jai.nder of U )"
Another posGibillty is that b. -aS^-\} '-'» ~''f '^'^-^'^'^/x'-'^ls'T'-ia cBtse
In snslogcus to the first ons, with the cniy difference that e^
is the mte of go orciirrenc;-; of -s J^"^ Injtsad of y^\ < The action
to bfS taken Is eKactly 'ohe orre of •'•he firet case, vith the only
difference that o',^ 2nd U rejilsce o*!! and U"^,, r6>.£psctivelyc
Finally, if neither '.:-f these two coablnetions of conditions is
sstiBfisd, D^ is not the mate of anything and no action is taken.
The basic difference betveen the operations of niating occiurrences
of open parentheses or corcaas on one side end occiirrences cf closedon the ot±<er
parentheses., <f' and }S\ /^^'^ imbedding purposes is that- in the first
case one should merxorii^e all addrssses of possible cstey and plQce
thea vhsn the occui'rence to be rrated is found^ vhile in the necond
caee, since the occiirrsncc- of closed parentheses and condition msrko
pr»-ct^'J'
; be:" •? known.
I:;• ,1. with -^fi.
!
i
I I
J/
,-"'
Date Due
3 TOfio 003 ata ito
3 Toao 003 ata E3t
7-^3
3 Toao 003 ata i7a
3 TOaO 003 a^T lib
iiiiiiiiiiiiiiiiiiiliiTfliiiniiiiiiiiiiiiiiii
3 Toao 003 flba n4
S-^
1^-4 '.^
3 Toao 003 ata eio
3 Toao 003 ata ms
3 loao DD3 a^T mo
