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