Юрий Владимирович Матиясевич. Десятая проблема Гильберта. Решение и применения в информатике. Лекция 03

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

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

Ïåðå÷èñëèìûå ìíîæåñòâà


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

a1 , . . . , an -

îñòàíîâêà, åñëè a1 , . . . , an ∈ M
R
-
âå÷íàÿ ðàáîòà â ïðîòèâíîì ñëó÷àå


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

a1 , . . . , an -

îñòàíîâêà, åñëè a1 , . . . , an ∈ M
R
-

Ýêâèâàëåíòíîå îïðåäåëåíèå. Ìíîæåñòâî M, ñîñòîÿùåå èç
n-îê íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿ ïåðå÷èñëèìûì, åñëè
ìîæíî íàïèñàòü ïðîãðàììó P êîòîðàÿ (ðàáîòàÿ áåñêîíå÷íî
äîëãî) áóäåò ïå÷àòàòü òîëüêî ýëåìåíòû ìíîæåñòâà M è
íàïå÷àòàåò êàæäîå èç íèõ, áûòü ìîæåò, ìíîãî ðàç.


Îïðåäåëåíèå. Ìíîæåñòâî M íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿ
ïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R, òàêóþ ÷òî
a

îñòàíîâêà, åñëè a∈M
R
- -

Ðåãèñòðîâûå ìàøèíû


Ðåãèñòðîâàÿ ìàøèíà èìååò êîíå÷íîå êîëè÷åñòâî ðåãèñòðîâ
n
R1, . . . ,R êàæäûé èç êîòîðûõ ìîæåò ñîäåðæàòü ïðîèçâîëüíî
áîëüøîå íàòóðàëüíîå ÷èñëî.


n
áîëüøîå íàòóðàëüíîå ÷èñëî. Ìàøèíà âûïîëíÿåò ïðîãðàììó
ñîñòîÿùóþ èç êîíå÷íîãî ÷èñëà èíñòðóêöèé ñíàáæåííûõ
ìåòêàìè m
S1, . . . ,S . Êîãäà ìàøèíà âûïîëíÿåò èíñòðóêöèþ ñ
k
ìåòêîé S , ìû ãîâîðèì, ÷òî ìàøèíà íàõîäèòñÿ â ñîñòîÿíèè k S .


n
ìåòêàìè m
k

Èíñòðóêöèè áûâàþò òð¼õ òèïîâ:


n
ìåòêàìè m
k

k
I. S : R + +;S i


n
ìåòêàìè m
k

k
I. S : R + +;S i
II. Sk : R − −; S ;Si j


n
ìåòêàìè m
k

k
I. S : R + +;S i
II. Sk : R − −; S ;Si j
III. Sk : STOP


ïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R äëÿ
ðåãèñòðîâîé ìàøèíû, òàêóþ ÷òî

a

R
- -


ïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R äëÿ

a

R
- -

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

Ïðèìåð

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

Óðàâíåíèÿ ñ ïàðàìåòðàìè

Ñåìåéñòâî äèîôàíòîâûõ óðàâíåíèé èìååò âèä
M (a1, . . . , an , x1, . . . , xm ) = 0,
ãäå M ìíîãî÷ëåí ñ öåëûìè êîýôôèöèåíòàìè, ïåðåìåííûå
êîòîðãî ðàçäåëåíû íà äâå ãðóïïû:


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


M (a1, . . . , an , x1, . . . , xm ) = 0,
íåèçâåñòíûå x1, . . . ,xm .
Ðàññìîòðèì ìíîæåñòâî M òàêîå, ÷òî

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


M (a1, . . . , an , x1, . . . , xm ) = 0,
íåèçâåñòíûå x1, . . . ,xm .
Ðàññìîòðèì ìíîæåñòâî M òàêîå, ÷òî

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

Ìíîæåñòâà, èìåþùèå òàêèå ïðåäñòàâëåíèÿ íàçûâàþòñÿ
äèîôàíòîâûìè.

Äèîôàíòîâû ìàøèíû

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



NDDM
âõîä - óãàäàòü
a P (a, x1, . . . , xm ) = 0
?
x1, . . . , xm
ÄÀ ÍÅÒ

a
-
ïðèíÿòü îòâåðãíóòü
?



NDDM
a P (a, x1, . . . , xm ) = 0
?
x1, . . . , xm
ÄÀ ÍÅÒ

a
-
?

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

Äèîôàíòîâà ñëîæíîñòü

NDDM
a P (a, x1, . . . , xm ) = 0
?
x1, . . . , xm
ÄÀ ÍÅÒ

a
-
?


NDDM
a P (a, x1, . . . , xm ) = 0
?
x1, . . . , xm
ÄÀ ÍÅÒ

a
-
?

x
SIZE(a)=ìèíèìàëüíî âîçìîæíîå çíà÷åíèå | 1 | + · · · + | m |, ãäå x
x
| | îáîçíà÷àåò äëèíó äâîè÷íîé çàïèñè . x


Leonard Adleman è Kenneth Manders [1975] ââåëè â
ðàññìîòðåíèå êëàññ D ñîñòîÿùèé èç ìíîæåñòâ M èìåþùèõ
ïðåäñòàâëåíèÿ âèäà

a ∈ M ⇐⇒
⇐⇒ ∃x1 . . . xm P (a, x1 , . . . , xm ) = 0 |x1 | + · · · + |xm | ≤ |a|k .


Leonard Adleman è Kenneth Manders [1975] ââåëè â
ðàññìîòðåíèå êëàññ D ñîñòîÿùèé èç ìíîæåñòâ M èìåþùèõ
ïðåäñòàâëåíèÿ âèäà

a ∈ M ⇐⇒
⇐⇒ ∃x1 . . . xm P (a, x1 , . . . , xm ) = 0 |x1 | + · · · + |xm | ≤ |a|k .

?
Îòêðûòàÿ ïðîáëåìà. D = NP.


ïåðå÷èñëèìûì , åñëè ìîæíî íàïèñàòü ïðîãðàììó R äëÿ

a

R
- -



a

R
- -

a



a

R
- -

a
Áåç îãðàíè÷åíèÿ îáùíîñòè ìû ïðåäïîëàãàåì, ÷òî èìååòñÿ
åäèíñòâåííàÿ êîìàíäà STOP êîìàíäà ñ ìåòêîé S m



a

R
- -

a
Áåç îãðàíè÷åíèÿ îáùíîñòè ìû ïðåäïîëàãàåì, ÷òî èìååòñÿ
åäèíñòâåííàÿ êîìàíäà STOP êîìàíäà ñ ìåòêîé S , à âm
ìîìåíò 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
1, åñëè íà øàãå t ìàøèíà áûëà â ñîñòîÿíèè k
sk ,t = 0 â ïðîòèâíîì ñëó÷àå

Ïðîòîêîë

q ... t +1 t ... 0
.
. .
. .
. .
. .
. .
. .
.
. . . . . . .
S k 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 = 0, âåñëè r ,t 0ñëó÷àå
1
ïðîòèâíîì

Ïðèìåð

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

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r1,t+1 =

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r1,t+1 = r1,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r1,t+1 = r1,t + s7,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r1,t+1 = r1,t + s7,t + s8,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r1,t+1 = r1,t + s7,t + s8,t − z1,t s1,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r1,t+1 = r1,t + s7,t + s8,t − z1,t s1,t − z1,t s2,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r1,t+1 = r1,t + s7,t + s8,t − z1,t s1,t − z1,t s2,t − z1,t s4,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r1,t+1 = r1,t + s7,t + s8,t − z1,t s1,t − z1,t s2,t − z1,t s4,t − z1,t s5,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 =

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 =

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 =

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s3,t+1 =

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s3,t+1 = z1,t s2,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s3,t+1 = z1,t s2,t + z1,t s5,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s3,t+1 = z1,t s2,t + z1,t s5,t s4,t+1 =

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s3,t+1 = z1,t s2,t + z1,t s5,t s4,t+1 = s3,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s3,t+1 = z1,t s2,t + z1,t s5,t s4,t+1 = s3,t s5,t+1 =

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s3,t+1 = z1,t s2,t + z1,t s5,t s4,t+1 = s3,t s5,t+1 = z1,t s4,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 =

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t + s7,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t + s7,t s7,t+1 =

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t + s7,t s7,t+1 = z2,t s6,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t + s7,t s7,t+1 = z2,t s6,t
s8,t+1 =

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t + s7,t s7,t+1 = z2,t s6,t
s8,t+1 = (1 − z1,t )s1,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t + s7,t s7,t+1 = z2,t s6,t
s8,t+1 = (1 − z1,t )s1,t + s8,t

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t + s7,t s7,t+1 = z2,t s6,t
s8,t+1 = (1 − z1,t )s1,t + s8,t s9,t+1 =

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t + s7,t s7,t+1 = z2,t s6,t
s8,t+1 = (1 − z1,t )s1,t + s8,t s9,t+1 = (1 − z1,t )s2,t
s1,0 =

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t + s7,t s7,t+1 = z2,t s6,t
s8,t+1 = (1 − z1,t )s1,t + s8,t s9,t+1 = (1 − z1,t )s2,t
s1,0 = 1

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t + s7,t s7,t+1 = z2,t s6,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

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t + s7,t s7,t+1 = z2,t s6,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 =

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t + s7,t s7,t+1 = z2,t s6,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

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t + s7,t s7,t+1 = z2,t s6,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

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t + s7,t s7,t+1 = z2,t s6,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

Ïðèìåð

S1:R1−−; S2; S8 S4:R1−−; S5; S6 S7:R1++; S6
S2:R1−−; S3; S9 S5:R1−−; S3; S8 S8:R1++; S8
S3:R2++; S4 S6:R2−−; S7; S1 S9:STOP
r2,t+1 = r2,t + s3,t − z2,t s6,t
s1,t+1 = (1 − z2,t )s6,t s2,t+1 = z1,t s1,t
s6,t+1 = (1 − z4,t )s4,t + s7,t s7,t+1 = z2,t s6,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 ,t sk ,t
+
ãäå -ñóììèðîâàíèå âåäåòñÿ ïî âñåì èíñòðóêöèÿì âèäà

Sk : R + +; S , i
−
à -ñóììèðîâàíèå ïî âñåì èíñòðóêöèÿì âèäà

Sk : R i j
− −; S ; S .

Íîâûå ñîñòîÿíèÿ

sd ,t+1 = d sk ,t +
+
d z ,t sk ,t +
−
d (1 − z ,t )sk ,t
0

+
ãäå d -ñóììèðîâàíèå âåäåòñÿ ïî âñåì èíñòðóêöèÿì âèäà
Sk : R + +; S ,d
−
d -ñóììèðîâàíèå âåäåòñÿ ïî âñåì èíñòðóêöèÿì âèäà
Sk : R d j
− −; S ; S ,
0
à d -ñóììèðîâàíèå ïî âñåì èíñòðóêöèÿì âèäà
Sk : R i d
− −; S ; S .

Ïðîòîêîë

q ... t +1 t ... 0
.
. .
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.
. . . . . . .
Sk sk ,q ... sk ,t+1 sk ,t ... sk ,0
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R r ,q ... r ,t + 1 r , t ... r ,0
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Z z ,q ... z ,t + 1 z , t ... z ,0
.
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.
. . . . . . .

Ïðîòîêîë

q ... t +1 t ... 0
.
. .
. .
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. .
. .
. .
.
. . . . . . .
Sk sk ,q ... sk ,t+1 sk ,t ... sk ,0 sk = q s bt
t = 0 k ,t
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R r ,q ... r ,t + 1 r , t ... r ,0
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Z z ,q ... z ,t + 1 z , t ... z ,0
.
. .
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.
. . . . . . .

Ïðîòîêîë

q ... t +1 t ... 0
.
. .
. .
. .
. .
. .
. .
.
. . . . . . .
Sk sk ,q ... sk ,t+1 sk ,t ... sk ,0 s
= k= q s bt
t = 0 k ,t
.
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R r ,q ... r ,t + 1 r , t ... r ,0
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Z z ,q ... z ,t + 1 z , t ... z ,0
.
. .
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. .
. .
. .
. .
.
. . . . . . .

Ïðîòîêîë

q ... t +1 t ... 0
.
. .
. .
. .
. .
. .
. .
.
. . . . . . .
Sk sk ,q ... sk ,t+1 sk ,t ... sk ,0 = k=s q s bt
t = 0 k ,t
.
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.
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R r ,q ... r ,t + 1 r , t ... r ,0 = r = q r bt
t = 0 ,t
.
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.
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Z z ,q ... z ,t + 1 z , t ... z ,0
.
. .
. .
. .
. .
. .
. .
.
. . . . . . .

Ïðîòîêîë

q ... t +1 t ... 0
.
. .
. .
. .
. .
. .
. .
.
. . . . . . .
Sk sk ,q ... sk ,t+1 sk ,t ... sk ,0 = k=s q s bt
t = 0 k ,t
.
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.
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R r ,q ... r ,t + 1 r , t ... r ,0 = r = q r bt
t = 0 ,t
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.
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Z z ,q ... z ,t + 1 z , t ... z ,0 = z = q z bt
t =0 ,t
.
. .
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. .
. .
. .
. .
.
. . . . . . .


q q q
sk = sk ,t bt r = r ,t bt z = z ,t b t
t =0 t =0 t =0

r ,t + 1 = r ,t + +
sk ,t − −
z ,t sk ,t


q q q
t =0 t =0 t =0

r ,t + 1 b t + 1 = r ,t b t + 1 + +
sk ,t bt+1 − −
z ,t sk ,t bt+1


q q q
t =0 t =0 t =0
q −1 q −1
r ,t + 1 b t +1 = r ,t b t + 1 + +
sk ,t bt+1 − −
z ,t sk ,t bt+1
t =0 t =0


q q q
t =0 t =0 t =0
q −1 q −1
r ,t + 1 b t +1 = r ,t b t + 1 + +
sk ,t bt+1 − −
z ,t sk ,t bt+1
t =0 t =0

r − r ,0 = br + b +
sk − b −
z ∧ sk )
(


q q q
t =0 t =0 t =0
q −1 q −1
r ,t + 1 b t +1 = r ,t b t + 1 + +
sk ,t bt+1 − −
z ,t sk ,t bt+1
t =0 t =0

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


q q
sk = sk ,t bt z = z ,t b t
t =0 t =0

sd ,t+1 =

d sk , t d z ,t sk ,t d (1 − z ,t )sk ,t
+ − 0
= + +


q q
t =0 t =0

sd ,t+1bt+1 =

d sk , t b d z ,t sk ,t b d (1 − z ,t )sk ,t b
= + t +1 + − t +1 + 0 t +1


q q
t =0 t =0
q−1
sd ,t+1bt+1 =
t =0
q −1
= + t +1 + − t +1 + 0 t +1
t =0


q q
t =0 t =0
q−1
sd ,t+1bt+1 =
t =0
q −1
= + t +1 + − t +1 + 0 t +1
t =0

sd − sd ,0 = b d sk + b d (z ∧ sk ) + b d ((e − z ) ∧ sk )
+ + 0


q q
t =0 t =0
q−1
sd ,t+1bt+1 =
t =0
q −1
= + t +1 + − t +1 + 0 t +1
t =0

+ + 0

q−1
e= b
1 · t +1 =
bq − 1
t =0 b−1

Èíäèêàòîðû íóëÿ

z ,t = r
0, åñëè ,t = 0
1 â ïðîòèâíîì ñëó÷àå


z ,t = r
0, åñëè ,t = 0

b = 2 c +1 r ,t ≤ 2c − 1


z ,t = r
0, åñëè ,t = 0

b = 2 c +1 r ,t ≤ 2c − 1 = 01 . . . 1


z ,t = r
0, åñëè ,t = 0

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


z ,t = r
0, åñëè ,t = 0

b = 2 c +1 r ,t ≤ 2c − 1 = 01 . . . 1
2c − 1 + r ,t = r
01 . . . 1, åñëè ,t = 0
1 ∗ · · · ∗ â ïðîòèâíîì ñëó÷àå


z ,t = r
0, åñëè ,t = 0

b = 2 c +1 r ,t ≤ 2c − 1 = 01 . . . 1
2c − 1 + r ,t = r
01 . . . 1, åñëè ,t = 0

2c ∧ (2c − 1 + r ,t ) = 2c z ,t


z ,t = r
0, åñëè ,t = 0

b = 2 c +1 r ,t ≤ 2c − 1 = 01 . . . 1
2c − 1 + r ,t = r
01 . . . 1, åñëè ,t = 0

2c ∧ (2c − 1 + r ,t ) = 2c z ,t
q q
(2c ∧ (2c − 1 + r ,t ))bt = 2c z ,t b t
t =0 t =0


z ,t = r
0, åñëè ,t = 0

b = 2 c +1 r ,t ≤ 2c − 1 = 01 . . . 1
2c − 1 + r ,t = r
01 . . . 1, åñëè ,t = 0

2c ∧ (2c − 1 + r ,t ) = 2c z ,t
q q
(2c ∧ (2c − 1 + r ,t ))bt = 2c z ,t b t
t =0 t =0
q
f f r
2c ∧ ((2c − 1) + ) = 2c z f= b
1 · t +1 =
bq+1 − 1
t =0 b−1

Âûáîð c

r ,t b

Âûáîð c

r , t b = 2 c +1

Âûáîð c

r , t b = 2 c +1 r ,t ≤ 2c − 1 = 01 . . . 1
2c − 1 + r ,t = r
01 . . . 1, åñëè ,t = 0

Âûáîð c

r , t b = 2 c +1 r ,t ≤ 2c − 1 = 01 . . . 1
2c − 1 + r ,t = r
01 . . . 1, åñëè ,t = 0

2c ∧ r ,t =0

Âûáîð c

r , t b = 2 c +1 r ,t ≤ 2c − 1 = 01 . . . 1
2c − 1 + r ,t = r
01 . . . 1, åñëè ,t = 0
q
(2 c ∧ r ,t ) b t = 0
t =0

Âûáîð c

r , t b = 2 c +1 r ,t ≤ 2c − 1 = 01 . . . 1
2c − 1 + r ,t = r
01 . . . 1, åñëè ,t = 0
q
(2 c ∧ r ,t ) b t = 0
t =0
q
f r
2c ∧ f= b
1 · t +1 =
b q +1 − 1
=0
t =0 b−1

Âñå óñëîâèÿ

b = 2 c +1


b = 2 c +1

r − r ,0 = br + b +
sk − b −
( z ∧ sk )


b = 2 c +1

r − r ,0 = br + b +
sk − b −
( z ∧ sk )

+ + 0


b = 2 c +1

r − r ,0 = br + b +
sk − b −
( z ∧ sk )

+ + 0

e = bb − 11
−q


b = 2 c +1

r − r ,0 = br + b +
sk − b −
( z ∧ sk )

+ + 0

e = bb − 11
−q

z f = b b −− 1
q +1
f f r
2c ∧ ((2c − 1) + ) = 2c
1


b = 2 c +1

r − r ,0 = br + b +
sk − b −
( z ∧ sk )

+ + 0

e = bb − 11
−q

z f = b b −− 1
q +1
f f r
2c ∧ ((2c − 1) + ) = 2c
1

f r
2c ∧ =0


b = 2 c +1

r − r ,0 = br + b +
sk − b −
( z ∧ sk )

+ + 0

e = bb − 11
−q

z f = b b −− 1
q +1
f f r
2c ∧ ((2c − 1) + ) = 2c
1

f r
2c ∧ =0 sm = bq

Ïðîòîêîë

q ... t +1 t ... 0
.
. .
. .
. .
. .
. .
. .
.
. . . . . . .
Sk sk ,q ... sk ,t+1 sk ,t ... sk ,0 = k=s Q s bt
t = 0 k ,t
.
. .
. .
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.
. . . . . . .
.
. .
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. .
. .
. .
. .
.
. . . . . . .
R r ,q ... r ,t + 1 r , t ... r ,0 = r = Q r bt
t = 0 ,t
.
. .
. .
. .
. .
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. .
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. . . . . . .
.
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. .
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. . . . . . .
Z z ,q ... z ,t + 1 z , t ... z ,0 = z = Q z bt
t =0 ,t
.
. .
. .
. .
. .
. .
. .
.
. . . . . . .

Юрий Владимирович Матиясевич. Десятая проблема Гильберта. Решение и применения в информатике. Лекция 03

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