Десятая проблема Гильберта. Решение и применения в информатике, весна 2010: Лекция 3

Ãèïîòåçà Martin'a Davis'à (=DPRM-òåîðåìà)

Ãèïîòåçà M. Davis'à (DPRM-òåîðåìà). Êàæäîåïåðå÷èñëèìîå ìíîæåñòâî ÿâëÿåòñÿ äèîôàíòîâûì.



Ïåðå÷èñëèìûå ìíîæåñòâà

Îïðåäåëåíèå. Ìíîæåñòâî M, ñîñòîÿùåå èç n-îê íàòóðàëüíûõ÷èñåë íàçûâàåòñÿ ïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòüïðîãðàììó R, òàêóþ ÷òî

R-〈a1, . . . , an〉 -îñòàíîâêà, åñëè 〈a1, . . . , an〉 ∈M

âå÷íàÿ ðàáîòà â ïðîòèâíîì ñëó÷àå

Ýêâèâàëåíòíîå îïðåäåëåíèå. Ìíîæåñòâî M, ñîñòîÿùåå èçn-îê íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿ ïåðå÷èñëèìûì, åñëèìîæíî íàïèñàòü ïðîãðàììó P êîòîðàÿ (ðàáîòàÿ áåñêîíå÷íîäîëãî) áóäåò ïå÷àòàòü òîëüêî ýëåìåíòû ìíîæåñòâà M èíàïå÷àòàåò êàæäîå èç íèõ, áûòü ìîæåò, ìíîãî ðàç.












Îïðåäåëåíèå. Ìíîæåñòâî M íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R, òàêóþ ÷òî

R-a -îñòàíîâêà, åñëè a ∈M



Îïðåäåëåíèå. Ìíîæåñòâî M íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R, òàêóþ ÷òî



Ðåãèñòðîâûå ìàøèíû

Ðåãèñòðîâàÿ ìàøèíà èìååò êîíå÷íîå êîëè÷åñòâî ðåãèñòðîâ

R1, . . . ,Rn êàæäûé èç êîòîðûõ ìîæåò ñîäåðæàòü ïðîèçâîëüíîáîëüøîå íàòóðàëüíîå ÷èñëî. Ìàøèíà âûïîëíÿåò ïðîãðàììó

ñîñòîÿùóþ èç êîíå÷íîãî ÷èñëà èíñòðóêöèé ñíàáæåííûõìåòêàìè S1, . . . ,Sm. Êîãäà ìàøèíà âûïîëíÿåò èíñòðóêöèþ ñìåòêîé Sk , ìû ãîâîðèì, ÷òî ìàøèíà íàõîäèòñÿ â ñîñòîÿíèè Sk .

Èíñòðóêöèè áûâàþò òð¼õ òèïîâ:

I. Sk : R` + +;Si

II. Sk : R`−−; Si ;SjIII. Sk : STOP



R1, . . . ,Rn êàæäûé èç êîòîðûõ ìîæåò ñîäåðæàòü ïðîèçâîëüíîáîëüøîå íàòóðàëüíîå ÷èñëî.

Ìàøèíà âûïîëíÿåò ïðîãðàììó



I. Sk : R` + +;Si







I. Sk : R` + +;Si







I. Sk : R` + +;Si







I. Sk : R` + +;Si







I. Sk : R` + +;Si

II. Sk : R`−−; Si ;Sj

III. Sk : STOP






I. Sk : R` + +;Si



Îïðåäåëåíèå. Ìíîæåñòâî M íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R äëÿðåãèñòðîâîé ìàøèíû, òàêóþ ÷òî



Ïðè çàïóñêå ìàøèíû ÷èñëî a ïîìåùåíî â ðåãèñòð R1, âñåîñòàëüíûå ðåãèñòðû ñîäåðæàò íóëè.











ÏðèìåðS1: R1−−; S2; S8S2: R1−−; S3; S9S3: R2++; S4S4: R1−−; S5; S6S5: R1−−; S3; S8S6: R2−−; S7; S1S7: R1++; S6S8: R1++; S8S9: STOP

Óðàâíåíèÿ ñ ïàðàìåòðàìè

Ñåìåéñòâî äèîôàíòîâûõ óðàâíåíèé èìååò âèä

M(a1, . . . , an, x1, . . . , xm) = 0,

ãäå M � ìíîãî÷ëåí ñ öåëûìè êîýôôèöèåíòàìè, ïåðåìåííûåêîòîðãî ðàçäåëåíû íà äâå ãðóïïû:

I ïàðàìåòðû a1, . . . ,an;

I íåèçâåñòíûå x1, . . . ,xm.

Ðàññìîòðèì ìíîæåñòâî M òàêîå, ÷òî

〈a1, . . . , an〉 ∈M⇐⇒∃x1 . . . xm{M(a1, . . . , an, x1, . . . , xm) = 0}.

Ìíîæåñòâà, èìåþùèå òàêèå ïðåäñòàâëåíèÿ íàçûâàþòñÿäèîôàíòîâûìè.



M(a1, . . . , an, x1, . . . , xm) = 0,





〈a1, . . . , an〉 ∈M⇐⇒∃x1 . . . xm{M(a1, . . . , an, x1, . . . , xm) = 0}.




M(a1, . . . , an, x1, . . . , xm) = 0,





〈a1, . . . , an〉 ∈M⇐⇒∃x1 . . . xm{M(a1, . . . , an, x1, . . . , xm) = 0}.




M(a1, . . . , an, x1, . . . , xm) = 0,





〈a1, . . . , an〉 ∈M⇐⇒∃x1 . . . xm{M(a1, . . . , an, x1, . . . , xm) = 0}.


Äèîôàíòîâû ìàøèíû

Leonard Adleman è Kenneth Manders [1976] ââåëè ïîíÿòèåíåäåòåðìèíèðîâàííîé äèîôàíòîâîé ìàøèíû, NDDM .

NDDM

P(a, x1, . . . , xm)?= 0 �-

? --

âõîäa

óãàäàòüx1, . . . , xm

ÄÀ ÍÅÒ

ïðèíÿòü a îòâåðãíóòü

DPRM-òåîðåìà: NDDM èìååþò òàêóþ æå âû÷èñëèòåëüíóþñèëó êàê, íàïðèìåð, ìàøèíû Òüþðèíãà, òî åñòü ëþáîåìíîæåñòâî, ïðèíèìàåìîå íåêîòîðîé ìàøèíîé Òüþðèíãà,ïðèíèìàåòñÿ íåêîòîðîé NDDM, è, î÷åâèäíî, íàîáîðîò.



NDDM

P(a, x1, . . . , xm)?= 0 �-

? --

âõîäa


ÄÀ ÍÅÒ





NDDM

P(a, x1, . . . , xm)?= 0 �-

? --

âõîäa


ÄÀ ÍÅÒ



Äèîôàíòîâà ñëîæíîñòü

NDDM

P(a, x1, . . . , xm)?= 0 �-

? --

âõîäa


ÄÀ ÍÅÒ


SIZE(a)=ìèíèìàëüíî âîçìîæíîå çíà÷åíèå |x1|+ · · ·+ |xm|, ãäå|x | îáîçíà÷àåò äëèíó äâîè÷íîé çàïèñè x .


NDDM

P(a, x1, . . . , xm)?= 0 �-

? --

âõîäa


ÄÀ ÍÅÒ


SIZE(a)=ìèíèìàëüíî âîçìîæíîå çíà÷åíèå |x1|+ · · ·+ |xm|, ãäå|x | îáîçíà÷àåò äëèíó äâîè÷íîé çàïèñè x .


Leonard Adleman è Kenneth Manders [1975] ââåëè âðàññìîòðåíèå êëàññ D ñîñòîÿùèé èç ìíîæåñòâ M èìåþùèõïðåäñòàâëåíèÿ âèäà

a ∈M ⇐⇒⇐⇒ ∃x1 . . . xm

[P(a, x1, . . . , xm) = 0& |x1|+ · · ·+ |xm| ≤ |a|k

].

Îòêðûòàÿ ïðîáëåìà. D?= NP.


Leonard Adleman è Kenneth Manders [1975] ââåëè âðàññìîòðåíèå êëàññ D ñîñòîÿùèé èç ìíîæåñòâ M èìåþùèõïðåäñòàâëåíèÿ âèäà

a ∈M ⇐⇒⇐⇒ ∃x1 . . . xm

[P(a, x1, . . . , xm) = 0& |x1|+ · · ·+ |xm| ≤ |a|k

].

Îòêðûòàÿ ïðîáëåìà. D?= NP.






Ïðè çàïóñêå ìàøèíû ÷èñëî a ïîìåùåíî â ðåãèñòð R1, âñåîñòàëüíûå ðåãèñòðû ñîäåðæàò íóëè.Áåç îãðàíè÷åíèÿ îáùíîñòè ìû ïðåäïîëàãàåì, ÷òî èìååòñÿåäèíñòâåííàÿ êîìàíäà STOP � êîìàíäà ñ ìåòêîé Sm, à âìîìåíò îñòàíîâêè âñå ðåãèñòðû ïóñòû.













Áåç îãðàíè÷åíèÿ îáùíîñòè ìû ïðåäïîëàãàåì, ÷òî èìååòñÿåäèíñòâåííàÿ êîìàíäà STOP � êîìàíäà ñ ìåòêîé Sm, à âìîìåíò îñòàíîâêè âñå ðåãèñòðû ïóñòû.






Ïðè çàïóñêå ìàøèíû ÷èñëî a ïîìåùåíî â ðåãèñòð R1, âñåîñòàëüíûå ðåãèñòðû ñîäåðæàò íóëè.Áåç îãðàíè÷åíèÿ îáùíîñòè ìû ïðåäïîëàãàåì, ÷òî èìååòñÿåäèíñòâåííàÿ êîìàíäà STOP � êîìàíäà ñ ìåòêîé Sm

, à âìîìåíò îñòàíîâêè âñå ðåãèñòðû ïóñòû.







Ïðîòîêîë

q . . . t + 1 t . . . 0

S1 s1,q . . . s1,t+1 s1,t . . . s1,0...

......

......

......

Sk sk,q . . . sk,t+1 sk,t . . . sk,0...

......

......

......

Sm sm,q . . . sm,t+1 sm,t . . . sm,0

sk,t =

{1, åñëè íà øàãå t ìàøèíà áûëà â ñîñòîÿíèè k

0 â ïðîòèâíîì ñëó÷àå

Ïðîòîêîë

q . . . t + 1 t . . . 0...

......

......

......

Sk sk,q . . . sk,t+1 sk,t . . . sk,0...

......

......

......

R1 r1,q . . . r1,t+1 r1,t . . . r1,0...

......

......

......

R` r`,q . . . r`,t+1 r`,t . . . r`,0...

......

......

......

Rn rn,q . . . rn,t+1 rn,t . . . rn,0

r`,t � ýòî ñîäåðæèìîå `-ãî ðåãèñòðà íà øàãå t

Ïðîòîêîë

q . . . t + 1 t . . . 0...

......

......

......

Sk sk,q . . . sk,t+1 sk,t . . . sk,0...

......

......

......

......

......

......

...R` r`,q . . . r`,t+1 r`,t . . . r`,0...

......

......

......

Z1 z1,q . . . z1,t+1 z1,t . . . z1,0...

......

......

......

Z` z`,q . . . z`,t+1 z`,t . . . z`,0...

......

......

......

Zn zn,q . . . zn,t+1 zn,t . . . zn,0

z`,t =

{1, åñëè r`,t > 00 â ïðîòèâíîì ñëó÷àå

ÏðèìåðS1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP

r1,t+1 = r1,t + s7,t + s8,t − z1,ts1,t − z1,ts2,t − z1,ts4,t − z1,ts5,t

r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t

s3,t+1 = z1,ts2,t + z1,ts5,t s4,t+1 = s3,t s5,t+1 = z1,ts4,t

s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0


r1,t+1 =

r1,t + s7,t + s8,t − z1,ts1,t − z1,ts2,t − z1,ts4,t − z1,ts5,t

r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0


r1,t+1 = r1,t

+ s7,t + s8,t − z1,ts1,t − z1,ts2,t − z1,ts4,t − z1,ts5,t

r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0


r1,t+1 = r1,t + s7,t

+ s8,t − z1,ts1,t − z1,ts2,t − z1,ts4,t − z1,ts5,t

r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0


r1,t+1 = r1,t + s7,t + s8,t

− z1,ts1,t − z1,ts2,t − z1,ts4,t − z1,ts5,t

r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0


r1,t+1 = r1,t + s7,t + s8,t − z1,ts1,t

− z1,ts2,t − z1,ts4,t − z1,ts5,t

r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0


r1,t+1 = r1,t + s7,t + s8,t − z1,ts1,t − z1,ts2,t

− z1,ts4,t − z1,ts5,t

r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0


r1,t+1 = r1,t + s7,t + s8,t − z1,ts1,t − z1,ts2,t − z1,ts4,t

− z1,ts5,t

r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 =

r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t

+ s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t

− z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 =

(1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t

s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 =

z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t

s3,t+1 =

z1,ts2,t + z1,ts5,t s4,t+1 = s3,t s5,t+1 = z1,ts4,t

s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t

s3,t+1 = z1,ts2,t

+ z1,ts5,t s4,t+1 = s3,t s5,t+1 = z1,ts4,t

s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t

s3,t+1 = z1,ts2,t + z1,ts5,t

s4,t+1 = s3,t s5,t+1 = z1,ts4,t

s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t

s3,t+1 = z1,ts2,t + z1,ts5,t s4,t+1 =

s3,t s5,t+1 = z1,ts4,t

s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t

s3,t+1 = z1,ts2,t + z1,ts5,t s4,t+1 = s3,t

s5,t+1 = z1,ts4,t

s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t

s3,t+1 = z1,ts2,t + z1,ts5,t s4,t+1 = s3,t s5,t+1 =

z1,ts4,t

s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 =

(1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t

+ s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t

s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 =

z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 =

(1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t

+ s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t

s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 =

(1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 =

1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1

s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 =

a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a

r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1

s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0



r2,t+1 = r2,t + s3,t − z2,ts6,t

s1,t+1 = (1− z2,t)s6,t s2,t+1 = z1,ts1,t


s6,t+1 = (1− z4,t)s4,t + s7,t s7,t+1 = z2,ts6,t

s8,t+1 = (1− z1,t)s1,t + s8,t s9,t+1 = (1− z1,t)s2,t

s1,0 = 1 s2,0 = · · · = sm,0 = 0

r1,0 = a r2,0 = · · · = rn,0 = 0

sm,q = 1 s1,q = · · · = sm−1,q = 0

r1,q = · · · = rn,q = 0

Íîâûå çíà÷åíèÿ ðåãèñòðîâ

r`,t+1 = r`,t +∑+

` sk,t −∑−

` z`,tsk,t

ãäå∑+

` -ñóììèðîâàíèå âåäåòñÿ ïî âñåì èíñòðóêöèÿì âèäà

Sk : R` + +; Si ,

à∑−

` -ñóììèðîâàíèå � ïî âñåì èíñòðóêöèÿì âèäà

Sk : R`−−; Si ; Sj .

Íîâûå ñîñòîÿíèÿ

sd ,t+1 =∑+

d sk,t +∑−

d z`,tsk,t +∑

0

d (1− z`,t)sk,t

ãäå∑+

d -ñóììèðîâàíèå âåäåòñÿ ïî âñåì èíñòðóêöèÿì âèäà

Sk : R` + +; Sd ,∑−d -ñóììèðîâàíèå âåäåòñÿ ïî âñåì èíñòðóêöèÿì âèäà

Sk : R`−−; Sd ; Sj ,

à∑

0

d -ñóììèðîâàíèå � ïî âñåì èíñòðóêöèÿì âèäà

Sk : R`−−; Si ; Sd .

Ïðîòîêîë

q . . . t + 1 t . . . 0

......

......

......

...Sk sk,q . . . sk,t+1 sk,t . . . sk,0

= sk =∑q

t=0sk,tb

t

......

......

......

...

......

......

......

...R` r`,q . . . r`,t+1 r`,t . . . r`,0

= r` =∑q

t=0r`,tb

t

......

......

......

...

......

......

......

...Z` z`,q . . . z`,t+1 z`,t . . . z`,0

= z` =∑q

t=0z`,tb

t

......

......

......

...

Ïðîòîêîë

q . . . t + 1 t . . . 0

......

......

......

...Sk sk,q . . . sk,t+1 sk,t . . . sk,0

=

sk =∑q

t=0sk,tb

t

......

......

......

...

......

......

......

...R` r`,q . . . r`,t+1 r`,t . . . r`,0

= r` =∑q

t=0r`,tb

t

......

......

......

...

......

......

......

...Z` z`,q . . . z`,t+1 z`,t . . . z`,0

= z` =∑q

t=0z`,tb

t

......

......

......

...

Ïðîòîêîë

q . . . t + 1 t . . . 0

......

......

......

...Sk sk,q . . . sk,t+1 sk,t . . . sk,0 = sk =

∑qt=0

sk,tbt

......

......

......

...

......

......

......

...R` r`,q . . . r`,t+1 r`,t . . . r`,0

= r` =∑q

t=0r`,tb

t

......

......

......

...

......

......

......

...Z` z`,q . . . z`,t+1 z`,t . . . z`,0

= z` =∑q

t=0z`,tb

t

......

......

......

...

Ïðîòîêîë

q . . . t + 1 t . . . 0

......

......

......


∑qt=0

sk,tbt

......

......

......

...

......

......

......

...R` r`,q . . . r`,t+1 r`,t . . . r`,0 = r` =

∑qt=0

r`,tbt

......

......

......

...

......

......

......

...Z` z`,q . . . z`,t+1 z`,t . . . z`,0

= z` =∑q

t=0z`,tb

t

......

......

......

...

Ïðîòîêîë

q . . . t + 1 t . . . 0

......

......

......


∑qt=0

sk,tbt

......

......

......

...

......

......

......

...R` r`,q . . . r`,t+1 r`,t . . . r`,0 = r` =

∑qt=0

r`,tbt

......

......

......

...

......

......

......

...Z` z`,q . . . z`,t+1 z`,t . . . z`,0 = z` =

∑qt=0

z`,tbt

......

......

......

...


sk =

q∑t=0

sk,tbt r` =

q∑t=0

r`,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

r`,t+1

bt+1

=

q−1∑t=0

(

r`,t

bt+1

+∑+

` sk,t

bt+1

−∑−

` z`,tsk,t

bt+1)

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

r1 − a = br1 + b∑+

` sk − b∑−

` (z` ∧ sk)r` = br` + b

∑+` sk − b

∑−` (z` ∧ sk), ` = 2, . . . , n


sk =

q∑t=0

sk,tbt r` =

q∑t=0

r`,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

r`,t+1

bt+1

=

q−1∑t=0

(

r`,t

bt+1

+∑+

` sk,t

bt+1

−∑−

` z`,tsk,t

bt+1)

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

r1 − a = br1 + b∑+

` sk − b∑−

` (z` ∧ sk)r` = br` + b

∑+` sk − b

∑−` (z` ∧ sk), ` = 2, . . . , n


sk =

q∑t=0

sk,tbt r` =

q∑t=0

r`,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

r`,t+1

bt+1

=

q−1∑t=0

(

r`,t

bt+1

+∑+

` sk,t

bt+1

−∑−

` z`,tsk,t

bt+1)

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

r1 − a = br1 + b∑+

` sk − b∑−

` (z` ∧ sk)r` = br` + b

∑+` sk − b

∑−` (z` ∧ sk), ` = 2, . . . , n


sk =

q∑t=0

sk,tbt r` =

q∑t=0

r`,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

r`,t+1

bt+1

=

q−1∑t=0

(

r`,t

bt+1

+∑+

` sk,t

bt+1

−∑−

` z`,tsk,t

bt+1)

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

r1 − a = br1 + b∑+

` sk − b∑−

` (z` ∧ sk)r` = br` + b

∑+` sk − b

∑−` (z` ∧ sk), ` = 2, . . . , n


sk =

q∑t=0

sk,tbt r` =

q∑t=0

r`,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

r`,t+1bt+1 =

q−1∑t=0

(

r`,tbt+1 +

∑+` sk,tb

t+1 −∑−

` z`,tsk,tbt+1

)

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

r1 − a = br1 + b∑+

` sk − b∑−

` (z` ∧ sk)r` = br` + b

∑+` sk − b

∑−` (z` ∧ sk), ` = 2, . . . , n


sk =

q∑t=0

sk,tbt r` =

q∑t=0

r`,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

r`,t+1bt+1 =

q−1∑t=0

(r`,tb

t+1 +∑+

` sk,tbt+1 −

∑−` z`,tsk,tb

t+1)

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

r1 − a = br1 + b∑+

` sk − b∑−

` (z` ∧ sk)r` = br` + b

∑+` sk − b

∑−` (z` ∧ sk), ` = 2, . . . , n


sk =

q∑t=0

sk,tbt r` =

q∑t=0

r`,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

r`,t+1bt+1 =

q−1∑t=0

(r`,tb

t+1 +∑+

` sk,tbt+1 −

∑−` z`,tsk,tb

t+1)

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

r1 − a = br1 + b∑+

` sk − b∑−

` (z` ∧ sk)r` = br` + b

∑+` sk − b

∑−` (z` ∧ sk), ` = 2, . . . , n


sk =

q∑t=0

sk,tbt r` =

q∑t=0

r`,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

r`,t+1bt+1 =

q−1∑t=0

(r`,tb

t+1 +∑+

` sk,tbt+1 −

∑−` z`,tsk,tb

t+1)

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

r1 − a = br1 + b∑+

` sk − b∑−

` (z` ∧ sk)r` = br` + b

∑+` sk − b

∑−` (z` ∧ sk), ` = 2, . . . , n


sk =

q∑t=0

sk,tbt r` =

q∑t=0

r`,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

r`,t+1bt+1 =

q−1∑t=0

(r`,tb

t+1 +∑+

` sk,tbt+1 −

∑−` z`,tsk,tb

t+1)

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

r1 − a = br1 + b∑+

` sk − b∑−

` (z` ∧ sk)r` = br` + b

∑+` sk − b

∑−` (z` ∧ sk), ` = 2, . . . , n


sk =

q∑t=0

sk,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

sd ,t+1

bt+1

=

=

q−1∑t=0

(

∑+d sk,t

bt+1

+∑−

d z`,tsk,t

bt+1

+∑

0

d (1− z`,t)sk,t

bt+1

)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =

q−1∑t=0

1 · bt+1 =bq − 1

b − 1


sk =

q∑t=0

sk,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

sd ,t+1

bt+1

=

=

q−1∑t=0

(

∑+d sk,t

bt+1

+∑−

d z`,tsk,t

bt+1

+∑

0

d (1− z`,t)sk,t

bt+1

)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =

q−1∑t=0

1 · bt+1 =bq − 1

b − 1


sk =

q∑t=0

sk,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

sd ,t+1

bt+1

=

=

q−1∑t=0

(

∑+d sk,t

bt+1

+∑−

d z`,tsk,t

bt+1

+∑

0

d (1− z`,t)sk,t

bt+1

)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =

q−1∑t=0

1 · bt+1 =bq − 1

b − 1


sk =

q∑t=0

sk,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

sd ,t+1

bt+1

=

=

q−1∑t=0

(

∑+d sk,t

bt+1

+∑−

d z`,tsk,t

bt+1

+∑

0

d (1− z`,t)sk,t

bt+1

)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =

q−1∑t=0

1 · bt+1 =bq − 1

b − 1


sk =

q∑t=0

sk,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

sd ,t+1bt+1 =

=

q−1∑t=0

(

∑+d sk,tb

t+1 +∑−

d z`,tsk,tbt+1 +

∑0

d (1− z`,t)sk,tbt+1

)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =

q−1∑t=0

1 · bt+1 =bq − 1

b − 1


sk =

q∑t=0

sk,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

sd ,t+1bt+1 =

=

q−1∑t=0

(∑+d sk,tb

t+1 +∑−

d z`,tsk,tbt+1 +

∑0


)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =

q−1∑t=0

1 · bt+1 =bq − 1

b − 1


sk =

q∑t=0

sk,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

sd ,t+1bt+1 =

=

q−1∑t=0

(∑+d sk,tb

t+1 +∑−

d z`,tsk,tbt+1 +

∑0


)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =

q−1∑t=0

1 · bt+1 =bq − 1

b − 1


sk =

q∑t=0

sk,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

sd ,t+1bt+1 =

=

q−1∑t=0

(∑+d sk,tb

t+1 +∑−

d z`,tsk,tbt+1 +

∑0


)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =

q−1∑t=0

1 · bt+1 =bq − 1

b − 1


sk =

q∑t=0

sk,tbt z` =

q∑t=0

z`,tbt

q−1∑t=0

sd ,t+1bt+1 =

=

q−1∑t=0

(∑+d sk,tb

t+1 +∑−

d z`,tsk,tbt+1 +

∑0


)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =

q−1∑t=0

1 · bt+1 =bq − 1

b − 1

Èíäèêàòîðû íóëÿ

z`,t =

{0, åñëè r`,t = 01 â ïðîòèâíîì ñëó÷àå

b = 2c+1 r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =

{01 . . . 1, åñëè r`,t = 01 ∗ · · · ∗ â ïðîòèâíîì ñëó÷àå

2c ∧ (2c − 1 + r`,t) = 2cz`,t

q∑t=0

(2c ∧ (2c − 1 + r`,t))

bt

=

q∑t=0

2cz`,t

bt

2c f ∧ ((2c − 1)f + r`) = 2cz`

f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1


z`,t =


b = 2c+1 r`,t ≤ 2c − 1

= 01 . . . 1

2c − 1 + r`,t =


2c ∧ (2c − 1 + r`,t) = 2cz`,t

q∑t=0

(2c ∧ (2c − 1 + r`,t))

bt

=

q∑t=0

2cz`,t

bt

2c f ∧ ((2c − 1)f + r`) = 2cz`

f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1


z`,t =


b = 2c+1 r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =


2c ∧ (2c − 1 + r`,t) = 2cz`,t

q∑t=0

(2c ∧ (2c − 1 + r`,t))

bt

=

q∑t=0

2cz`,t

bt

2c f ∧ ((2c − 1)f + r`) = 2cz`

f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1


z`,t =


b = 2c+1 r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =


2c ∧ (2c − 1 + r`,t) = 2cz`,t

q∑t=0

(2c ∧ (2c − 1 + r`,t))

bt

=

q∑t=0

2cz`,t

bt

2c f ∧ ((2c − 1)f + r`) = 2cz`

f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1


z`,t =


b = 2c+1 r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =


2c ∧ (2c − 1 + r`,t) = 2cz`,t

q∑t=0

(2c ∧ (2c − 1 + r`,t))bt =

q∑t=0

2cz`,tbt

2c f ∧ ((2c − 1)f + r`) = 2cz`

f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1


z`,t =


b = 2c+1 r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =


2c ∧ (2c − 1 + r`,t) = 2cz`,t

q∑t=0

(2c ∧ (2c − 1 + r`,t))bt =

q∑t=0

2cz`,tbt

2c f ∧ ((2c − 1)f + r`) = 2cz` f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1


z`,t =


b = 2c+1 r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =


2c ∧ (2c − 1 + r`,t) = 2cz`,t

q∑t=0

(2c ∧ (2c − 1 + r`,t))bt =

q∑t=0

2cz`,tbt

2c f ∧ ((2c − 1)f + r`) = 2cz` f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1


z`,t =


b = 2c+1 r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =


2c ∧ (2c − 1 + r`,t) = 2cz`,t

q∑t=0

(2c ∧ (2c − 1 + r`,t))bt =

q∑t=0

2cz`,tbt

2c f ∧ ((2c − 1)f + r`) = 2cz` f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1

Âûáîð c

r`,t < b

= 2c+1 r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =


q∑t=0

(

2c ∧ r`,t

)bt

= 0

2c f ∧ r` = 0

f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1

Âûáîð c

r`,t < b = 2c+1

r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =


q∑t=0

(

2c ∧ r`,t

)bt

= 0

2c f ∧ r` = 0

f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1

Âûáîð c

r`,t < b = 2c+1 r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =


q∑t=0

(

2c ∧ r`,t

)bt

= 0

2c f ∧ r` = 0

f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1

Âûáîð c

r`,t < b = 2c+1 r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =


q∑t=0

(

2c ∧ r`,t

)bt

= 0

2c f ∧ r` = 0

f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1

Âûáîð c

r`,t < b = 2c+1 r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =


q∑t=0

(

2c ∧ r`,t

)bt

= 0

2c f ∧ r` = 0

f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1

Âûáîð c

r`,t < b = 2c+1 r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =


q∑t=0

(

2c ∧ r`,t

)bt

= 0

2c f ∧ r` = 0

f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1

Âûáîð c

r`,t < b = 2c+1 r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =


q∑t=0

(

2c ∧ r`,t

)bt

= 0

2c f ∧ r` = 0

f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1

Âûáîð c

r`,t < b = 2c+1 r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =


q∑t=0

(2c ∧ r`,t)bt = 0

2c f ∧ r` = 0

f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1

Âûáîð c

r`,t < b = 2c+1 r`,t ≤ 2c − 1 = 01 . . . 1

2c − 1 + r`,t =


q∑t=0

(2c ∧ r`,t)bt = 0

2c f ∧ r` = 0 f =

q∑t=0

1 · bt+1 =bq+1 − 1

b − 1

Âñå óñëîâèÿ

b = 2c+1

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =bq − 1

b − 1

2c f ∧ ((2c − 1)f + r`) = 2cz` f =bq+1 − 1

b − 1

2c f ∧ r` = 0 sm = bq


b = 2c+1

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =bq − 1

b − 1

2c f ∧ ((2c − 1)f + r`) = 2cz` f =bq+1 − 1

b − 1

2c f ∧ r` = 0 sm = bq


b = 2c+1

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =bq − 1

b − 1

2c f ∧ ((2c − 1)f + r`) = 2cz` f =bq+1 − 1

b − 1

2c f ∧ r` = 0 sm = bq


b = 2c+1

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =bq − 1

b − 1

2c f ∧ ((2c − 1)f + r`) = 2cz` f =bq+1 − 1

b − 1

2c f ∧ r` = 0 sm = bq


b = 2c+1

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =bq − 1

b − 1

2c f ∧ ((2c − 1)f + r`) = 2cz` f =bq+1 − 1

b − 1

2c f ∧ r` = 0 sm = bq


b = 2c+1

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =bq − 1

b − 1

2c f ∧ ((2c − 1)f + r`) = 2cz` f =bq+1 − 1

b − 1

2c f ∧ r` = 0 sm = bq


b = 2c+1

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =bq − 1

b − 1

2c f ∧ ((2c − 1)f + r`) = 2cz` f =bq+1 − 1

b − 1

2c f ∧ r` = 0

sm = bq


b = 2c+1

r` − r`,0 = br` + b∑+

` sk − b∑−

` (z` ∧ sk)

sd − sd ,0 = b∑+

d sk + b∑+

d (z` ∧ sk) + b∑

0

d ((e − z`) ∧ sk)

e =bq − 1

b − 1

2c f ∧ ((2c − 1)f + r`) = 2cz` f =bq+1 − 1

b − 1

2c f ∧ r` = 0 sm = bq

Ïðîòîêîë

q . . . t + 1 t . . . 0

......

......

......

...

Sk sk,q . . . sk,t+1 sk,t . . . sk,0 = sk =∑Q

t=0sk,tb

t

......

......

......

...

......

......

......

...

R` r`,q . . . r`,t+1 r`,t . . . r`,0 = r` =∑Q

t=0r`,tb

t

......

......

......

...

......

......

......

...

Z` z`,q . . . z`,t+1 z`,t . . . z`,0 = z` =∑Q

t=0z`,tb

t

......

......

......

...

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Десятая проблема Гильберта. Решение и применения в информатике, весна 2010: Лекция 3