IAeANUeEEE AeA?AEAAIEE IIE?OAOI?*IEE OI?AA?NEOAO

EssOIAAOeUeEEE IEAENAIAe? IEOAEEIAE*

OAeE 681.325.5

IAOIAeE OOIEOe?IIAEUeII-EIA?*IIAI I?IAEOOAAIIss OAEAeEIAe?TH*EO
A?EOIAOE*IEO I?ENO?I?A IA INIIA? NEIAO?E*IEO AOEAAEO OOIEOe?E

05.13.05 “Aeaiaioe oa i?eno?i?

ia/enethaaeueii? oaoi?ee oa nenoai ea?oaaiiy”

AAOI?AOA?AO

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa oaoi?/ieo iaoe

Iaeana — 1999

Aeena?oaoe??th ? ?oeiien.

?iaioa aeeiiaia ia eaoaae?? “Nenoaiia i?ia?aiia caaacia/aiiy”

Iaeanueeiai aea?aeaaiiai iie?oaoi?/iiai oi?aa?neoaoo i?i?noa?noaa ina?oe
Oe?a?ie.

Iaoeiaee ea??aiee

eaiaeeaeao oaoi?/ieo iaoe, aeioeaio

Iaoe?i Ieaa Ieeieaeiae/, Iaeanueeee aea?aeaaiee iie?oaoi?/iee
oi?aa?neoao, aeioeaio eaoaae?e

“Nenoaiia i?ia?aiia caaacia/aiiy”

Io?oe?ei? iiiiaioe:

aeieoi? oaoi?/ieo iaoe, i?ioani? ssuoe Eaii?ae Iiaeyiiae/, Iaeanueea
aea?aeaaia aeaaeai?y ca’yceo, caa?aeoaa/ eaoaae?ith ia?aae ? nenoai
iiooiaiai ca’yceo

eaiaeeaeao oaoi?/ieo iaoe, aeioeaio Iaua?yeia Aieiaeeie? ?aaiiae/,
Iaeanueea aea?aeaaia aeaaeai?y oieiaeo, caa?aeoaa/ eaoaae?ith NAI?

I?ia?aeia onoaiiaa

?inoeooo i?iaeai iaoaiaoe/ieo iaoei oa nenoai IAI Oe?a?ie, i. Ee?a

Caoeno a?aeaoaeaoueny 23. 09. 1999 ?ieo i 13 aiaeei? 30 oaeeei ia
can?aeaii? niaoe?ae?ciaaii? a/aii? ?aaee Ae 41.052.01 Iaeanueeiai
aea?aeaaiiai iie?oaoi?/iiai oi?aa?neoaoo ca aae?anith: 270044, i.
Iaeana, i?iniaeo Oaa/aiea, 1.

C aeena?oaoe??th iiaeia iciaeiieoenue a a?ae?ioaoe? Iaeanueeiai
aea?aeaaiiai iie?oaoi?/iiai oi?aa?neoaoo ca aae?anith: 270044, i.
Iaeana, i?. Oaa/aiei, 1.

Aaoi?aoa?ao ?ic?neaiee 20.08.1999 “_____”___________ 1999 ?.

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee ssiiieueneee TH.N.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. A aeaiee /an iaeiei ?c cania?a ca?eueoaiiy
i?iaeoeoeaiino? ia/enethaaeueieo i?eno?i?a ? i?aeaeuaiiy oaeaeeiae??
eiia?iaoe?eieo a?eoiaoe/ieo i?eno?i?a (AI). ?icaeoie ia/enethaaeueii?
oaoi?ee eaea ii oeyoo aacia?a?aiiai oaeineiiaeaiiy oaoiieia??
ae?iaieoeoaa, ui aaaea aei i?aeaeuaiiy nooiaith ?ioaa?aoe?? aeaeo?iiieo
eiiiiiaio?a ? c?inoaiiy oaeoiai? /anoioe. Aeey no/anieo i?ioeani??a
oa?aeoa?i? noia?neaey?ia aaaaoiiioi/ia a?o?oaeoo?a ? aeniea oaeoiaa
/anoioa. Oea aenoaa? aei?noe? aeiiae a ia?oo /a?ao aei oaei?
oa?aeoa?enoeee AI ye oaeaeeiae?y. Oaeaeeiae?y AI eiiaae??iiai oeio
iaiaaeaia oaeaeeiae??th naiiai iia?eueiiai ia/enethaa/a a eiiaae???, ai
a?i iiaeiai ni?aoeueiaoaaoe ca iaeei oaeo, ? aaee/eia oaeoo ia iiaea
aooe iaioa cao?eiee ?iciianthaeaeaiiy neaiaeo o naiiio iia?eueiiio
ia/enethaa/?. Cao?eiea ?iciianthaeaeaiiy neaiaeo caeaaeeoue a?ae ?aiao
oeueiai ia/enethaa/a, oiaoi a?ae e?eueeino? aaioee?a, ui
ni?aoeueiaothoue iine?aeiaii. Oaeei /eiii, i?i?i?caoe?y ?aiao noaie AI
aeicaiey? ca?eueoeoe oaeoiao /anoioo i?ioeani?a ? i?aeiyoe eiai
i?iaeoeoeai?noue.

No/ani? caniae iiaoaeiae ia/enethaaeueieo i?eno?i?a i???ioothoueny ia
niaoe?ae?caoe?th ye naieo i?eno?i?a, oae ? iaoiae?a ?o i?iaeooaaiiy. I?e
oeueiio caniaii i?aeaeuaiiy i?iaeoeoeaiino? ia/enethaaeueieo i?eno?i?a
? aeei?enoaiiy niaoe?ae?ciaaieo aia?aoii ?aae?ciaaieo ia/enethaa/?a.
Iai?eeeaae, iaa?oue o no/anieo i?ioeani?ao caaaeueiiai i?ecia/aiiy
cai?iaaaeaeothoueny ?ai?oa aeanoea? eeoa aeey noia?-AII niaoe?aeuei?
eiiaiaee aeey aeeiiaiiy iia?aoe?e iaae iao?eoeyie ? ianeaaie /enae, ui
?aae?cothoueny niaoe?ae?ciaaieie AI. Oaeei /eiii, niinoa??aa?oueny
oaiaeaioe?y a?aeiiae a?ae oi?aa?naeueieo aia?aoieo ia/enethaa/?a ia
ei?enoue niaoe?ae?ciaaieo.

?nioth/? iaoiaee iiaoaeiae AI caniiaai? ia o?aaeeoe?eiiio iaoaiaoe/iiio
aia?ao? aoeaai? aeaaa?e ? ia a?aeiia?aeathoue aeiiaai, ui
i?aae’yaeythoueny aei aenieii?iaeoeoeaieo ia/enethaaeueieo i?eno?i?a.
sse i?aaeei, on? oae? AI aoaeothoueny ia aac? a?ia?ieo ia/enethaaeueieo
no?oeoo?. Iiaeeeai i?aeaeuaiiy i?iaeoeoeaiino? ca ?aooiie ca?eueoaiiy
e?eueeino? iaeii/anii ia?iaeth?ieo iia?aiae?a. I?e oeueiio ciaioo?oueny
e?euee?noue iine?aeiaii aeeiiaieo eiiaiae aeey cae?eniaiiy iaei???
iia?aoe?? iaae aaeeeith e?euee?noth iia?aiae?a (aaeoi?i? iia?aoe??), aea
oea aeinyaa?oueny oneeaaeiaiiyi noaie a?aeiia?aeiiai ia/enethaa/a. C
aeuaneacaiiai aeaeii, ui caaea/a ciaioaiiy ?aiao ? ca?eueoaiiy
oaeaeeiae?? ia/enethaa/a yaey?oueny iniaeeai aeooaeueiith.

Oaeei /eiii, ?ic?iaeaiiy iiaiai i?aeoiaeo aei i?iaeooaaiiy
oaeaeeiae?th/eo AI, ui aeei?enoiao? niaoeeo?/i? aeanoeaino? aeeiioaaieo
oeeie AI iia?aoe?e, yaey?oueny aeooaeueiith caaea/ath, oiio ui aeicaiey?
iaaeeceoeny aei iaeneiaeueii aeinyaeieo iieaciee?a oaeaeeiae?? oa
aeiiii?/iino? iiaoaeiaaieo i?eno?i?a.

Ca’ycie ?iaioe c iaoeiaeie i?ia?aiaie, ieaiaie, oaiaie. Aeine?aei?
?iaioe, cie?aia, ?ic?iaea iaeaoa i?eeeaaeieo i?ia?ai aeey aaoiiaoecaoe??
i?iaeooaaiiy a?eoiaoe/ieo i?eno?i?a, ca’ycai? c oaiith ieaiiai?
iaoeiai-aeine?aeii? ?iaioe eaoaae?e nenoaiiiai i?ia?aiiiai caaacia/aiiy
IAeIO ? 329-73 “I?ia?aii? caniae aaoiiaoeciaaieo nenoai. ?ic?iaea oa
aeine?aeaeaiiy iaoiae?a oa cania?a aaoiiaoeciaaieo nenoai”.

Iaoa ? caaea/? aeine?aeaeaiiy. Iaoith aeena?oaoe?eii? ?iaioe ?
?ic?iaeaiiy ? ?icaeoie iaoiae?a i?iaeooaaiiy oaeaeeiae?th/eo
iaeiioaeoieo AI, ui a?oioothoueny ia aeei?enoaii? aeanoeaino? neiao???
ao?aeieo ?ioi?iaoe?eieo neaiae?a. Oe? iaoiaee iiaeii? caaacia/oaaoe
i?aeaeuaiiy oaeaeeiae?? AI, onoiaiiy aia?aoii? iaaei??iino?
ci?iaeoiaaieo i?eno?i?a ca ?aooiie a?aooaaiiy niaoeeo?ee aeeiioaaieo
ieie iia?aoe?e, ni?iuaiiy oa iaeiiiai?oi?noue i?ioeano i?iaeooaaiiy ?
iiaeeea?noue oi?iae?caoe?? oa aaoiiaoecaoe?? ie?aieo eiai aoai?a.

Aeey aeinyaiaiiy iinoaaeaii? iaoe ae??oothoueny oae? caaea/?:

1. Nenoaiaoecaoe?y no?oeoo?, oa?aeoa?enoee ? cania?a iiaoaeiae AI
??ciiiai?oieo eean?a, oaeeo ye noiaoi??a ? iiiiiaeoaa/?a, ?
aeine?aeaeaiiy ?o niaoeeo?/ieo aeanoeainoae, cie?aia, neiao??? aoiae?a.

2. ?icaeoie ? aaeaioaoe?y aeaaa?e neiao?e/ieo ooieoe?e (NO) ? /anoeiai
neiao?e/ieo aoeaaeo ooieoe?e (*NO) aeey i?iaeooaaiiy AI.

3. ?ic?iaeaiiy oi?iae?ciaaiiai iaoiaeo aeiaecia/aiiy iaiiai?noth
caaeaii? aeia?eueii? aoeaai? ooieoe?? aei neiao?e/ii? ooieoe??.

4. ?ic?iaeaiiy iaoiaeo iieno no?oeoo?e AI o aaceno NO ? *NO oa
i?ioeaaeo?, i?eaeaoieo aeey aaoiiaoeciaaiiai neioaco no?oeoo?e AI ia
iniia? oeueiai iaoiaeo.

5. ?ic?iaeaiiy iaeaoo i?aei?ia?ai ooieoe?iiaeueii-eia?/iiai i?iaeooaaiiy
AI, ui aeeth/a? a naaa:

( i?ia?aio aeyaeaiiy neiao?e/ii? /anoeie a aeia?euei?e aoeaa?e ooieoe??;

( i?ia?aio ?ic?aooieo ?ic?yaeieo ?iaeaen?a aeey i?iaeooaaiiy noiaoi??a
?c aeia?eueiith e?euee?noth aeiaeaie?a o ?ic?yaeao.

Iaoiaee aeine?aeaeaiue aacothoueny ia i?eeeaaei?e oai??? oeeo?iaeo
aaoiiao?a, aoeaa?e aeaaa??, oai??? a?ao?a, oai??? ooieoe?e e?eueeio
ci?iieo.

Iaoeiaa iiaecia iaea?aeaieo ?acoeueoao?a iieyaa? a ianooiiiio:

1. Cai?iiiiiaaii iaoiae i?aaenoaaeaiiy ? i?i?i?caoe??
/anoeiaeo-neiao?e/ieo ooieoe?e ? ?ic?iaeaii iaoiae ? aeai?eoi
aeiaecia/aiiy iaiiai?noth caaeaii? aeia?eueii? aoeaai? ooieoe?? aei
neiao?e/ii? ooieoe??, o ?acoeueoao? /iai aeaaa?a /anoeiai-neiao?e/ieo
aoeaaeo ooieoe?e iaea?aeaea iiaeaeueoee ?icaeoie a aaeoc? i?aaenoaaeaiiy
? i?i?i?caoe?? ooieoe?e ?c e?eueeiia eiiieaeoaie neiao?e/ieo a?aoiaio?a,
ui aeicaiey? canoinoaaoe ?? aeey aoaeoeaiiai neioaco aaaaoi?ic?yaeieo
noiaoi??a.

2. ?ic?iaeaii iaoiae i?iaeooaaiiy noiaoi??a ?c aeia?eueiith e?euee?noth
aeiaeaie?a o ?ic?yaeao ? noai i?enei?aiiai ia?aiino a oa?i?iao aeaaa?e
neiao?e/ieo ooieoe?e, ui caaacia/o? i?aeaeuaiiy oaeaeeiae?? noiaoi??a o
1,3 ?ace ? a?eueo, a oaeiae aeai?eoi ? i?ia?aia aaoiiaoe/ii? aaia?aoe??
?ic?yaeieo ?iaeaen?a aeey i?iaeooaaiiy oaeeo noiaoi??a oa aeine?aeaeaia
?o aoaeoeai?noue.

3. Aia?oa cai?iiiiiaaii ocaaaeueiaio no?oeoo?o ? iaoiae i?iaeooaaiiy
noaie aecia/aiiy ciaea ?acoeueoaoo i?aenoiiaoaaiiy N /enae o
aeiaeaoeiaiio eiae? aeey aeia?eueiiai N, ui caaacia/othoue ca?eueoaiiy
oaeaeeiae?? a 1,3 ?ace ? a?eueo oa ciaioaiiy neeaaeiino? noaie a 1,4
?ace ? a?eueo.

I?aeoe/ia cia/aiiy iaea?aeaieo ?acoeueoao?a. ?ic?iaeai? iaoiaee
i?iaeooaaiiy oaeaeeiae?th/eo iaeiioaeoieo AI aeicaieythoue ni?inoeoe
i?ioean i?iaeooaaiiy noiaoi??a oa iiiiiaeoth/eo i?eno?i?a, a oaeiae
oi?iae?coaaoe ? aaoiiaoecoaaoe ie?ai? eiai aoaie. ?ic?iaeai? ia iniia?
oeeo iaoiae?a AI iathoue i?i?iaeueio aia?aoio iaaei??i?noue ca ?aooiie
aeei?enoaiiy i?e i?iaeooaaii? aeanoeainoae neiao??? aoiae?a oeeo AI, ui
o naith /a?ao i?aeaeuo? oaeaeeiae?th cai?iaeoiaaieo ae?ia?a. Ia iniia?
cai?iiiiiaaieo iaoiae?a ?ic?iaeaii iaeao i?eeeaaeieo i?ia?ai.
?acoeueoaoe ?iaioe ?aeiiaiaeo?oueny aeei?enoiaoaaoe i?e ?ic?iaeaii?
a?eoiaoeei-eia?/ieo aeie?a oeeo?iai? ia?iaee neaiae?a, ia?iaee
ia?aeaeueieo iioie?a aeaieo o ?aaeueiiio /an?, i?eno?i?a eiaeoaaiiy ?
aeaeiaeoaaiiy oa ?ioeo caaea/, e?eoe/ieo aei /ano ?aaeoe?? nenoaie.

?ic?iaeaiee a aeena?oaoe?ei?e ?iaio? iaeao i?eeeaaeieo i?ia?ai aeey
ooieoe?iiaeueii-eia?/iiai i?iaeooaaiiy niaoe?ae?ciaaieo
aaaaoiiia?aiaeieo noiaoi??a ?c aeia?eueiith e?euee?noth aeiaeaie?a o
?ic?yaeao i?eeiyoi aei aeei?enoaiiy o ae?iaie/i-eiia?oe?ei?e o??i?
«OAEAEA?O» ? canoiniao?oueny i?e i?iaeooaaii? aeieo eiaeoaaiiy i?eno?ith
?i?oe?ae?caoe?? oa caieno oaeaoiiieo /?i-ea?o «OAEAEA?O-5NE» aeey
oaeaeeiae?th/i? aia?aoii? ?aae?caoe?? e?eioia?ao?/iiai aeai?eoio.

Iaoeia? ?acoeueoaoe ?iaioe ai?iaaaeaeai? a iaa/aeueiee i?ioean IAeIO ia
eaoaae?ao «Nenoaiia i?ia?aiia caaacia/aiiy» ? «Eiii’thoa?i? oa
?ioaeaeooaeuei? nenoaie oa ia?aae?». Iaoiae i?aaenoaaeaiiy ?
i?i?i?caoe?? /anoeiai-neiao?e/ieo aoeaaeo ooieoe?e ? iaoiae
aeiaecia/aiiy iaiiai?noth caaeaii? aoeaai? ooieoe?? aei neiao?e/ii?
aeeeaaeathoueny a oai? «Aoeaaa aeaaa?a» aeenoeeie?ie «Aeene?aoia
iaoaiaoeea» aeey nooaeaio?a niaoe?aeueiino? 7.080403. Iaoiaee
i?iaeooaaiiy aaaaoiiia?aiaeieo aaaaoi?ic?yaeieo noiaoi??a, noaie
i?enei?aiiai ia?aiino aeey oaeeo noiaoi??a ? noaie aecia/aiiy ciaea
?acoeueoaoo i?aenoiiaoaaiiy e?eueeio /enae aeeeaaeathoueny a oai?
«A?eoiaoe/i? i?eno?i? AII» aeenoeeie?ie «Eiii’thoa?ia aeaeo?ii?ea» aeey
nooaeaio?a niaoe?aeueiino? 7.091501.

Iniaenoee aianie caeiaoaa/a iieyaa? a:

1. Aiae?co iaoiae?a iiaoaeiae oa no?oeoo? oaeaeeiae?th/eo iiiiiaeoaa/?a
[1].

2. Ocaaaeueiaii? aeaaa?e /anoeiai-neiao?e/ieo aoeaaeo ooieoe?e ia
aeiaaeie e?eueeio eiiieaeo?a neiao?e/ieo a?aoiaio?a [9].

3. ?ic?iaeaii? iaoiaeo oa aeai?eoio aeiaecia/aiiy iaiiai?noth aecia/aii?
aeia?eueii? aoeaai? ooieoe?? aei neiao?e/ii? ooieoe?? [5].

4. ?icaeoeo iaoiaeo i?iaeooaaiiy noiaoi??a ?c aeia?eueiith e?euee?noth
aeiaeaie?a o ?ic?yaeao [8] ? ?ic?iaeaii? iaoiaeo i?iaeooaaiiy noai
i?enei?aiiai ia?aiino [4] a oa?i?iao aeaaa?e NO, a oaeiae noai?aii?
aeai?eoio ? iiaeoe?a i?ia?aie aaoiiaoe/ii? aaia?aoe?? ?ic?yaeieo
?iaeaen?a aeey i?iaeooaaiiy oaeeo noiaoi??a [3].

5. Noai?aii? i?ia?aie aeyaeaiiy neiao?e/ii? /anoeie a aeia?euei?e
ooieoe?? [3].

6. ?ic?iaeaii? ocaaaeueiaii? no?oeoo?e ? iaoiaeo i?iaeooaaiiy noaie
aecia/aiiy ciaeo ?acoeueoaoo i?aenoiiaoaaiiy N /enae o aeiaeaoeiaiio
eiae? [2].

7. ?ic?iaeaii? oaaee/ii-aeai?eoi?/iiai iiiiiaeoaa/a ia iniia?
o?eiia?aiaeiiai noiaoi?a [6] oa ?icaeoeo oaeaeeiae?th/iai
oaaee/ii-aeai?eoi?/iiai iaoiaeo iiiaeaiiy ia aac? aaaaoiiia?aiaeieo
ia/enethaaeueieo no?oeoo? [7].

8. ?ic?iaeaii? iiaeeo?eiaaiiai iaoiaeo cai?oee aaaaoi?yaeiiai eiaeo ia
iniia? aaaaoiiia?aiaeieo noiaoi??a.

Ai?iaaoe?y ?acoeueoao?a aeena?oaoe??. ?acoeueoaoe ?iaioe aeieeaaeaeeny
oa iaaiai?thaaeenue ia ui??/ieo iaoeiaeo eiioa?aioe?yo nooaeaio?a ?
iieiaeeo aeine?aeiee?a (1996-1998 ?.) oa ia i?aeia?iaei?e
iaoeiai-oaoi?/i?e eiioa?aioe?? «I?eeaaeiaoaeoaaiiy-98» o i. ?aiaoi??y.

Ioae?eaoe??. Ca oaiith aeena?oaoe?eii? ?iaioe iioae?eiaaii 9 iaoeiaeo
?ia?o, na?aae ieo iaoaio Oe?a?ie, iiaeai? 2 cayae ia iaea?aeaiiy iaoaioo
a Aea?aeiaoaio Oe?a?ie ? io?eiai? i??i?eoaoi? aeia?aeee oa iioae?eiaai?
??oaiiy uiaei i?eeiyooy caya aei ?icaeyaeo (Atheaoaiue “I?iieneiaa
aeani?noue”, 1998, ? 3/1. ( N. 333-334).

No?oeoo?a oa ianya aeena?oaoe??. Aeena?oaoe?eia ?iaioa neeaaea?oueny c
anooio, /ioe?ueio ?icae?e?a, aeniiae?a, aeiaeaoe?a. ?iaioa aeeeaaeaia ia
209 noi??ieao, i?noeoue 31 ?enoiie ia 17 noi??ieao, 55 oaaeeoeue ia 13
noi??ieao ? 4 aeiaeaoee, ui caeiathoue 34 noi??iee. Nienie aeei?enoaieo
aeaea?ae caeia? 7 noi??iie ? neeaaea?oueny c 80 iaeiaioaaiue.

INIIAIEE CI?NO

O ia?oiio ?icae?e? ?icaeyiooi no/aniee noai caaea/? neioaco
a?eoiaoe/ieo i?eno?i?a, oai??? neiao?e/ieo aoeaaeo ooieoe?e ? a?aeii?
iaoiaee i?iaeooaaiiy a?eoiaoe/ieo i?eno?i?a ? ?o oa?aeoa?enoeee.

, oi oeae aeae neiao??? iicia/a?oueny xi ( xj ? iaceaa?oueny ci?oaiith
neiao???th. sseui ? xi~xj, ? xi(xj, oi aiai?youe i?i iie?ii?oio
neiao??th ? ieoooue xi(xj. Caien xi(xj icia/a?, ui xi ? xj yaeythoueny
NA: xi ~xj, xi ( xj aai xi ( xj.

, aai Sn(a1, a2, … , ak). Aeia?eueia NO y=f(xn—1, …, x1, x0)
iaeiicia/ii i?aaenoaaey?oueny oa?aeoa?enoe/iei (n+1)-?ic?yaeiei
aea?eeiaei eiaeii ((f)=((0, (1,…, (n), aea (s ( cia/aiiy f ia
aoaeue-yeiio iaai?? cia/aiue a?aoiaio?a xn--1, …, x1, x0, ui i?noeoue
??aii s iaeeieoeue (0 ( s ( n). Oaeei /eiii, NO aecia/a?oueny eieaeueiei
eiaeii, aeiaaeeia eio?iai e?i?eii caeaaeeoue a?ae n. E?euee?noue
iao?ea?aeueieo NO aei??aith? 2n-1. NO c iaeiei ?iaeaenii iaceaa?oueny
aeaiaioa?iith NO (ANO).

iaceaa?oueny eii’thieoe?y an?o a?oi NO. AeAO ia? aeaeyae:

, (1)

aea ai ( ?iaeaene NO y; G1, … , Gp ( a?oie j-ai iaai?o a?oi; li (
e?euee?noue NI aaae ai.

Ii?yae ?c neiao?e/ieie ooieoe?yie, ?nio? eean /anoeiai neiao?e/ieo
aoeaaeo ooieoe?e (*NO), o yeeo o?eueee /anoeia a?aoiaio?a m (2 ( m (
n-2) ? NA. Aeey *NO aaaeeeaei ? iiiyooy neiao?e/iiai eiiieaeoo
a?aoiaio?a (NE). Iiiaeeia a?aoiaio?a X ={xi1, xi2, … , xim} ooai?th?
NE, yeui (xik , xij : xik(xij (1 ( k, j ( m ). NE aoaathoue i?ino? ?
iie?ii?oi?. O i?inoeo NE a?aoiaioe ca’ycai? o?eueee a?aeiioaiiyie i?yii?
? ci?oaii? neiao???. I?inoee NE caaea?oueny iaai?ii k a?aoiaio?a ?
k-?ic?yaeiei aaeoi?ii iiey?ecaoe??, ui caaea? ?iaa?niaai? a?aoiaioe. O
iie?ii?oieo NE a?aoiaioe ca’ycai? o?eueee a?aeiioaiiyi iie?ii?oii?
neiao???. Iie?ii?oiee NE caaea?oueny iaai?ii m a?aoiaio?a aac
o?aooaaiiy ?o ?iaa?n?e.

*NO o caaaeueiiio aeiaaeeo neeaaea?oueny c e?eueeio NE, ui ia
ia?aoeiathoueny. *NO ?c /eneii NE a?eueo iaeiiai ? aac ianeiao?e/ieo
a?aoiaio?a (INA) caaoueny eaac?neiao?e/iith ooieoe??th (ENO). ENO
a?athoue aaaeeeao ?ieue i?e iien? aaaaoi?ic?yaeieo aaaaoiiia?aiaeieo
noiaoi??a.

Iaea?eueo iiooaei? iaoiaee aeyaeaiiy neiao??? caniiaai? ia aeanoeaino?
o?aiceoeaiino? NA ? ia ?iceeaaeaii? Oaiiia ooieoe?? y=f(xn-1,…, x1,
x0) ii aeaii a?aoiaioai, yea aeaeyaea? oae:

(2)

aea caeeoeia? ooieoe?? R(X) caeaaeaoue a?ae a?aoiaio?a xn-1,…,
xi+1,xi-1,…, xj+1, xj-1,…, x0. C ??aiyiiy (2) aeieeaathoue i?aaeea
aecia/aiiy a?aeiioaiiy neiao??? xi(xj.

(R1 (X) = R2 (X) ) ( xi ~ xj . (3)

(R0 (X) = R3 (X) ) ( xi ( xj . (4)

(R1 (X) = R2 (X) ) ? (R0 (X)= R3 (X) ) ( xi ( xj . (5)

Aoaeoeaiei caniaii aiae?co caeeoeiaeo ooieoe?e yaeythoueny
aeaeiiiiceoe?ei? ea?oe. Aeaeiiiiceoe?eia ea?oa (AeE) Dij(y) ooieoe??
y=f(X) ii a?aoiaioao xi ? xj ( oea oaaeeoey c 4 ?yaeeaie oa 2n-2
noiai/eeaie. O oaaeeoeth caienothoueny aeanyoeia? aea?aaeaioe iaai??a
cia/aiue n a?aoiaio?a, i?e oeueiio ?yaeeia? 0 a?aeiia?aeathoue iaai?e, o
eio?eo xi=0 ? xj=0, oiaoi caeeoeiaa ooieoe?y R0(X), ?yaeeia? 1 ( xi=0 ?
xj=1 ? R1 (X), ?yaeeia? 2 ( xi=1 ? xj=0 ? R2 (X), ?yaeeia? 3 ( xi=1 ?
xj=1 ? R3(X). Aecia/aiiy ??aiino? caeeoeiaeo ooieoe?e caiaeeoueny aei
iiaeaiaioiiai ii??aiyiiy ?yaee?a AeE.

Oe? iaoiaee oi?aa?naeuei?, ai aeathoue iiaio ?ioi?iaoe?th uiaei neiao???
ooieoe??, aecia/athoue iiaio, /anoeiao neiao??th, neiao??th uiaei
?iaa?niaaieo a?aoiaio?a ? aeyaeythoue i?ino? ? iie?ii?oi? NE. Iaoiaeeea
aiae?co ooieoe?? ia neiao?e/i?noue caiaeeoueny aei iaea?aeaiiy
a?aeiioaiue i?ae on?ia ia?aie a?aoiaio?a ca aeiiiiiaith AeE ? iiaoaeiae
ni?yiiaaiiai a?aoa a?aeiioaiue. Eiaeiee oeeee o oeueiio a?ao? aea? na?e
NE.

A aaeoc? i?iaeooaaiiy AI iaea?eueo ?ic?iaeaieie ? iaoiaee i?iaeooaaiiy
noiaoi??a ? iiiiiaeoaa/?a ia iniia? a?ia?ieo i?aenoiiaoth/eo no?oeoo?.
Iaeiei c aoaeoeaieo iaoiae?a i?aeaeuaiiy i?iaeoeoeaiino? AI ?
aeei?enoaiiy aaaaoiiia?aiaeieo ia/enethaaeueieo no?oeoo? ia?iaee
?ioi?iaoe?? ye aeueoa?iaoeaa ia?aeaeueiei no?oeoo?ai. Aaaaoi?yaei? eiaee
aeicaieythoue ?icia?aeaeeoe ia/enethaaeueiee i?ioean ia ??ai? ia?iaee
ie?aieo ?ic?yae?a ? aeeeth/eoe i?ae?ic?yaei? ia?aiine ocaeiaae
?ic?yaeii? n?oee. ?o ia?aaaaa ( aeniea iaeii??aei?noue io?eiaieo noai.

Oaeaeeiae?th/? ?aae?caoe?? iia?aoe?e aaaaoiiia?aiaeiiai i?aenoiiaoaaiiy
? iiiaeaiiy caniiaai? ia iia?aoe?? cai?oee aaaaoi?yaeiiai eiaeo. Iaoiaee
cai?oee aaaaoi?yaeiiai eiaeo iio?aaothoue aaaaoiiia?aiaeieo
iaeii?ic?yaeieo noiaoi??a (AIN) ? aaaaoi?ic?yaeieo noiaoi??a. Nenoaie
aoeaaeo ooieoe?e, ye? iienothoue ?iaioo AIN, iiaaii i?i?i?cothoueny a
eeanao ii?iaeueieo oi?i, aiane?aeie /iai neioac AIN o?aaeeoe?eieie
iaoiaeaie o?oaeii?noeee. Oiio iaea?eueo iioe?aiee can?a ?oiuei?
iiaoaeiae ( aeeth/aiiy ca i??ai?aeaeueiith noaiith iaeii?ic?yaeieo
aea?eeiaeo noiaoi??a, ye? aeeiiothoue aeiaeaaaiiy aea?eeiaeo neaiae?a
iaeiaeiai? aaae, ui ca?eueoo? aeeaeio noaie, i?eaiaey/e aei cieaeaiiy
aoaeoeaiino? aaaaoiiia?aiaeieo AI ia AIN.

?niothoue iaoiaeeee i?iaeooaaiiy AIN ? aaaaoiiia?aiaeieo iaei?ic?yaeieo
(aei aeaio ?ic?yae?a) noiaoi??a ia iniia? eaac?neiao?e/ieo aoeaaeo
ooieoe?e ? caniao iie?iii?aeueiiai ?iceeaaeaiiy NO. I?ioa canoinoaaiiy
oeeo iaoiaeee i?e aaeeeiio /ene? ?ic?yae?a oneeaaeiaii, ai aeiaaa?
aaaaoie?aoii? ?o/ii? aeaeiiiiceoe?? ooieoe?e ca aeiiiiiaith aiae?oe/ieo
ae?ac?a, ui i?ecaiaeeoue aei aaeeei? oaeoi?eciaaiino? ?acoeueoao?a,
oneeaaeiaiith neioaciaaii? noaie ? iia??oaiith ?? oaeaeeiae??.

A ?acoeueoao? e?eoe/iiai aiae?co aeyaeai? oae? iaaeie?ee ?nioth/eo
o?aaeeoe?eieo iaoiae?a i?iaeooaaiiy AI: 1) ?niothoue aoaeoeai? iaoiaee
i?iaeooaaiiy o?eueee a?ia?ieo AI, a aaaaoiiia?aiaei? no?oeoo?e, cie?aia
iiiiiaeoaa/?, aoaeothoueny ia iniia? a?ia?ieo no?oeoo?; 2) caaea/a
i?iaeooaaiiy oaeaeeiae?th/eo N-?ic?yaeieo aaaaoiiia?aiaeieo noiaoi??a
ia? ?ici??i?noue 2MN, aea M ( e?euee?noue iia?aiae?a, ui iaiaaeo?
iiaeeea?noue ?? ae??oaiiy o?aaeeoe?eieie iaoiaeaie aoeaai? aeaaa?e i?e
aaeeeeo M, N; 3) ia ?niothoue noaie i?enei?aiiai ia?aiino aeey
aaaaoiiia?aiaeieo noiaoi??a.

Aeyaeaii oae? iaaeie?ee ?nioth/eo iaoiae?a i?iaeooaaiiy AI ia aac? NO:
1) ?nioth/? iaoiaee aeicaieythoue i?iaeooaaoe AI eeoa a iaiaaeaiiio
aacen? NO, cie?aia, ii?iaiaeo ooieoe?e; 2) ?nioth/ee iaoaiaoe/iee aia?ao
NO ? *NO aeicaiey? aeei?enoiaoaaoe eeoa iaeei eiiieaeo neiao?e/ieo
a?aoiaio?a, ui aea? iiaeeea?noue i?iaeooaaoe aaaaoiiia?aiaei? no?oeoo?e
eeoa c eaioethaii iioe?aiiy i?ae?ic?yaeiiai aioo??oiueiai ia?aiino; 3)
a?aeii? caniae iiaoaeiae ? no?oeoo?e aaaaoiiia?aiaeieo noiaoi??a
iaiaaeothoueny aeai?ic?yaeieie no?oeoo?aie; 4) ia ?niothoue iaoiaee
aeiaecia/aiiy iaiiai?noth caaeaii? aoeaai? ooieoe?? aei NO, ui ia
aeicaiey? aeei?enooaaoe aeey oaeeo ooieoe?e iaoaiaoe/iee aia?ao NO ?
*NO.

Ia i?aenoaa? i?iaaaeaiiai aiae?co iinoaaeaii oae? caaea/?: 1) aei?iaeoe
aeaaa?o *NO aeey i?aaenoaaeaiiy ? i?i?i?caoe?? *NO ?c iiiaeeiith
iacaeaaeieo eiiieaeo?a neiao?e/ieo a?aoiaio?a; 2) ?ic?iaeoe iaoiae
aeiaecia/aiiy iaiiai?noth caaeaii? aoeaai? ooieoe?? aei NO aai *NO; 3)
?ic?iaeoe iaoiae i?iaeooaaiiy aaaaoiiia?aiaeiiai aaaaoi?ic?yaeiiai
noiaoi?a ?c aeia?eueiith e?euee?noth aeiaeaie?a o ?ic?yaeao ? noaie
i?enei?aieo ia?aiin?a aeey iueiai; 4) aea/eoe caeiiii??iino? ia/eneaiiy
?ic?yae?a ia?aiiaiaiiy ? cia/aiiy ciaeiaiai ?ic?yaeo i?e
i?aenoiiaoaaii? N iia?aiae?a ? ?ic?iaeoe iaoiae i?iaeooaaiiy noaie
aecia/aiiy ciaeo ?acoeueoaoo i?aenoiiaoaaiiy N ciaeiaeo iia?aiae?a; 5)
?ic?iaeoe iaoiae i?iaeooaaiiy iiiiiaeoaa/a ia iniia? aaaaoiiia?aiaeieo
ia/enethaaeueieo no?oeoo? ? aeaaa?e NO; 6) ?ic?iaeoe iaeao i?aei?ia?ai
ooieoe?iiaeueii-eia?/iiai i?iaeooaaiiy AI, cie?aia: a) i?ia?aio
aeyaeaiiy neiao?e/ii? /anoeie a aeia?euei?e aoeaa?e ooieoe??; a)
i?ia?aio ?ic?aooieo ?ic?yaeieo ?iaeaen?a aeey i?iaeooaaiiy noiaoi??a ?c
aeia?eueiith e?euee?noth aeiaeaie?a o ?ic?yaeao.

A ae?oaiio ?icae?e? oai??y NO ?icaeaa?oueny oa ocaaaeueith?oueny c
iaoith aaeaioaoe?? aei ae??oaiiy caaea/? i?iaeooaaiiy aeayeeo eean?a AI.
Iiiyooy /anoeiai? neiao??? ocaaaeueith?oueny ia aeiaaeie aeae?eueeio
NE, ui ?icoe?th? eean NO. Cai?iiiiiaaii caniae i?aaenoaaeaiiy ?
i?i?i?caoe?? *NO, ui neeaaeathoue iniiao iaoiae?a neioaco noai ia iniia?
*NO. ?ic?iaeaii iaoiae aeiaecia/aiiy iaiiai?noth caaeaii? ooieoe?? aei
NO, ui ?icoe?th? aaeocue canoinoaaiiy iaoaiaoe/iiai aia?aoo NO ? *NO
aeey neioaco noai, ye? iienothoueny oaeeie ooieoe?yie.

I?aaenoaaeaiiy *NO caniiaaia ia aeei?enoaii? oaaeeoe? iaai??a a?oi (OIA)
? ia aeanoeainoyo /anoeiai neiao?e/ieo iaai??a (*NI). ?icaeyiaii
i?ioeaaeo?o iaea?aeaiiy OIA. I?ney aecia/aiiy neiao?e/ieo a?aoiaio?a
(NA) ia?aa?oio?ii ci?ii? a oaaeeoe?, uia ?icae?eeoe neiao?e/io (N*) ?
ianeiao?e/io /anoeio (IN*), ? ca?oio?ii iayai? noeoiiino? NA. Iio?i
cai?iy?ii io?eiai? noeoiiino? NA a?oiaie ? aecia/a?ii cia/aiiy a?oi.
I?e oeueiio oaaeeoey ?noeiiino? ia?aoai?th?oueny a OIA, ye?e aeanoea?
cae/aei? aea?eeia? ci?ii? ? ?oi? cia/aiiy. sseui *NO ia? e?eueea NE, oi
ci?ii?, ui neeaaeathoue eiaeiee NE, ? a?aoiaioaie a?aeiia?aeii? a?oie.

?icaeyiaii aeanoeaino? /anoeiai neiao?e/ieo iaai??a. Noeoii?noue *NI
ii?yaeeo r ?c nooiaiyie eiiieaeo?a mi (1( i ( r) oaeeo, ui e?euee?noue
iaeeieoeue a i-i eiiieaeo? N* aei??aith? ai, iie?eaa?oueny noeoii?noth
*NO-eiinoeoo?io aeaeo

(6)

( on? iiaeeea? IN*-eiinoeoo?ioe, i?e yeeo aeeiio?oueny oiiaa
aeanoeaino?. Oaeei /eiii, ooieoe?y, ui a?aeiia?aea? oe?e noeoiiino?
*NI

(7)

aea y(x1, x2 , … , xk ) ( aeayea ooieoe?y a?ae ianeiao?e/ieo
a?aoiaio?a x1 , x2, … xk. .

Oai?aia 1. Aoaeue-yeo *NO iiaeia i?aaenoaaeoe o aeaeyae? eia?/ii? noie
/eai?a (7).

I?i?i?caoe?y *NO caniiaaia ia ?iceeaaeaii? *NO ia N* ? IN* ? iia?aoe?yo
a?oiiaiai ? a?ia?iiai neea?iiy. ?icaeyaeathoueny aeaa niiniae
i?aaenoaaeaiiy *NO.

1. A?ia?ii-a?oiiaa i?i?i?caoe?y. Nii/aoeo cae?enith?oueny a?ia?ia
neea?iiy a OIA ne??cue, aea oea iiaeeeai, a iio?i a?oiiaa neea?iiy.
?icaeyaeath/e IN* ie?aii a?ae N* eiaeiiai aeaiaioa OIA, i?iaiaeeii
an?eye? neea?iiy INA eeane/ieie iaoiaeaie. I?ney oeueiai i?iaiaeeii
a?oiiaa neea?iiy i?ae oeie aeaiaioaie oaaeeoe?, o yeeo IN* ??ai?.

2. A?oi-a?ia?ia i?i?i?caoe?y. Nii/aoeo i?iaaaeeoueny a?oiiaa neea?iiy
ne??cue, aea oea iiaeeeai, a iio?i a?ia?ia. ?icaeyaeath/e N* ie?aii a?ae
IN* eiaeiiai aeaiaioa OIA, i?iaiaeeii an?eye? a?oiia? neea?iiy. I?ney
oeueiai i?iaiaeeii a?ia?ia neea?iiy i?ae oeie aeaiaioaie, o yeeo
?acoeueoaoe a?oiiaiai neea?iiy aeyaeeeny iaeiaeia?.

sseui ooieoe?iioaaiiy AI iieno?oueny iaiiai?noth aecia/aiith ooieoe??th,
yeo iiaeia ae?aceoe o aeaeyae? NO, oi aeey iieno AI canoiniao?oueny
aia?ao NO, ui ni?iuo? i?ioean i?iaeooaaiiy ? aeicaiey? iaea?aeaoe
aeiiii?/i? noaie. Iaoiae aeei?enoiao? AeE aeey aiae?co oiia, ui
iaeeaaeathoueny ia iaaecia/ai? iaai?e aeey ?nioaaiiy neiao??? aeaio
a?aoiaio?a. Cae?enith?oueny aiae?c on?o oaeeo oiia ca aeiiiiiaith
?ic?iaeaiiai eiia?iaoi?iiai aeai?eoio iiooeo NE ia iniia? ?oa?aoe?eiiai
i?ioeano iiaoaeiae aea?aaa ??oaiue, a iio?i aeey eiaeiiai ciaeaeaiiai NE
ca aeiiiiiaith iiaoaeiae a?aoa, ui a?aoiao? iaiaaeaiiy AeE aeey eiaeiiai
a?aoiaioo oeueiai NE, ciaoiaeyoueny on? aa??aioe aeiaecia/aiiy.

Ioe?iea neeaaeiino? cai?iiiiiaaiiai iaie aeai?eoio aea?oueny oi?ioeith

. (8)

O aeiaaeeo iiaiiai ia?aai?o on?o aa??aio?a aeiaecia/aiiy oa aiae?co
io?eiaieo ooieoe?e ia neiao?e/i?noue a?aeiieie iaoiaeaie (aeea. ?icae?e
1), neeaaei?noue caaea/? noaiiaeoue

. (9)

Ia?aaaaith cai?iiiiiaaiiai iaoiaeo ? eiai caeaoi?noue ciaoiaeeoe an?
iiaeeea? aa??aioe aeiaecia/aiiy aei NO c o?aooaaiiyi ?iaa?n?e a?aoiaio?a
? /anoeiai? neiao???. Aeei?enoaiiy iaoiaeo aeicaiey? ciaioeoe
neeaaei?noue caaea/? a?ae aeniiiaioe?aeueii? aei e?i?eii? noiniaii
e?eueeino? iaaecia/aieo iaai??a. Iaoiae ia? aenieee nooi?iue
oi?iae?caoe??, ui aeicaiey? ?aae?coaaoe eiai ia AII. Aeaeiiiiceoe?y
caaea/? aeicaiey? nei?ioeoe ia?aa?? aa??aio?a, aea a ??cieo a?eeao
caaea/? iiaeooue cono??/aoeny iiaoi?thaai? ??oaiiy. Iaoiae iiaea
?icaeaaoeny a iai?yieo niaoe?ae?caoe?? c iaoith iiooeo i?aeiiiaeei
aa??aio?a aeiaecia/aiy, ye? aeathoue NE, ui caaeiaieueiythoue caaeaiei
aeiiaai. I?eeeaae oaei? iiaeeo?eaoe?? iienaii. Oaeei /eiii,
cai?iiiiiaaiee iaoiae iiaea neoaeeoe aacith aeey canoinoaaiiy a nenoaiao
neioaco AI, ui iienothoueny iaiiai?noth caaeaieie ooieoe?yie.

O o?aoueiio ?icae?e? ?icaeyaeathoueny ieoaiiy i?iaeooaaiiy
aaaaoiiia?aiaeieo no?oeoo? oaeaeeiae?th/eo noiaoi??a oa ?o canoinoaaiiy
a iao?e/ieo iiiiiaeoaa/ao. Io?eiaii iaoiaeeeo i?iaeooaaiiy eaneaaeo
aaaaoiiia?aiaeiiai noiaoi?a c aeia?eueiith e?euee?noth aeiaeaie?a o
?ic?yaeao, ui caniiaaia ia iiaeae? c aa?oeeaeueiith ia?iaeith
?ioi?iaoe?? oa ia ?oa?aoe?eiiio ia/enethaaeueiiio aeai?eoi?, yeee
iiaeaeth? iioe?aiiy ia?aiino ocaeiaae ?ic?yaeii? n?oee. Io?eiai? ca
aeiiiiiaith iienaii? iaoiaeeee noiaoi?e a?ae??ciythoueny a?ae
o?aaeeoe?eieo noiaoi??a ?c ia?aeaeueiei ia?aiinii a?aenooi?noth
eaioethaa aioo??oiueiai ia?aiino. Ca ?aooiie aeei?enoaiiy iaiiceoe?eiiai
ia?aiino c a?oaie iaeiaeiai? aaae ? ?ic?iaeaiiai a aeai?e ?iaio?
iaoaiaoe/iiai aia?aoo i?aaenoaaeaiiy ? i?i?i?caoe?? *NO oey iaoiaeeea
aeicaieeea io?eiaoe /ioe?ea?oiee eaneaae o?eiia?aiaeiiai noiaoi?a, ui
ca?eueoo? oaeaeeiae?th a 1,3 ?ace a ii??aiyii? c o?aaeeoe?eiei caniaii
i?aenoiiaoaaiiy o?ueio iia?aiae?a, oi/a ? ia? ia 30% a?eueoo i?eaaaeaio
neeaaei?noue. *a?ac oa, ui ?oa?aoe?eiee aeai?eoi eaaei ?aae?co?oueny ia
AII, iaoiaeeea canoiniao?oueny aeey aaoiiaoecaoe?? i?iaeooaaiiy
oaeaeeiae?th/eo noiaoi??a ?c iiaeeea?noth aa??thaaiiy oa?aeoa?enoeeaie
neeaaeiino? oa oaeaeeiae??.

C iaoith i?aeaeuaiiy oaeaeeiae?? aaaaoi?ic?yaeieo aaaaoiiia?aiaeieo
noiaoi??a, ye? neeaaeathoueny c aeae?eueeio eaneaae?a (iiaeeeai,
??ciioeiieo), ?ic?iaeaii iaoiae i?iaeooaaiiy noaie i?enei?aiiai
ia?aiino, yea aaia?o? ao?aei? ia?aiine eiaeiiai eaneaaeo. Oey iaoiaeeea
caniiaaia ia iiaeeo?eaoe?? ?oa?aoe?eiiai aeai?eoio ? aeicaiey?
iaea?aeaoe noaie, ui caaacia/othoue ca?eueoaiiy oaeaeeiae?? a 1,5-2 ?ace
i?e ca?eueoaii? neeaaeiino? i?eno?ith ia 5-13% o ii??aiyii? c
iine?aeiaiei c’?aeiaiiyi eaneaae?a.

?ic?iaeaii iao?e/iee oaaee/ii-aeai?eoi?/iee iiiiiaeoaa/, o yeiio
oao?aaee /anoeiaeo aeiaooe?a i?aenoiiaothoueny o?eiia?aiaeiei noiaoi?ii.
Aeei?enoaiiy aaaaoiiia?aiaeieo no?oeoo? ia iniia? NO aeicaiey?
ca?eueoeoe oaeaeeiae?th ia 30%.

?ic?iaeaii i?eioeeiiai iiaee aoaeoeaiee iaoiae i?iaeooaaiiy ? no?oeoo?o
i?eno?ith aecia/aiiy ciaeo ? ?ic?yae?a ia?aiiaiaiiy ?acoeueoaoo
i?aenoiiaoaaiiy N /enae o aeiaeaoeiaiio eiae?, ui caaacia/o? aeiiii?th
aeo?ao onoaoeoaaiiy a aeae?eueea ?ac?a i?e aaeee?e e?eueeino? iia?aiae?a
o ii??aiyii? c o?aaeeoe?eiei niiniaii ?icoe?aiiy ?ic?yaeii? n?oee
iia?aiae?a ia iaiao?aeio e?euee?noue ?ic?yae?a.

Iieacaii, yea i?noea caeiathoue ?ic?iaeai? iaoiaee i?iaeooaaiiy AI o
caaaeuei?e no?oeoo?? eiiieaenii? nenoaie i?iaeooaaiiy oeeo?iaeo
i?eno?i?a (OeI).

Aaaaoi?yaeiee eiae ?c ci?iiei /eneii aeiaeaie?a o ?ic?yaeao iicia/eii MN
MN-1 … M2M1, aea Mi ( e?euee?noue aeiaeaie?a a i-io ?ic?yaeiiio c??c?.
Eiie?aoiee iaa?? cia/aiue aeiaeaie?a i?e ii?ac?yaeiiio i?aaenoaaeaii?
iicia/a?ii N-?ic?yaeiei eiaeii AN=aN aN-1… a2 a1. Eiaeiee oaeee iaa??
i?aaenoaaey? iiiaeeio iaai??a aeiaeaie?a ?c /eneii iaeeieoeue a i-io
?ic?yaeiiio c??c? ai, ui a?ae??ciythoueny eeoa ii?yaeeii ?icoaooaaiiy
iaeeieoeue o eiaeiiio ?ic?yaeiiio c??c?. Oaeei /eiii, aeei?enoaiiy
?iaeaen?a, ??aieo e?eueeino? iaeeieoeue o eiaeiiio ?ic?yaeiiio c??c?,
aeicaiey? cia/ii nei?ioeoe ianya aeo?aeieo aeaieo aeey iieno
ooieoe?iioaaiiy noiaoi?a ? iaaeae? ni?inoeoe eiai i?iaeooaaiiy.

, a i?aenoieiaith aaaith aoiae?a (IAA) ( aaee/eio Q’, ui aecia/a?oueny
ni?aa?aeiioaiiyi

,

aea N ( /eneiaee aea?aaeaio ao?aeiiai ia?aiino.

Iaoae S=?k?k-1…?1 SN… .S2S1 ( ?acoeueoao i?aenoiiaoaaiiy, iaaeaiee
o aea?eeia?e nenoai? /eneaiiy, S = CaeoSN… S2S1 ( ?acoeueoao
i?aenoiiaoaaiiy c aeo?aeiei ia?aiinii ?c a?oaie iaeiaeiai? aaae, aea
SN.…S2S1 ( a?oe noie S, ?k?k-1…?1 ( a?oe ia?aiino. Aeo?aei? ooieoe??
AI caaeathoueny ENO. Aeey eiaeii? aeo?aeii? ooieoe?? noie Si aai
ia?aiino Caeo j aecia/athoueny cia/aiiy IAAe Q, i?e yeeo caaeaia
ooieoe?y aei??aith? iaeeieoe?, ca oi?ioeaie

Si = 1 i?e (Q mod 2 i) ( 2i-1 — (C mod 2 i) (1 ( i ( N), (10)

Caeo j = 1 i?e (Q ( j*2N- C). (11)

Eiaeiiio cia/aiith IAAe Q a?aeiia?aea? iiiaeeia iaai??a AN=aN aN-1 …
a2 a1. Eiaeiee iaa?? AN a?aeiia?aea? eiie?aoiiio cia/aiith iaai?o a?oi
G1, G2, …GN ? iiaea aooe iiaeaiee o aeaeyae? eii’thieoe?? ANO, ui
a?aeiia?aeathoue ?iaeaenai iaai?o AN = aN

aN-1…a2a1. Oaeei /eiii, eiaeia c aeo?aeieo ENO Si ? Cauo j
i?aaenoaaey?oueny aeec’thieoe??th iaai??a cia/aiue a?oi G1, G2, …GN ,
ui a?aeiia?aeathoue oei AN ? IAAe Q, ye? caaeiaieueiythoue oi?ioeai
(10-11). Ioaea, oe? ooieoe?? iiaeooue aooe caaeai? o aeaeyae? AeAO,
caaeaii? a oaaee/iiio aeaeyae?. Aoaeaii caienoaaoe a oaaeeoeth eeoa
iiiaeeie ?iaeaen?a NO ? iaceaaoe ?? oaaeeoeath ?ic?yaeieo ?iaeaen?a
(O??).

O?? aeey eiaei?e c aeo?aeieo ooieoe?e iiaeooue aooe io?eiai? ca
aeiiiiiaith ?oa?aoe?eiiai aeai?eoio. Aeey oeueiai aeei?enoiaothoueny
?oa?aoe?ei? oi?ioee, ye? iienothoue i?ioean ia/eneaiue o noiaoi??, ui
eaaei ?aae?cothoueny aeai?eoiii. Oi?ioea aeey ia/eneaiiy /eneiaiai
aea?aaeaioo ia?aiino:

Ci+1 = (Ci+ai)/2, N1 = N0, 1 ( i ( N, (12)

aea Ci+1 ( ia?aiin o (i+1)-e ?ic?yae c i-ai ?ic?yaeo, N0 ( ao?aeiee
ia?aiin. Oey oi?ioea iiaeaeth? i?ioean iioe?aiiy ia?aiino ocaeiaae
?ic?yaeii? n?oee ? aeieeaa? c oiai, ui eiaeiee ?ic?yae noiaoi?a ?
iiae?eueieeii e?eueeino? iaeeieoeue ia 2.

Oi?ioea aeey ia/eneaiiy aeo?aeieo ooieoe?e noie:

Si+1=(Ci+1 + ai) mod 2. (13)

Oi?ioea aeey ia/eneaiiy aeo?aeieo ooieoe?e ia?aiino:

C(i+1) j = 1 i?e (Ci+1 ( j), 1 ( j ( Ci+1(max), 1 ( i ( N, (14)

aea C(i+1) j ( j-e ?ic?yae ia?aiino c i-ai a (i+1)-e ?ic?yae.

Iaoiae ?ic?aooieo O?? iieyaa? a ianooiiiio.

O?eno?ii ao?aeiee ia?aiin Cao=N0 ? ?icaeyaea?ii on? eiia?iaoe?? iaai??a
AN=aN

aN-1…a2a1. Aeey eiaeiiai iaai?o AN ?ic?aoiao?ii aeo?aeio ooieoe?th
F=Si+1 /e F=C(N+1)j. O? iaai?e AN, aeey eio?eo F=1, caiineii a O??.
Oaeei /eiii, aeey eiaeii? aeo?aeii? ooieoe?? ia?ii ii iaei?e O?? aeey
eiaeiiai cia/aiiy ao?aeiiai ia?aiino Cao=C0. Ianooiiei aoaiii ?
i?i?i?caoe?y io?eiaieo O?? ia iniia? iia?aoe?e a?oiiaiai neea?iiy.

N-?ic?yaeiee eaneaae noiaoi?a neeaaea?oueny c aeiea aeiiii?aeieo
ooieoe?e (AAeO), yeee ae?iaey? ENO, ui a?aeiia?aeathoue on?i io?eiaiei
O?? (eiaei?e O?? a?aeiia?aea? iaeia ENO), a oaeiae aeie?a noie (AN) ?
ia?aiino (AI), ui ae?iaeythoue aeo?aei? ooieoe?? noie ? ia?aiino
a?aeiia?aeii ? i?aoeththoue ye eiiooaoi?e (?en. 1). Oaeei /eiii, oea
o?e?aiaiaa noaia, ai AAeO neeaaea?oueny c aaia?aoi??a NO (ANO) a?o?a
eiaeiiai ?ic?yaeiiai c??co ? eia?/ii? iao?eoe?, ui ?aae?co? ooieoe?th,
caaeaio O?? (?en. 2).

Cai?iiiiiaaiee iaoiae i?iaeooaaiiy noiaoi?a aeicaiey? aaoiiaoe/ii
iaea?aeoaaoe iien ??cieo aa??aio?a ooieoe?iioaaiiy i?eno?ith a
oaaee/iiio aeaeyae? c iiaeeea?noth aa??thaaiiy oa?aeoa?enoeeaie
neeaaeiino? oa oaeaeeiae??.

I?e ?aae?caoe?? aaaaoiiia?aiaeieo no?oeoo? aeieea? caaea/a aecia/aiiy
ciaeo ?acoeueoaoo i?aenoiiaoaaiiy N /enae o aeiaeaoeiaiio eiae? aeey
N-iia?aiaeiiai noiaoi?a. O?aaeeoe?eii oey caaea/a ae??oo?oueny oeyoii
?icoe?aiiy ?ic?yaeii? n?oee iia?aiae?a ia k=[log2(N-1)]+1 ?ic?yae?a
oeyoii aeoaethaaiiy ciaeiaiai ?ic?yaeo, uia ?acoeueoao i?aenoiiaoaaiiy
oeeaaeaany oe?eeii o ?ic?yaeio n?oeo iia?aiae?a. Oea i?ecaiaeeoue aei
oneeaaeiaiiy noaie i?aenoiiaoaaiiy ca ?aooiie aeiaeaoeiaeo ?ic?yae?a
aaaaoiiia?aiaeieo noiaoi??a.

O ?iaio? i?iaaaeaia aeine?aeaeaiiy caeaaeiino? cia/aiue ciaeiaiai
?ic?yaeo ? ?ic?yae?a ia?aiiaiaiiy ?acoeueoaoo i?aenoiiaoaaiiy N /enae o
aeiaeaoeiaiio eiae? a?ae cia/aiue ciaeiaeo ?ic?yae?a iia?aiae?a ?
ia?aiin?a c iniiaii? ?ic?yaeii? n?oee iia?aiae?a, oa io?eiaia
caeaaei?noue, ui aeicaiey? ?aae?coaaoe aoaeoeai? noaie aeey aoaeue-yeiai
/enea iia?aiae?a:

(15)

aea Rk…R2R1 — aea?eeiaee aea?aaeaio noie ciaeiaeo ?ic?yae?a iia?aiae?a,
oiaoi

Rk…R2R1 = (Zi (i?aenoiiaoaaiiy cae?enith?oueny ii an?i iia?aiaeai);
Qk…Q1 — iiceoe?ei? ia?aiine c o?aooaaiiyi N ciaeiaeo ?ic?yae?a, oiaoi
Qk…Q1Q0 = Rk…R2R1 + Pk…P2P1;

Z — ae?eniee ciaeiaee ?ic?yae ?acoeueoaoo; U1… Uk — ae?eni? ?ic?yaee
ia?aiiaiaiiy;

( iaa?iaiee eiae /enea Rk…R2R1.

a iaa?iaiiio eiae?, ? k-?ic?yaeiee aeaiiia?aiaeiee noiaoi? ( aeey
i?aenoiiaoaaiiy iaa?iaiiai eiaeo c ia?aiinaie Qk …Q2Q1 ca oi?ioeith
(15). Oaeaeeiae?y i?eno?ith caeaaeeoue eeoa a?ae aeaiiia?aiaeiiai
noiaoi?a. Io?eiaii oi?ioee ? noaia aeey N=3 (?en. 3):

.

?en. 1. N-?ic?yaeiee ?en. 2. No?oeoo?a ?en. 3. Noaia aecia/aiiy

eaneaae AAeO ciaeo

I?e i?iaeooaaii? aaaaoi?ic?yaeieo noiaoi??a aeieea? caaea/a ?icaeaee
iia?aiae?a ia naaiaioe, ui i?aenoiiaothoueny eiaeiee nai?i eaneaaeii. C
iaoith ca?eueoaiiy oaeaeeiae?? i?eno?ith, ui neeaaea?oueny c e?eueeio
eaneaae?a, ?ic?iaeaii iaoiae i?iaeooaaiiy noaie i?enei?aiiai ia?aiino,
ui ia/eneth? oa iaeii/anii aeaea? ia aoiaee ia?aiin?a an?o eaneaae?a
cia/aiiy ao?aeieo ia?aiin?a Nao(i) ia iniia? ao?aeieo ia?aiin?a
i?eno?ith Nao(1) ? aeayeeo aeiiii?aeieo ooieoe?e AOi, ui caeaaeaoue a?ae
iia?aiae?a ? aaia?othoueny AAeO eiaeiiai eaneaaeo. Oeae iaoiae
caniiaaiee ia ?ioa?i?aoaoe?? eaneaaeo ye iaeiiai ?ic?yaeiiai c??co c
e?euee?noth iaeeieoeue ia aoiae? Mi a?ae 0 aei Qmax(i)= M1(i)+
2M2(i)+…+2k-1Mk(i), ? ia oiio, ui k-?ic?yaeiee eaneaae noiaoi?a
aeeiio? ooieoe?th ae?eaiiy e?eueeino? iaeeieoeue ia 2k. O ?acoeueoao?
canoinoaaiiy iiaeeo?eiaaiiai iaoiaeo ooai?ththoueny O??, oa?iaie yeeo ?
AOi , ui caeaaeaoue a?ae Q(i). Canoinoaaiiy noaie i?enei?aiiai ia?aiino
caaacia/o? ca?eueoaiiy oaeaeeiae?? i?eno?ith a 1,5-2 ?ace a ii??aiyii? c
iine?aeiaiei c’?aeiaiiyi eaneaae?a i?e ca?eueoaii? neeaaeiino? i?eno?ith
ia 5-15%. Ia?aaaaith oeueiai iaoiaeo ? iiaeeea?noue i?iaeooaaiiy aeiea
i?enei?aiiai ia?aiino aeey ??ciioeiieo eaneaae?a.

Ca aeiiiiiaith cai?iiiiiaaiiai iaoiaeo ?ic?iaeaii o?eiia?aiaeiee
16-?ic?yaeiee /ioe?eeaneaaeiee noiaoi? ?c noaiith i?enei?aiiai ia?aiino,
yeee oaeaeoa i?ioioeio ia 30%. O?aaiaio O??, ui iieno? eiai
ooieoe?iioaaiiy, i?eaaaeaii a oaae. 1.

Oaaeeoey 1

O?aaiaioe O?? /ioe?ea?oiiai eaneaaeo o?eiia?aiaeiiai noiaoi?a

Ao?aei? ia?aiine Cao2C ao1 Ao?aei? ia?aiine Cao2C ao1

0 0 0 1 1 1 0 0 0 1 1 1

?ic?yaei? ?iaeaene aeey S3 ?ic?yaei? ?iaeaene aeey S2

3 2 1 3 2 1 3 2 1 2 1 2 1 2 1

0,2 1,2 2,3 0,2 1,2 1,2 0,2 0,1 2,3 0,2 2,3 0,2 1,2 0,2 0,1

0,2 2,3 0,1 1,3 0,3 1,2 0,2 1,2 0,1 1,3 0,1 1,3 0,3 1,3 2,3

1,3 0,1 0,1 0,2 2,3 0 1,3 0,3 0,1

1,3 0,3 2,3 1,3 0,1 0 1,3 2,3 2,3

0,2 0,1 3

1,3 2,3 3

Ia iniia? o?eiia?aiaeiiai noiaoi?a ?ic?iaeaii oaaee/ii-aeai?eoi?/iee
iiiiiaeoaa/ 4n*4n, ui o 1,3 ?ace oaeaeoa i?ioioeio, yeee aeei?enoiao?
aeey i?aenoiiaoaaiiy /anoeiaeo aeiaeaoe?a aeaiiia?aiaei? noiaoi?e (?en.
4).

?en. 4. Oaaee/ii-aeai?eoi?/iee iiiiiaeoaa/ 12*12.

Ia i?aenoaa? io?eiaieo ye?nieo oa?aeoa?enoee aaaaoiiia?aiaeieo noiaoi??a
cai?iiiiiaaii iiaeeo?eiaaiee iaoiae cai?oee aaaaoi?yaeiiai eiaeo, ui
aeicaiey? ciaioeoe aeo?aoe onoaoeoaaiiy. Oaeaeeee eiia?iiaaiee
aa?oeeaeueii-naeoe?eiee o?eaoaiiee niin?a cai?oee i?eionea? ia ia?oiio
aoai? noene m-?yaeiiai eiaeo a m’-?yaeiee eiae ca aeiiiiiaith AIN, ia
ae?oaiio aoai? naeoe?iioaaiiy m’-?yaeiiai eiaeo c iaoith cai?oee a
aeai?yaeiee, i?e/iio eiaeia naeoe?y ia?iaey?oueny m’-iia?aiaeiei
k-?ic?yaeiei noiaoi?ii, aea m’=k=[log2(m’-1)]+1. C o?aooaaiiyi
iiaeeeaino? i?iaeooaaiiy aoaeoeaieo oaeaeeiae?th/eo aaaaoiiia?aiaeieo
noiaoi??a cai?iiiiiaaii iiaeeo?eaoe?th iaoiaeo naeoe?iioaaiiy, i?e yeiio
naeoe?iiiaaiee m-?yaeiee eiae noeneo?oueny ia a aeai?yaeiee eiae, a a
t-?yaeiee (t>2) ? io?eiaiee eiae ia o?aoueiio aoai? i?aenoiiao?oueny
t-iia?aiaeiei ia?aeaeueiei noiaoi?ii. I?e oeueiio ?ic?yaei?noue naeoe??
k ciaioo?oueny a t ?ac?a (k=[(1/t)log2(m-1)]+1), oa aia?aoo?i? aeo?aoe
ia?aiinyoueny c aeieo ia?iaee naeoe?e o aeie cai?oee t-?yaeiiai eiaeo.
Aeine?aeaeaiiy aoaeoeaiino? oe??? iiaeeo?eaoe?? iieaco?, ui ia?ao?ae aei
o?e?yaeiiai eiaeo aea? ciaioaiiy onoaoeoaaiiy i?e ?yaeiino? eiaeo m>4.

O /aoaa?oiio ?icae?e? cae?enith?oueny aiae?oe/ia oa aenia?eiaioaeueia
aeine?aeaeaiiy oa?aeoa?enoee ? aeanoeainoae ?ic?iaeaieo a
aeena?oaoe?ei?e ?iaio? iaoiae?a i?iaeooaaiiy AI. Aiae?oe/ia
aeine?aeaeaiiy caniiaaia ia i?ia?aiiiio ?ic?aooieo aii??e/ieo oi?ioe, ui
ae?aaeathoue caeaaei?noue oa?aeoa?enoee i?eno?ith a?ae eiai ia?aiao??a
(e?eueeino? iia?aiae?a ? ?ic?yae?a aaaaoi?yaeiiai eiaeo).
Aenia?eiaioaeueia aeine?aeaeaiiy caaacia/o?oueny noai?aiiyi
iiaiiooieoe?iiaeueii? i?ia?aiii? iiaeae? i?eno?ith, ui i?aeaea?oueny
i?ioeaaeo?ai neioaco, aa??o?eaoe?? ? iiaeaethaaiiy ia ??cieo ??aiyo
aeaoae?caoe?? iieno.

Noai?aii VHDL-iiaeaeue /ioe?ea?oiiai eaneaaeo o?eiia?aiaeiiai noiaoi?a,
io?eiaiiai ca aeiiiiiaith ?ic?iaeaiiai a aeena?oaoe?? iaoiaeo
i?iaeooaaiiy aaaaoiiia?aiaeieo no?oeoo?, ? i?iaaaeaii
ooieoe?iiaeueii-eia?/ia iiaeaethaaiiy i?eno?ith aeey ia?aa??ee
i?aaeeueiino? ooieoe?iioaaiiy noiaoi?a. A ?acoeueoao? i?iaaaeaiiai
iiaeaethaaiiy aeiaaaeaia i?aaeeuei?noue ooieoe?iioaaiiy iiaeae? ?
iieacaia a??ia?aei?noue aeuaaeacaiiai iaoiaeo. O?aaiaioe ae?aa?ai ?iaioe
noiaoi?a iaaaaeai? ia ?en. 5, aea Aop, Bop, Cop ( iia?aiaee,
sExpectedSum ( i/?eoaaia noia, sGotSum ( io?eiaia noia.

?en. 5. Ae?aa?aie ?iaioe eaneaaeo o?ueioiia?aiaeiiai noiaoi?a

Ca aeiiiiiaith i?ia?aie Orcad Express 7.0 i?iaaaeaii eia?/iee neioac
iiaeae? noiaoi?a ia aeaiaioi?e aac? ianeao eia?/ieo aaioee?a (FPGA)
o??ie Xilinx ? ooieoe?iiaeueii-eia?/ia iiaeaethaaiiy noiaoi?a. Eia?/iee
neioac iaaeaa iiaeeea?noue aeaanoe aa?enoe/io oi?ioeo aaioeeueii?
neeaaeiino? neioaciaaieo ii cai?iiiiiaaiiio a oe?e ?iaio? iaoiaeo noai
aaaaoiiia?aiaeieo noiaoi??a i?e aeia?eueieo cia/aiiyo ia?aiao??a
noiaoi?a ? ?ic?aooaaoe aaee/eio iioeaee oe??? oi?ioee. Aiae?oe/ia
ni?aa?aeiioaiiy aeey /enaeueiiai cia/aiiy neeaaeiino? noaie, ae?aaeaii?
o aaioeeyo:

Z = Z(AAeO) + 4(N-1) + 5(N-1)(M-1) + 2N+2 M + (N-1) N M / 2, (16)

aea Z(AAeO) ( aaioeeueiee eiao?oe??io neeaaeiino? AAeO, ui
?ic?aoiao?oueny iaoiaeii Eaaeia, N ( e?euee?noue iia?aiae?a noiaoi?a, I
( e?euee?noue eiai ?ic?yae?a.

Aeaye? ?ic?iaeai? iaoiaee aeiaaaeai? aei i?ia?aiii? ?aae?caoe?? ia AII
oa i?eaaaeai? ?aeiiaiaeaoe?? uiaei ?o aeei?enoaiiy. I?ia?aia aeae?eaiiy
neiao?e/ii? /anoeie c aeia?eueii? ooieoe?? aea? ae/a?iio ?ioi?iaoe?th c
i?eaiaeo neiao??? aoeaai? ooieoe?? ? iiaea aooe iniiaith i?aenenoaie
ooieoe?iiaeueii-eia?/iiai i?iaeooaaiiy OeI ia iniia? *NO. I?ia?aia
?ic?aooieo O??, yea a?aeiia?aea? ?ic?iaeaiiio iaoiaeo i?iaeooaaiiy
aaaaoiiia?aiaeieo noiaoi??a, aeicaiey? a aaoiiaoe/iiio ?aaeei?
iaea?aeoaaoe iiiaeeio aa??aio?a, ui a?ae??ciythoueny oa?aeoa?enoeeaie
neeaaeiino? oa oaeaeeiae?? ? aeei?enoo?oueny aeey aiae?co aeanoeainoae
cacia/aiiai iaoiaeo, cie?aia neeaaeiino? iaea?aeaieo noai.

I?iaaaeaiee aiae?oe/iee ? aenia?eiaioaeueiee aiae?c oa?aeoa?enoee
iaoiaeo i?iaeooaaiiy aaaaoiiia?aiaeieo noiaoi??a aeicaieea ?ic?iaeoe
?aeiiaiaeaoe?? aeey eiai aeei?enoaiiy. Aeey aeai?o iioeiaeueieo
ia?aiao??a eaneaaeo, ye io e?eueeino? iia?aiae?a N ? ?ic?yaeiino? M, a
oaeiae oeio ia?aiino, cae?enith?oueny aeine?aeaeaiiy ooieoe?e
neeaaeiino? Ziau(N, M) oa i?eaaaeaii? aei iaeiiai a?oa neeaaeiino?
Zaeo(N, M) N-iia?aiaeiiai M-?ic?yaeiiai noiaoi?a (N(M). Aiae?c a?ao?e?a
ia ?en. 6 iieaco?, ui ii/eiath/e c aeayei? aaee/eie M0, eaea ??cea
aeniiiaioe?aeueia c?inoaiiy neeaaeiino?. Aiaeia?/ii c?iaeaii aeniiaie,
ui aeei?enoaiiy aeo?aeieo ia?aiin?a c a?oaie iaeiaeiai? aaae a?eueo
aoaeoeaii, i?ae iiceoe?eiee aea?eeiaee ia?aiin, i?e e?eueeino?
iia?aiae?a N<9. Ia ?en. 7 i?eaaaeaii ii??aiyeuei? a?ao?ee neeaaeiino? noai aecia/aiiy ciaeo. ?en. 6. Ia?aiao?e/i? a?ao?ee neeaaeiino? noiaoi?a ?en. 7. Aecia/aiiy ciaeo AENIIAEE 1. Aeei?enoaiiy NO ? *NO aeey iieno AI i?e neioac? N-?ic?yaeieo oaeaeeiae?th/eo no?oeoo? ciaioo? ?ici??i?noue caaea/? neioaco a ii??aiyii? c aeei?enoaiiyi o?aaeeoe?eii? aoeaai? aeaaa?e a?ae 2MN aei (M+1)N, aea M ( e?euee?noue iia?aiae?a noiaoi?a. Oea aeicaiey? ?icoe?eoe canoinoaaiiy iaoiae?a neioaco ia aaaaoi?ic?yaei? no?oeoo?e. 2. Iien ooieoe?iioaaiiy AI o oa?i?iao aeaaa?e NO ? *NO aeicaiey? cia/ii caoceoe ciio iiooeo iioeiaeueiiai ??oaiiy caaea/? neioaco AI, a?aeeeioaoe aaeeeo e?euee?noue iaei?enieo aa??aio?a neioaco, ? iaaeeceoe i?eno??e aei iioeiaeueiiai. 3. Canoinoaaiiy aia?aoo NO caaacia/o? iaeiiiai?oi?noue iieno ooieoe?iioaaiiy AI oa iiaeeea?noue oi?iae?caoe?? i?ioeano i?iaeooaaiiy AI, ui aeaei iiaeeea?noue noai?eoe aoaeoeai? iaoiaee aaoiiaoeciaaiiai i?iaeooaaiiy noiaoi??a. 4. Iniiai? ?acoeueoaoe ?iaioe: 1) cai?iiiiiaaii iaoiae i?aaenoaaeaiiy ? i?i?i?caoe?? *NO ?c aeae?eueeiia eiiieaeoaie NA, ui aeicaiey? canoinoaaoe *NO aeey aoaeoeaiiai neioaco aaaaoi?ic?yaeieo noiaoi??a; 2) ?ic?iaeaii iaoiae ? aeai?eoi aeiaecia/aiiy iaiiai?noth caaeaii? aeia?eueii? aoeaai? ooieoe?? aei NO, ui aeicaiey? ?icoe?eoe eean NO; 3) ?ic?iaeaii iaoiae ? i?ia?aia eia?/iiai i?iaeooaaiiy N-iia?aiaeiiai noiaoi?a ?c aeia?eueiith e?euee?noth iia?aiae?a o ?ic?yaeao oa aaoiiaoeciaaii? aaia?aoe?? eiai iieno a oa?i?iao aeaaa?e NO, a oaeiae iaoiae i?iaeooaaiiy noaie i?enei?aiiai ia?aiino aeey aaaaoiiia?aiaeiiai noiaoi?a; 4) ia iniia? cai?iiiiiaaiiai iaoiaeo neioaciaaii noiaoi? o?ueio iia?aiae?a ia aac? /ioe?ea?oiiai eaneaaeo ? noaie i?enei?aiiai ia?aiino; 5) ?ic?iaeaii iaoiae i?iaeooaaiiy noaie aecia/aiiy ciaeo ?acoeueoaoo i?aenoiiaoaaiiy N /enae o aeiaeaoeiaiio eiae? ia aac? oi?aa?naeueii? aiae?oe/ii? caeaaeiino?, ui caaacia/o? i?aeaeuaiiy oaeaeeiae?? a 1,3 ?ace ? a?eueo ? ciaioaiiy neeaaeiino? noaie a 1,4 ?ace ? a?eueo; 6) ?ic?iaeaii iaoiae iiaoaeiae oaaee/ii-aeai?eoi?/iiai iiiiiaeoaa/a ia iniia? o?eiia?aiaeiiai noiaoi?a, ui aea? i?aeaeuaiiy oaeaeeiae?? i?eno?ith a 1,3 ?ace; 7) cai?iiiiiaaii iiaeeo?eiaaiee iaoiae cai?oee aaaaoi?yaeiiai eiaeo ia iniia? aaaaoiiia?aiaeieo noiaoi??a, ui aea? ciaioaiiy neeaaeiino? noaie a 1,2 ?ace ? a?eueo. NIENIE IIOAE?EIAAIEO AAOI?II I?AOeUe CA OAIITH AeENA?OAOe?? 1. Iaoeei I.I., Eyoiaaoeeee A.I., Neiaaoa I.E. Iiauoaiea auno?iaeaenoaey eiiaeiaoeeiiiuo oiiiaeeoaeae // O?oaeu Iaeanneiai iieeoaoie/aneiai oieaa?neoaoa. ( Iaeanna. ( 1996. ( ? 2. ( N. 17–18. 2. Iaoeei I.I., Eyoiaaoeeee A.I. Ii?aaeaeaiea cia/aiey ciaeiaiai ?ac?yaea ?acoeueoaoa noiie?iaaiey N /enae // O?oaeu Iaeanneiai iieeoaoie/aneiai oieaa?neoaoa. ( Iaeanna. ( 1997. ( ? 2. ( N. 37–40. 3. Iaoeei I.I., Eyoiaaoeeee A.I. Aaoiiaoecaoeey ooieoeeiiaeueii-eiae/aneiai i?iaeoe?iaaiey iaeioi?uo eeannia oeeo?iauo ono?ienoa // O?oaeu Iaeanneiai iieeoaoie/aneiai oieaa?neoaoa. ( Iaeanna. ( 1998. ( ? 2(6). ( N. 31–33. 4. Eyoiaaoeeee A.I. Aaoiiaoecaoeey i?iaeoe?iaaiey aeiea onei?aiiiai ia?aiina iiiaineiaiiai noiiaoi?a ia aaca neiiao?e/aneeo aoeaauo ooieoeee // I?eaeiii?ianueeee iaoeiaee ainiee. Oaoii/ii iaoee. ( Aeiai?iiao?iane. ( 1998. ( ? 90 (157). ( N. 43-48. 5. Iaoeei I.I., Eyoiaaoeeee A.I. Neioac neiiao?e/aneie aoeaaie ooieoeee ia aaca iaiieiinoueth caaeaiiie aoeaaie ooieoeee // I?eaeiii?ianueeee iaoeiaee ainiee. Oaoii/ii iaoee. ( Aeiai?iiao?iane. ( 1998. ( ? 108 (175). ( N. 49-55. 6. Iaoeei I.I., Eyoiaaoeeee A.I. Oaaee/ii-aeai?eoie/aneee oiiiaeeoaeue 4n*4n ia iniiaa noiiaoi?a o?ao iia?aiaeia // Iao/iua o?oaeu iieiaeuo o/aiuo. ( Iaeanna: Iaeanneee ainoaea?noaaiiue iieeoaoie/aneee oieaa?neoao. ( 1997. ( N. 97 – 101. 7. Iaoeei I.I., Eyoiaaoeeee A.I. Oaaee/ii-aeai?eoie/aneee iaoiae oiiiaeaiey // O/aiua caienee Neioa?iiieueneiai ainoaea?noaaiiiai oieaa?neoaoa. Nai?iee o?oaeia iaaeaeoia?iaeiie iao/ii-oaoie/aneie eiioa?aioeee “I?eai?ino?iaiea-98”. ( Aeiieoea-Neioa?iiieue. ( 1998. ( N. 156-159. 8. Iaoaio 94023276 Oe?aeiu, IEE G06F 7/50. I?eno?ie aeey iiaenoiiaoaaiiy o?ueio n-?ic?yaeieo iia?aiaeia / Iaoeei I.I., Eyoiaaoeeee A.I., Neiaaoa I.E.; Iaeanneee iieeoaoie/aneee oieaa?neoao. — ? 23235A; Cayae. 14.02.1994; Iioae. 19.05.1998; Athe. “I?iieneiaa aeani?noue”, 1998, ? 4/2. — N. 272. 9. Neioac eiiaeiaoeeiiiuo noai, iienuaaaiuo /anoe/ii neiiao?e/aneeie aoeaauie ooieoeeyie / Iaoeei I.I., Eyoiaaoeeee A.I.; Iaeanneee ainoaea?noaaiiue iieeoaoie/aneee oieaa?neoao. ( Iaeanna, 1995. ( 23 n. ( ?on. ( Aeai. a AIOA Oe?aeiu 13.02.95, ? 350 ( Oe95 // Aiio. a ae. AEIEOE ?AI “Aeaiiie?iaaiiua iao/iua ?aaiou”, ? 5 (282), 1995, a/i ? 238. Eyoiaaoeueeee I.I. Iaoiaee ooieoe?iiaeueii-eia?/iiai i?iaeooaaiiy oaeaeeiae?th/eo a?eoiaoe/ieo i?eno?i?a ia iniia? neiao?e/ieo aoeaaeo ooieoe?e. ( ?oeiien. Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa oaoi?/ieo ca niaoe?aeuei?noth 05.13.05 ( “Aeaiaioe oa i?eno?i? ia/enethaaeueii? oaoi?ee oa nenoai ea?oaaiiy”. ( Iaeanueeee aea?aeaaiee iie?oaoi?/iee oi?aa?neoao, Iaeana, 1999. A ?iaio? ?ic?iaey?oueny iiaee i?aeo?ae aei i?iaeooaaiiy oaeaeeiae?th/eo a?eoiaoe/ieo i?eno?i?a (AI), ui a?oioo?oueny ia aeei?enoaii? aeanoeaino? neiao??? ao?aeieo ?ioi?iaoe?eieo neaiae?a, yeee caaacia/o? i?aeaeuaiiy oaeaeeiae?? AI, onoiaiiy aiia?aoiiai iaaeeeoeo AI ca ?aooiie aeei?enoaiiy niaoeeo?ee ?o iia?aoe?e, ni?iuaiiy oa iaeiiiai?oi?noue i?ioeano i?iaeooaaiiy ? iiaeeea?noue oi?iae?caoe?? oa aaoiiaoecaoe?? aeayeeo eiai aoai?a. Cai?iiiiiaaii iaoiaee ooieoe?iiaeueii-eia?/iiai i?iaeooaaiiy oaeaeeiae?th/eo aaaaoiiia?aiaeieo noiaoi??a oa iiiiiaeoaa/?a, ui a?oioothoueny ia aeei?enoaii? neiao?e/ieo aoeaaeo ooieoe?e n /anoeiaith neiao???th aeey iieno ooieoe?iioaaiiy AI. ?acoeueoaoe ?iaioe ?aeiiaiaeiaai? aeey aeei?enoaiiy i?e ?ic?iaoe? a?eoiaoeei-eia?/ieo aeie?a, i?ecia/aieo aeey ae??oaiiy caaea/ oeeo?iai? ia?iaee neaiae?a, ia?iaee ia?aeaeueieo iioie?a aeaieo o ?aaeueiiio /an?, i?eno?i?a eiaeoaaiiy oa aeaeiaeoaaiiy oa ?ioeo caaea/, e?eoe/ieo aei /ano ?aaeoe?? nenoaie. Eeth/ia? neiaa: a?eoiaoe/i? i?eno?i?, neiao?e/i? aoeaa? ooieoe??, /anoeiaa oa iiaia neiao??y, aaoiiaoecaoe?y, i?iaeooaaiiy, oaeaeeiae?y. Lyakhovetsky A.M. Methods of functional-logical design of fast arithmetical devices based on symmetric boolean functions. ( Manuscript. The thesis for Master of technical Science degree by speciality 05.13.05 ( “Elements and devices of computers equipment and control systems”. ( Odessa State Politechnical University, Odessa, 1999. A new approach to fast arithmetical devices (AD) design has been developed, that is based on using of specific particularities of data input symmetry, and can provide considerable speedup, hardware abundunce reduction of developed devices due to considering of some specific properties of performed operations, simplification of design procedures, unification of design process and possibility of automation of some design procedures. Methods of functional-logical design of fast multi-operand adders and multipliers are suggested, that are based on usage of symmetric boolean functions with partial symmetry for defining functionality of these devices. The results of research work are recommended to use for design of arithmetical-logical units, contained in special processing devices, intended for digital signal processing, parallel data flow real time processing, coding and decoding devices and other tasks, that are critical to system response time. Key words: arithmetical device, symmetric boolean functions, partial and total symmetry, automation, design, speed. Eyoiaaoeeee A.I. Iaoiaeu ooieoeeiiaeueii-eiae/aneiai i?iaeoe?iaaiey auno?iaeaenoaothueo a?eoiaoe/aneeo ono?ienoa ia iniiaa neiiao?e/aneeo aoeaauo ooieoeee. ( ?oeiienue. Aeenna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa oaoie/aneeo iaoe ii niaoeeaeueiinoe 05.13.05 ( “Yeaiaiou e ono?ienoaa au/eneeoaeueiie oaoieee e nenoai oi?aaeaiey”. ( Iaeanneee ainoaea?noaaiiue iieeoaoie/aneee oieaa?neoao, Iaeanna, 1999. A ?aaioa ?ac?aaaouaaaony iiaue iiaeoiae e i?iaeoe?iaaieth auno?iaeaenoaothueo a?eoiaoe/aneeo ono?ienoa (AO), iniiauaathueeny ia eniieueciaaiee naienoaa neiiao?ee aoiaeia AO e iaania/eaathuee iiauoaiea auno?iaeaenoaey e ono?aiaiea ecauoi/iinoe AO, oi?iuaiea, aaeeiiia?acea e aiciiaeiinoue oi?iaeecaoeee e aaoiiaoecaoeee i?ioeanna i?iaeoe?iaaiey. I?aaeeiaeaiu iaoiaeu ooieoeeiiaeueii-eiae/aneiai i?iaeoe?iaaiey iiiaiiia?aiaeiuo noiiaoi?ia e oiiiaeeoaeae, iniiaaiiua ia eniieueciaaiee /anoe/ii-neiiao?e/aneeo aoeaauo ooieoeee (*NO). Auyaeaiu iaaeinoaoee ecaanoiuo iaoiaeia i?iaeoe?iaaiey auno?iaeaenoaothueo AO: a) nenoaiu aoeaauo ooieoeee, iienuaathuea iiiaiiia?aiaeiua iaeii?ac?yaeiua noiiaoi?u (IIN), neaai ieieiece?othony a eeannao ii?iaeueiuo oi?i, iiyoiio neioac IIN o?aaeeoeeiiiuie iaoiaeaie cao?oaeieoaeai; a) i?eiaiaiea nouanoaothueo iaoiaeee iiiai?ac?yaeiuo noiiaoi?ia ia iniiaa *NO i?e aieueoii /enea ?ac?yaeia oneiaeiaii, oae eae o?aaoao iiiaie?aoiie aiaeeoe/aneie aeaeiiiiceoeee ooieoeee, /oi i?eaiaeeo e oooaeoaieth auno?iaeaenoaey noai; a) nouanoaotho yooaeoeaiua iaoiaeu i?iaeoe?iaaiey oieueei aeia?iuo a?eoiaoe/aneeo no?oeoo?, a iiiaiiia?aiaeiua no?oeoo?u no?iyony ia iniiaa aeia?iuo; a) caaea/a neioaca N–?ac?yaeiuo noiiaoi?ia eiaao ?acia?iinoue 2MN, aaea M ( eiee/anoai iia?aiaeia, /oi ia?aie/eaaao aiciiaeiinoue aa ?aoaiey o?aaeeoeeiiiuie iaoiaeaie aoeaaie aeaaa?u; ae) nouanoaothuee aiia?ao NO iicaieyao eniieueciaaoue oieueei iaeei neiiao?e/aneee eiiieaeo a?aoiaioia (NE), /oi aeaao aiciiaeiinoue no?ieoue iiiaiiia?aiaeiua no?oeoo?u oieueei n oeaiueth ?ani?ino?aiaiey iaae?ac?yaeiiai ia?aiina. A aeenna?oaoeeiiiie ?aaioa oai?ey NO ?acaeaaaony e iaiauaaony n oeaeueth aaeaioaoeee e i?iaeoe?iaaieth iiiai?ac?yaeiuo noiiaoi?ia. Iiiyoea /anoe/iie neiiao?ee iaiauaaony ia neo/ae ianeieueeeo NE. I?aaeeiaeaiu niiniau i?aaenoaaeaiey e ieieiecaoeee *NO, ninoaaeythuea iniiao iaoiaeia neioaca noai ia iniiaa *NO. ?ac?aaioai iaoiae aeiii?aaeaeaiey iaiieiinoueth caaeaiiie ooieoeee aei NO, niiniaiue iaoiaeeoue ana aa?eaiou aeiii?aaeaeaiey n o/aoii eiaa?nee e /anoe/iie neiiao?ee. Aai eniieueciaaiea nie?auaao neiaeiinoue caaea/e io yeniiiaioeeaeueiie aei eeiaeiie. Iaoiae iaeaaeaao aunieie noaiaiueth oi?iaeecaoeee, /oi iicaieyao ?aaeeciaaoue aai ia YAI. Iieo/aia iaoiaeeea i?iaeoe?iaaiey iiiaiiia?aiaeiiai auno?iaeaenoaothuaai noiiaoi?a n i?iecaieueiui eiee/anoaii neaaaaiuo a ?ac?yaeao, iniiaaiiay ia aa?oeeaeueiie ia?aaioea eioi?iaoeee e ia eoa?aoeeiiiii aeai?eoia, iiaeaee?othuai ?ani?ino?aiaiea ia?aiina. Iieo/aiiua noiiaoi?u ioee/athony io o?aaeeoeeiiiuo ionoonoaeai oeaie aioo?aiiaai ia?aiina. Ca n/ao eniieueciaaiey iaoiaea i?aaenoaaeaiey e ieieiecaoeee *NO aeaiiay iaoiaeeea iicaieeea iieo/eoue o?aoiia?aiaeiue noiiaoi?, eioi?ue auno?aa a 1,3 ?aca ii n?aaiaieth n o?aaeeoeeiiiui niiniaii noiie?iaaiey o?ao iia?aiaeia. N oeaeueth iiauoaiey auno?iaeaenoaey iiiaiiia?aiaeiuo noiiaoi?ia ?ac?aaioai iaoiae i?iaeoe?iaaiey noaiu onei?aiiiai ia?aiina, oaaee/eaathuee auno?iaeaenoaea a 1,5-2 ?aca ii n?aaiaieth n iineaaeiaaoaeueiui niaaeeiaieai eaneaaeia. ?ac?aaioai iao?e/iue oaaee/ii-aeai?eoie/aneee oiiiaeeoaeue, a eioi?ii oao?aaeu /anoe/iuo i?iecaaaeaiee noiie?othony n iiiiuueth o?aoiia?aiaeiiai noiiaoi?a, /oi oaaee/eaaao auno?iaeaenoaea ia 30%. ?ac?aaioai iaoiae i?iaeoe?iaaiey ono?ienoaa ii?aaeaeaiey ciaea ?acoeueoaoa noiie?iaaiey N /enae, /oi aeaao yeiiiieth cao?ao iai?oaeiaaiey a ianeieueei ?ac ii n?aaiaieth n o?aaeeoeeiiiui niiniaii ?anoe?aiey ?ac?yaeiie naoee iia?aiaeia. I?eaaaeai i?eia? ?ac?aaioee noaiu ii?aaeaeaiey ciaea aeey o?ao iia?aiaeia. I?aaeeiaeai iiaeeoeoee?iaaiiue iaoiae naa?oee iiiai?yaeiiai eiaea, iicaieythuee oiaiueoeoue cao?aou iai?oaeiaaiey, ninoiyuee a i?eiaiaiee ia iineaaeiai yoaia naa?oee iiiaiiia?aiaeiiai noiiaoi?a aianoi aeaooiia?aiaeiiai. I?iaaaeaii aiaeeoe/aneia e yenia?eiaioaeueiia enneaaeiaaiea oa?aeoa?enoee i?aaeeiaeaiiuo iaoiaeia i?iaeoe?iaaiey AO. Aiaeeoe/aneia enneaaeiaaiea iniiaaii ia ?an/aoa yiie?e/aneeo oi?ioe, au?aaeathueo caaeneiinoue oa?aeoa?enoee ono?ienoaa io aai ia?aiao?ia. Yenia?eiaioaeueiia enneaaeiaaiea iaania/eaaaony nicaeaieai iieiiooieoeeiiaeueiie i?ia?aiiiie iiaeaee, eioi?ay iiaeaa?aaaony i?ioeaaeo?ai neioaca, aa?eoeeaoeee e iiaeaee?iaaiey ia ?aciuo o?iaiyo aano?aeoeee. Nicaeaia VHDL-iiaeaeue ?ac?aaioaiiiai o?aoiia?aiaeiiai noiiaoi?a, e i?iaaaeaii aa ooieoeeiiaeueii-eiae/aneia iiaeaee?iaaiea. I?iecaaaeai eiae/aneee neioac iiaeaee ia aaca i?ia?aiie?oaiiai ianneaa eiae/aneeo aaioeeae (FPGA) e ooieoeeiiaeueii-eiae/aneia iiaeaee?iaaiey iiaeaee ia eiae/aneii o?iaia, i?eaaaeaiu aiaeeoe/aneea ioeaiee aaioeeueiie neiaeiinoe noiiaoi?ia. Iaeioi?ua iaoiaeu aeiaaaeaiu aei i?ia?aiiiie ?aaeecaoeee. I?ia?aiia auaeaeaiey neiiao?e/aneie /anoe ec aoeaaie ooieoeee aeaao iieioth eioi?iaoeeth ii iiaiaeo aa neiiao?ee. I?ia?aiia ?an/aoa oaaeeoe ?ac?yaeiuo eiaeaenia ii iaoiaeo i?iaeoe?iaaiey iiiaiiia?aiaeiuo noiiaoi?ia, iicaieyao iieo/aoue iiiaeanoai no?oeoo? noiiaoi?ia e eniieuecoaony aeey aiaeeca naienoa oeacaiiiai iaoiaea. ?acoeueoaou ?aaiou ?aeiiaiaeoaony eniieueciaaoue i?e ?ac?aaioea a?eoiaoeei-eiae/aneeo aeieia aeey caaea/ oeeo?iaie ia?aaioee neaiaeia, ia?aaioee ia?aeeaeueiuo iioieia aeaiiuo a ?aaeueiii ?aaeeia a?aiaie. Eeth/aaua neiaa: a?eoiaoe/aneea ono?ienoaa, neiiao?e/aneea aoeaaua ooieoeee, /anoe/iay e iieiay neiiao?ey, aaoiiaoecaoeey, i?iaeoe?iaaiea, auno?iaeaenoaea. PAGE 22

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