a ~t~ematical approac m to - uni-saarland.de · 2019-01-10 · a ~t~ematical approac m to program...
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![Page 1: A ~T~EMATICAL ApPROAC M TO - uni-saarland.de · 2019-01-10 · a ~t~ematical approac m to program bemav!ou r ahd loo k-ame.o.d replac£iiemf algor ltmms otto spa nio l a 7h6](https://reader033.vdocument.in/reader033/viewer/2022041809/5e569e17fdb8f71e44045952/html5/thumbnails/1.jpg)
A ~T~EMATICAL ApPROAC M TO
PROGRAM BEMAV!OUR AHD
LoO K-AME.o.D REPLAC£IIEMf ALGOR lT MMS
OTTO SPA NIO L A 7H6
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A a ode l for p.oq'" b.I",io •• is d ... lo • • d ~ lI ie ' h ~ . l l
.uiUble f o r lI i~1I0' 1 ... 1 h . ~ • • ~o p,.' '' ~ ''
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t urn , out to b. '.poei" to til •• ,u.l d . ~ . n d 010 1. 0
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I. lIo ll .. en P,oo" .~ l e r .pr lC ' e . b, . tiel, l e't l ol .
~. f d • • G. u.d l •• • diu .. ' Iod , l l, wl.d ,i . loo ' -"o' d
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in . 1. di e Ub l lch." n ... nd_P'9 I na _ ~ l oar i t!l .e •.
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2. lME MII~~H _~ ~ 'ERFD _.~UCE C"HRIO~ FO~
LOOK·AH[~D ~ LCORl l~~~
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P.gu I . ~", o r If th e ,.nto nt< o f XJ(i) hav. not , h • • Qed
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by. r ead_only bit ) .
~ .eplo« ••• t .Igorltn. t . dete •• lne. by the followl'g
u n] "u'" : 1. look- , h ... <r H .. t.n : Oohrol nltlO. o f t h .... _nlen
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b ~ ,.pl".d 10 ' ... u, ., tnt t "to. log 0 '", ?
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- , -!. ,id •• tho '~ol<. 0 1 • 'p.,I,1 ' )90rtt~ • • tho t.,ffl, ~ . t w •• "
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" , 10 of orog • • _ l . 9 . 1<. ( cf [~l ).
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uppin,
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[ H Lpt 100. thu H.h~<t.iu. o<H<~ . hold bL •• Hq<~d
I"' ".g ': .tupu.ut'-.~ 0; • cut.' ~. ppi.9" u~ b.
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?r><t i « t,. ro? lo co • • nt ,o , to t . n. t o b • • Iho. f 'ICt i on of
tno l ood1 ' 9 ( o. t, (s]. ~ n ",act oon. id",Hion of r . lo odlng
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1. 2 Clo .. Hlcotlon of rephc .... . I . l go r ll o. ,
~ <cor dln 9 t o th e " Ite r l on wll ie n , ou' " th e 100 dln9 ( " 0 re -
100dl ng. 11 ~M 10 full ) of on . or . or. ~ ' 9 ' S ot • tin" t ho
f o llo wing c l .. u •• f ' e p i ocoMn' .1 90 , 110 .' . 111 h e dlH ln 9uhhed
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C1o" lo.~ing <.ndiU.n ~AII) I
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for •• n« ,,11"'0' for ORA .hon for no.·O~~.
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- , -_no t he r • ••••• f. , I_e """<tl o. "" O~~ I. the f ol l~I.,
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-. Th. U •• nd P'g ' nS >'g.dtll • ••• 17 dUrn 10 the ,oplace-eo,
.. H rd.R ( l~U , rlro , Rl~nOM, AG
, Bo ' e l CLE '" •. ).
ThUt He '.,"'U l I09 ,."<1H H' o f d • • tnd p . glng, • . ~ .
• tae' , 1goriU ••• ,"0 priority .1 go,l\nw, [5] . It ~ Ill tu rn
•• , thot for t he lOOk- t hUd . lgo , Hn . l H , whl e !> B der l.U
In ,n,. p. pH, the , tao, ",oputy 15 not '. lf111 .d.
1.<.< 0 .... 4 pr . pnh9 and 100,-.h .. d .12"Ul>w,
To t h . .. UO" kftOwlod go O"Y ' ''. po. e .. « ( 3] • [10] ) dul
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. n ' l t .. first stor'g" 1 ... 1 i. d leocll , . U ruubh ) .
h~ .. . ... Ion or on ( Ooe Bloc k loO k-thud ) .nd Hs .odl
f teo,I •• , Sl ( Sp,"O ' Loc ollty ) ,nd I t ( ', •• or ,1 toe. l lly )
oco d U Ud " 'P'9' 0 9 "gorHll w, ,,n,, ... Jo.Uh . O~l and t h
.. ,.,.10. Sf I 510.10 Predlctl •• 1 or o!t H. , •• • -. hud
.l gorltll_S.
In ' .W' HY. e • • " os, worts " follow, ; ~.,I~ .. to •• 1"lft9
P'g"! ( . nic~ I , ~llCod Oft t~o top of c ~ " LI U· SU , ' ) Its • •• 1 9 ~ ~our !q. 1 (If not y o t pro , o n' I n ~M ) I, 10'0"0 ( o.
th ~ot.u pI ... o f t~. uac t ) .
Thu s 8'1" 1 Ii r . phno .. t h O'U'".''' o f I ~ .... t p ... h vlt
if tho.o . .. . 0 r oforueo Oft t h i> p.~o I . Uo ..... \ 1~o.
8, U h 0'91.1,.110. "hich i, clo .. ly r ., l.tod to t ~o llU·
~ r ln<lp l • • "" II .0' bur d" . od wi th p. ~., s wh ic h ~ I ' O b"".
pr"p'9"d IopodluOu , ly, but on t~. othor ,10 ' t ho off."t of
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Jo.eph, •• c, I •• Of OR, i , • 100k-,I> •• d- 019 •• II ' _ , bOCoyS '
~M Is '."TVO. to ~ .ld p'g", ~h\ch are 1" .0. I n .d •• n c~ .
It wi ll turn 0. ' th.t DOt' ORt .nd SP Ho ,poe l ,1 c"", o f
LH w,ien will b. c.nst , .c t e. In 'H tio n l by • •••• o f •
•• t h ... t l ,, 1 • • de1. I t .hould bo not e d I h " 08 L. $P , 5l
ond H '.H b ... d.,he. ~.( "11 by he u,h Ue.) •• g • • on t<.
1.3 toooll tl, Mod e l. of pr.'], .. I> . h •• lo u,
A du ,"d P'9fn9 . llarHI> . I, "HI., ) if it , t p l. ,< . ,>t. p. g ..
wit" .... 11." 10 ' ''''. dl st .. , o , I. e. 1> 1 U • • 1"lnq
p. ~. ( 01 90rHI>. 8. of Be l . d1 !'1 ).
Co " o.pondlng Jy o . opt h, 1 d ,~,"d p,.p.gln~ . loo r IU. whic h 10
.bl. to load. po g .. at the sue t\ ". " Ill , . plo n tho •• 'Q" o f .~H "Ith grutes t fO'wor d dhunc. ' I th e \ pog .. .. htc~ ...
not 1 0 ' pre ••• t In ~~ .nd wh i ch h ••••• • 11.'t forwa , d ~I't • • c ••.
t o pr e dict the fulur~ bo~ .. lo u r of tho re f .,"nc .. by t~~ r .Cut
put ond to reph« pa9"' .. ~IC" we upoet to ~ ••• grutnt f o r _
wHd d(Honeo •. Tn" .... applies for I"'_a~ .. d .Igoritn • ••
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• • • 1.4 O .. niptl0. of . "ot~.r no •• l of P,"'i,U boll .. ; •• ,.
~''iton . of • Rro9" .
In tho fo ll. w ln~ , .. t nuot l c. ) Md,1 I, ~r • • " t . d wM e l!
I.\ ,.d., .. oropor t l • • of lo " l i t~ whj , ,, H~ ob .oe • • d 'or
... y porto of I F OUR~~-
. od . l wn l," .'Y h • • ,, 11 ., ~" CI . I ' y for l no pl n. proq" ~'.
dulO O PrtPogl ' 9 "" 100 ' -' '''. "~orlt " .' ,dl1 '" d.d,.d.
1.4.1 ~ ' 'll o",
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. n I ,
Is d. o to prog ..... l'g n." " "d tho ,t,.c t """ of ~ro9r . ~~ I "g
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, '" . ,, ~~ 1 • •• < "YO • ,.,I . n I of t ~ . I'Hru<~ lo ", of .
orog' u , , 'o 'l 'on V of • • ; ~ol. vor i" , ,,, " ' ;0" ~ l' ' l""
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n. nl " 00 " In [~ . I I] I ,to.
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for ,,.n, 0'"9" "' •• H, ttl on of ~~ 'n to • or S .. glo'. "i ll
b ••• ffle i , .t. 10 ,.~ •• • t . nt, tnl' ~o l d, I.dooodo.tl , of
th o pogo 'Izo.
Ih. n u~b.' of P09" .. ~I ,h • rooloo ""upi . , '0 .. " • .,i ..
.. I'~ th. " ~. t. ~ •• Io •• " , " to . nu , ~ ", of . O ~H 'n . ,~
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In t kh .. ,tlon, ho on or , t h o f o ' l.wln, ,i .. l il1o,tl0" HO
I.,rodu co d
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Of M~. ~o fr ' O ~'"IH'on orobh~. , . , con,' d' rod.
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If I~ . ,.1o tl o .. In quutl .. h.' ~ f or ~R • h' > 1 " "1
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If u , q t hn Ih e loUI.; . f no 19hbou ro of ~q "Oy b. "T l
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1. r > k/2 + 1 :
min(k-r,r-1) e 3 =.L (Y- J. a - Y-k+J· a )
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J=l 'r ' r . L . (Y- J. a -Y-k+J· a ) J=k-r+1 'r . ' r
= min(k-r,r-1)
L (Y- J. a - Y- k+J· a ) j=l ' r ' r
Thus e2 + e 3 = T ~ 0 concluding the proof.
Remarks:
a. Except from the assumption T ~ 0 all the conditions
of theorem 4 are consequences of the relations
b.
y. ~ Y l,a r i+sign(i),a r i + 0
Yi a = Y-i a , r ' r ( 0 ~ i ~ min(a'k-a a-I) ) "" r' r
which have been motivated in 1.4.3.2. for region V.
If Y i a ~ Yi+sign(i),a r for ; E (- k+ 1, k -1]
, r
then only the term d2 (or e2 ) of T might be negative.
In all cases of practical interest this term should be
compensated by the other terms. Moreover, these conside
rations indicate that our assumptions are satisfied
rather for larger k than for small values of the
page size.
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- 2~ -
<. I h • •• 1Idtt1 Of tU rehtlon Q 0, '
d. A •• ' U "it of t hoo,"_ 4 In tddttton to p.~, 80
of
.,. n •• d.d to .'ttblt,h
0t i •• ~ Q~I . , etc . .
2.3 . 2. 109tO", st"uctu,U H~. I
c. 'J.c •• > ~< .•
OI~t,>I , I , I t ·. I
'" '"
'"
jll
pt • • -p t •• . -. [I~ ' 1 - I.~ . ! " j . o /.4 r -J .4~ - 1,A t j 1 1 . ~,
jjl iHli- ll .l,li.IIO-I ]
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• l6 .
b. Ltt Yj," ~_" ,j" j. _ < 1· .. 1 ... , . «) .. lo .n
J.ri~ , '" , , , , .j"'\ .ilo,j!11
1/ . I ~
• I p~ "I~ ... ' Yr .. - Vk "- - V •• r .. I ~ , r' '~' ~ '. . •
" , . "
Proo "
•. °0 , 0 Ql,o
k k-, I P .( I y,
r.1 r,. j._ •• 1 " r ,
. -1 ~ -, (i .I ) _ -r > 19 . ( I y .! y - I J )
r.l r,< j. - r'\ j,'. j 1 J " r j o r t_ .. 1 j ,or
'" .-, 'Plo'(I1,. , j oo ' I
.-, .. , ' (11,
.s j - o " 1
'" • • P . ( i 1 1 ,0 j.- l. 1 J , ' ~
" - I , ) jo(r_I) ~ '1 j" k
(1+1) k_1
I " I j .n "1
" I " I j _ (r _ I) I. 1 ' Ok
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• 11 •
{l ol )k'\
. I' -j \ J " \ j! \k 11 , . \
• 1 , j •. k. 1 j '0l:
,-, , ~ \ .'{ L (1, , .,_, 0 ) • 1_1. "
' j. e ' I ''<
... '" • J<[(I .l t hl. ( hl ) " 1]1J, O\ )
Jfi
b .H I;o t hon
Q I , I Q\ , I
, lk·, • 1 P . ( I y"
,_I T.' j' k. , .1 'r
(hll '_' - ,) j .I, . ,. \ J,',
• P, " • ( 1 I "k - ' (I .I ) k. I",
) .
" • • , tho n , L " ,
• • ,., QI, O • " , foil< •
" 1{l.1) "1" . • • • , . •• ". , , { · .. I, .. , ·]).
If • ! , tn .n by ,.,._ptlon
• QJ fo , ,11 ••
! ,-. • P . ( I Y
• • 1 ", j .... 1 J,o,
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·u
••• . I,. ,'( ' •• " 1 . p . I' • • • •
• •• S t. « r !l~r • • • , . Q • • ' >
~.IH u'" " .. ~ .. ... • or o.
,_ . " , . 1 .• ,' . ,
0, • " ! • • , ,
• pr .b l billtl ..
p •• '
, , ...
b. Ihurto S ,how. t h .. b •• ldn the ·.u~.I· .. ,I I •
• f "t- I Iu ftft, hbOo' 'Q ' \ h .. lhe ~ t.~'H pr t .. It,
of be l _, 1 •• 01001 I n Idnnce ( I " <00 .... 1<0 . .. hh ,11
t he othe r .'QU).
In .... , .. t h fy l ng:
I f t he .I?' Ilu ' t. no t l ... !_I I t~t •• ~o"ld
i . cl ... o ... n < .... of Int . . .. t) t hen t. add it ion
t o I. t ho ,., ••• • • 1' . (.) . ~o ul' b. 1 •• d.d , •
• druel.
I f . , I_I the "UH i"9 of 'q. ,lln(I)-l." • IJ
.. , b, ... t h ated.
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- t~ -
d. ~on Of th e .. ,~.p'lo., of tho •• ~ ~ , ro oo . <lu,lo.,
of the dl ..... lon' ~I d. In I.~.J.I. , they 'hould be
uU, fl .d I. ""I t ... of prHt t .,1 I.torest. Th
t •• dltlo., .'1 b. w •• ~ ••• d I. ~'nl c • ••• bu •• On"
of th ••• '1 b ••• tIt ed wlthou, "Ore , do ; Ihl' " n l
b. >eO" fr o. ,he foll0. lnq ""or "'llIt1l1 .... pie,
'"
r • '" " ,.,
" . • . . , .n " J. I • k f o • • - t, .... I • •
• otnorwl ..
( d > • , , 1 , )
ThuS "," ,.,u"p. lo. (3) of th •• ' e" ~ Is .1.I,ted,
~. obt o ln' ... .... . ', .. ' " ..... " .. , .. - t p •• •• 1 ',' It"l
• (I_p , . )'(1, • -11 \ • ) - P, .·'I k • • 'I 'I ' ' I
" " > 2df(l+d ) .' Hu. 1f. • • 'he. thor ... Is" d > • "lth , .' Q." < Q, , s .
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•• •• ••••• •• ••• •• ••• ••••••••• •••••• ••• ••••
• ~ ov l d .. 10 ' d, d " t I ., ,-, " . ddIt I O. " P' g, " " 't _ 1 :
• 8~_ 1 , 8 ~.1 ( '"91O. • (
• B q + I ( , .gl •• (
B q. I ( reg l •• " • • ( JI M) , pogo,i , . , ,- Y. ,."Y j " <
• Sq_ 1 ( < • < , • <
H. , vb, .... b. - d. or . ,on Ulnod I n th e 1 00d i n~ ,vI.
-p ' e "gl.9 or t ne .elghbo"" (P~)' . hlch h" bo •• p,op."d
, ,w i l l b. p, .. ""t In ,-th", tho I OUln9 'ulo' for I . nd V ,. i .,l d • • • ry oft ••.
S.lut1." .f th o , . pl " .... ,t orObl • • , nd the l •• • ·.h .. d
pr.~l .. (I. e. the de tenl."I • • of • 'vb .. ! 0(1) of x(t)
<hHHteri,I'9 l oo ' -.hud .e.h, ... ,t.) wh l e" 0" boslng
o. P~ ... l oadl.g r vle .. Il l be ,hdl.d In the fo llowing
•
-. )
)
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• 31 •
II II .1'1 10 I •• th.t the d •• ,., o.~,.,I". " 90" 10' O' A
v "I<~ t, ,ooe • • ,d b, 'N I I • 101,t . , .~ 1 ••• 4 l ~U "' •
•• , he'"ft t polt <, Is .... to . ,\o •• , ....... .,Ioft 0'
OIL ( ~". 'loc ~ loo t.hud . of . OJ ). If .(t) ,. (, (i l •...• ,(!) ) a •• tn , contento of I "e
UU-ltoc t o f t n l t •• ge I.e' l 0 ' " ,Ion R ( wOn. , ( : 1 ...
t. t . ,ob,r, of tho p' g" v.lth . ro p.,.,.t I n l . t t l~1 t )
thl. O,~ .Or~ 1 I. f.llo • • :
.. " • " " here , • . (t) •
, , • " . • , , ( 10 1) ,. (~. a ( p . ( t ) ( t ) • (t), . . . ··.-1·· •• 1· • • •
". " . " , , I A ( t ) ,
( t ) ( t) I -I) • •• I ·····o-J· Q' .q " ,,,
[( t ) I t ' (t /) q •• I .. .. ·o·~ ·· "· " ",
(q .1(: 1 .... , .~~l ,.( ~IJ " '" .( tol l ., [q •• (I),· .. ,. ~~~ "i:1 ,I ( !)) " , " {(t! (t) 1
q •• I , .. . ""_2'~" " 1"
r~. I (}l .. ,. ,. i ~ l .'(!' ,~ . 01 " '" t ( t ) ( t ) ( t ) •• 1 ~ . , I ... . " . - 1 " 0-1'. - " PI
(1 ) , q'l , ~ . l •• ( Il
( t. , • • d,. o f the .,I,ObC ur pogu 11 .. tt .. tU b1 ,n,
to« t "1I fo . ....
.. 11 1 bl ' 0, 11 11, . )
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, " , In 0" , q. l , ~ (t) , [.(tl) , P I 0" • • ( t) , 0" , ~ I , ) , I ,(t)) " , 1'1 0" • , I t) , 0" • • ( t 1 '" , l 'l 0"
, A It ) 0"' , A ( t) , (.It) ,it)) n_ I • "
1" ,,, , A It )
'"' • • ( t I , I (-l,>l l ,
A I t) • ( t I " ''I 0" ,
'"' •
'"' , , '" or9ool .. \ lo. " " . ",.~bl ' "' r . ~lon' '" , " ,,' , , , , " I., 9' " ... , , ' , '. (~h • •• .. c~ • • 1 " ('I ' ~I I h
9n .. r" . ~ " 10<> 111' "' , ""'01'9 D~A o. P , .
~R A ) h .1 ~. l Hte d oon, l d . .. , I 1 "' , tJ R I"~x" , . , "' II ~I ("g lo b . l" , . ,
" O~A I .
1~ ;, ~od ; f; " tlon do . <no t pr . u.t • • y pr ob l ,. f or~ •• inc. t HO
R( i. gOYH.od ~ y tn • .... I •• dl'g ' ul " 'h i< h Ind . p,nd.n'
of UI< '"iion conoorn . d. "
I f ~ i, t. b. "9. nl .. d 910b,lly I n .gr •• """ wtH, lO U, t n • • •
1.00eo" o f I or 2 "« ttl,., for H cn of In . 'i we ,nou ld •
r ••• r • • n po,itlon, ( ~ ~ n ~ l ~ ) of ! to ~ • • o I n • • 'g" , , ~hlcn or" ' o. d . d In .d,,"CO fro ~ , h. <u b r~ 9 t .n. I) •. . .• !. , I "". ! <p llt. jn'o a LPU-part All) (contol .I'9 tn . p<g"
.hie n "H' boo n r .fo r . n c.d In th ~ c" c< nt ~'H) "' ~ • "Too,"
.h.-d " _PHt S(t)·
11 ... conn., u' t. , ., .... ~ 'Ich , l l.~, th o ., ;".,co o t "
po go to '0 ~ . prep ' god ... tt • •.
Inu . it no .u l tlp,.~c ... l ng t. , l lowod and If . Hog cn ~' 1
be por tttionod Int. 4 oc S c.~lon' of ot thl t . [9,11} , 100.
.no.d -P.ct co n.l 'ttn ~ ot ~ " ~ P' Q' S s houl d b • •• ft \cl " ,.
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• ) 1 •
8" t t.". i •• • "I'lp,"~" .. 1"1 .. ,Iron. tnt It ~ t11 b. "VH\'~
\~ H It h o ... lble to .. o ' ~ . ff lcl, n tly wHh 18 ( 1)1 • 8 i.
H • • , g, .l ult o. 0' Alt) ,n. e ( t) ~ i T1 be dh<"".d tn d . ,.I I
for th e 1.ok-.h.,d-,190,lth. lA~ .. h l<" Is p' ..... '"d In th e
... t 'Htlon.
1.1. Th. loo '- ' h .. d .hod t " . lU
A, pr .p .... tn 1.1., ~~ (t) ,pitt, Into. l ~ U- •• rt Al l) Hd
i n to • loo ~- 'h.,d .'rt e(t).
Fo, ' . ... nt.nce , D(t) : . e(1) ( t. •. loo k- •• ud , •• loce •• • t ,
will b. ~u. onl, IF t he dl \O , . tea.cod h or ... H I n l~'
100 k- . hOl. porI . , ~~ ll) ) .
~.r . o •• r It wlT I b. , ,," •• d that l e( I ) I . lal • con ,l.
Thu. , I f th~ ,I •• 01 "~(t) I •• ,0.,I.nl Ih e ,I,. or ~ ( t)
do.,not . ,ry In tl ••.
A(t) : . [,'i) .... "PIJ, B( t ) : . [ b(:)" .. ,bHI] whe •• 'i ' 0t < (I, . .. , . ) .
f " ,th. rw.,., eft) " III b. 9 ••• , n • • b, Ihe flfD·prl.cipl •
• I . ce S(I) 0.1, eontains PlY" whle " hH, nol bt u rote ' , ne,d
I. the roc .. t .....
CI."ly Ihue . .. othH po"tbl lt llo< .f o'9,.I,ln9 . (I)"d
5(11. ~o .... i .o, p' ob l u. wi l l not~. " 9"d , ~ i. th o foll ._
wl"9 •• . 9 . f.' ~ ul .. For IAI < l. I ~' lo . dl ftg o , oblo .. Fo'
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- 3' -
. acgln.l ~'9" of ~ M He. ; tt ,,111 ~e oll olou, to eUQ1He t h~
oT~.nl .. tlon.
To ,hortoo the 00101100 tho t ollo. hi .bbr .. totIOM w111 bo und,
O.i~ " it~o. 4
ttt ~~ oe (h "ub ... 0 , t h pAg e .,Ue~ i. U' <H.Hd .t (,." t .
, " t +1 " ,. •
" " I (I • •• " ',.
J.1.1.
AIHI) : . ( r t •• q), ..... ~ ~I , . l! l ... . , . j ~lJ
elt+l) : . e ltl
3.1.2.
~ccordln9 to t he l~U - p r ln c lpl . we h. .. ,It; l ) :. r , .
The . , lghbOU" of TI ITO lo.d , d InlO , 1" 1) If t h ey ITO no<
1"' ~r","1 I n A(I) v t( t ). If rt
• elt) t his will b. 0 r .. 1
loo k_. h.od . t op which . ' y bo do • • without In t . r r"otln~ th e
P"og ' '', ThU, If co,h ... . 9he . by "t he - t ll o d. fl.HI •• o f
the co'" of , rtphce-ent ,lqo .. lt h .. (cf. deflnttlon I)
,hould b . .. odHled In ,uch • " ' 1 th .. only "on - Io.~- .h"d
r.~l . ce •• nt co.t, . re .. eg.rded. Thl. p .. e •• nt •• n Intore ,tlni
"9U_ .. t In f .. our of non - de ... d-p.glng OI 90TH"", .
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- lS -
1~. t op of ,(t+l) <on.l.t. o f I~' PIg, 'il) w~ I< ~ Is •• p ll " ~
fro. A(I) (If . (~) < R.) Ind .f tho 1 " <o.ln~ n. l g~bo" , (S)
of " W~U ... t~o bo tto. p.gu of a(t) .. e .. placed.
l.l.r.l. Qrql.luU ••• f A(t+! )
lRU ! (ropl"utn t o f ·(1) . '!~l ): Alt+!) . • ( t) (t) 1 't·1 I · .. ·· · IAI · l .
If .(~) < RI • ,he. b('i l ) :. "!~l (e " J.).2.2.).
lRU l ( .. ph.o.e. ' of t ~ . pIg. of Alt ) wOld. is .0' I ... ber
.f ~ , I nd ~hl<h h .. bun 10 .. , .. «nt l y " .. d):
." ( • • '" • '" '1'11 t' I"'" IAI -2 ' A
H.ltleR . , if ' IH tRt
( (t) (t) I'll ' " "."IIAI-l' "IAI -I" l ._h~. w ls.
Ihe .rglnl,.'I •• of ,(,+1) dOp •• ds on wh. t he. 't • ,(')
0, 't ~ ,t tl IS w.ll .. on ~h.,"or .Ip • R, (w~lch is 1 _ _
p.sslbl. if A(') Is g ..... . d b.Y lRU 2) or .Ip , ~"
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. 36 .
1 . 1.1,1.1.
.. ~.r. •
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" • ,- " " .(1 ) • " " •
J.].l.l . l. . ,'" , .'" 'I • •
Co , • ( t ) • • " • , • , (., .».
U. I"9 '" 1~.r , .I.t10" '
( -
" • • " " • " -• " ,- • " • -" " • "
-, obta i " '
3. 3. Z.Z.3.
, ,
.11) j " . ,
~" ( t)
1M( t) -"
.' • ,," (I) ,
, "~ ( I )
~~ It)
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-
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l.l.I.I.'.
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-, , t
.' " , . -
I { t I
( ' -(I) ( t It ' •••• 'OI·····bI 11 _
C ' (<I {tJJ 'I , b 1 ····· ~I II·I
( ' (I) (t) ( t )~ ",b, I " b I , .. , b l i l •
[ . ( <) ', .b I , .. ,b It) (01 BI _r , bl • 1
- ,,"It, " ,
- IIN{ t ) "
, -
" '" " '" " " , " '" " '"
""I t) • " , , .. .MI", (bf~11
,,~(t) "
,,~It) "
~~ (t I " '
-, -,
- b ( t)
" • ~ j ~ 1
• - " ( o cr' ,- l)
A ( t)
• A ( t)
• 1'1 {t } (t ) ... [ 't· b • . b I ,····. ' U_IJ
or9·"I .. ~ ~hh
o '~'"I"~ ~ 1U
If (I")
If (I')
If 0')
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- 18 -
{L' I , " • • :'IN( t l
(Z- I " • • ""( t l .... { b l~11
(n • b {t I " • ,. ~ ( I) or9u1 ud wHO '"" ,
1") " • -~m A (I I .rg. "I ... ,,110 I.~U 2
J.l.l. S . ~ • • " . l'plu,o t Hlo, .f LH
8, the lo.k - • • •• d . 19,,10 •• LAA • PH" ( . , . , J I
,'e louu .1"lto ..... ly I f 't <', . ~~{tl~ .( t ).
To .lo,lIf, , he 0 ' 9 •• 1,,11 •• '0"" of th e ' eol,u ... nl , .1.,
... , be ._Ht.d without . educl.g ,19.lFlcuU , th oedor'
','ce of tho .190.lt~ •.
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