Ãèïîòåçà Martin'a Davis'à (=DPRM-òåîðåìà)
Ãèïîòåçà M. Davis'à (DPRM-òåîðåìà). Êàæäîåïåðå÷èñëèìîå ìíîæåñòâî ÿâëÿåòñÿ äèîôàíòîâûì.
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.
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.
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.
Ïåðå÷èñëèìûå ìíîæåñòâà
Îïðåäåëåíèå. Ìíîæåñòâî M íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R, òàêóþ ÷òî
R-a -îñòàíîâêà, åñëè a ∈M
âå÷íàÿ ðàáîòà â ïðîòèâíîì ñëó÷àå
Ïåðå÷èñëèìûå ìíîæåñòâà
Îïðåäåëåíèå. Ìíîæåñòâî M íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R, òàêóþ ÷òî
R-a -îñòàíîâêà, åñëè a ∈M
âå÷íàÿ ðàáîòà â ïðîòèâíîì ñëó÷àå
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
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
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
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
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
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 ;Sj
III. Sk : STOP
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
Ïåðå÷èñëèìûå ìíîæåñòâà
Îïðåäåëåíèå. Ìíîæåñòâî M íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R äëÿðåãèñòðîâîé ìàøèíû, òàêóþ ÷òî
R-a -îñòàíîâêà, åñëè a ∈M
âå÷íàÿ ðàáîòà â ïðîòèâíîì ñëó÷àå
Ïðè çàïóñêå ìàøèíû ÷èñëî a ïîìåùåíî â ðåãèñòð R1, âñåîñòàëüíûå ðåãèñòðû ñîäåðæàò íóëè.
Ïåðå÷èñëèìûå ìíîæåñòâà
Îïðåäåëåíèå. Ìíîæåñòâî M íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R äëÿðåãèñòðîâîé ìàøèíû, òàêóþ ÷òî
R-a -îñòàíîâêà, åñëè a ∈M
âå÷íàÿ ðàáîòà â ïðîòèâíîì ñëó÷àå
Ïðè çàïóñêå ìàøèíû ÷èñëî a ïîìåùåíî â ðåãèñòð R1, âñåîñòàëüíûå ðåãèñòðû ñîäåðæàò íóëè.
Ïåðå÷èñëèìûå ìíîæåñòâà
Îïðåäåëåíèå. Ìíîæåñòâî M íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R äëÿðåãèñòðîâîé ìàøèíû, òàêóþ ÷òî
R-a -îñòàíîâêà, åñëè a ∈M
âå÷íàÿ ðàáîòà â ïðîòèâíîì ñëó÷àå
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.
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.
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.
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.
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.
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.
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.
Äèîôàíòîâà ñëîæíîñòü
NDDM
P(a, x1, . . . , xm)?= 0 �-
? --
âõîäa
óãàäàòüx1, . . . , xm
ÄÀ ÍÅÒ
ïðèíÿòü a îòâåðãíóòü
SIZE(a)=ìèíèìàëüíî âîçìîæíîå çíà÷åíèå |x1|+ · · ·+ |xm|, ãäå|x | îáîçíà÷àåò äëèíó äâîè÷íîé çàïèñè x .
Äèîôàíòîâà ñëîæíîñòü
NDDM
P(a, x1, . . . , xm)?= 0 �-
? --
âõîäa
óãàäàòüx1, . . . , xm
ÄÀ ÍÅÒ
ïðèíÿòü a îòâåðãíóòü
SIZE(a)=ìèíèìàëüíî âîçìîæíîå çíà÷åíèå |x1|+ · · ·+ |xm|, ãäå|x | îáîçíà÷àåò äëèíó äâîè÷íîé çàïèñè x .
Äèîôàíòîâà ñëîæíîñòü
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.
Äèîôàíòîâà ñëîæíîñòü
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.
Ãèïîòåçà Martin'a Davis'à (=DPRM-òåîðåìà)
Ãèïîòåçà M. Davis'à (DPRM-òåîðåìà). Êàæäîåïåðå÷èñëèìîå ìíîæåñòâî ÿâëÿåòñÿ äèîôàíòîâûì.
Îïðåäåëåíèå. Ìíîæåñòâî M íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R äëÿðåãèñòðîâîé ìàøèíû, òàêóþ ÷òî
R-a -îñòàíîâêà, åñëè a ∈M
âå÷íàÿ ðàáîòà â ïðîòèâíîì ñëó÷àå
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.
Ãèïîòåçà Martin'a Davis'à (=DPRM-òåîðåìà)
Ãèïîòåçà M. Davis'à (DPRM-òåîðåìà). Êàæäîåïåðå÷èñëèìîå ìíîæåñòâî ÿâëÿåòñÿ äèîôàíòîâûì.
Îïðåäåëåíèå. Ìíîæåñòâî M íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R äëÿðåãèñòðîâîé ìàøèíû, òàêóþ ÷òî
R-a -îñòàíîâêà, åñëè a ∈M
âå÷íàÿ ðàáîòà â ïðîòèâíîì ñëó÷àå
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.
Ãèïîòåçà Martin'a Davis'à (=DPRM-òåîðåìà)
Ãèïîòåçà M. Davis'à (DPRM-òåîðåìà). Êàæäîåïåðå÷èñëèìîå ìíîæåñòâî ÿâëÿåòñÿ äèîôàíòîâûì.
Îïðåäåëåíèå. Ìíîæåñòâî M íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R äëÿðåãèñòðîâîé ìàøèíû, òàêóþ ÷òî
R-a -îñòàíîâêà, åñëè a ∈M
âå÷íàÿ ðàáîòà â ïðîòèâíîì ñëó÷àå
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.
Ãèïîòåçà Martin'a Davis'à (=DPRM-òåîðåìà)
Ãèïîòåçà M. Davis'à (DPRM-òåîðåìà). Êàæäîåïåðå÷èñëèìîå ìíîæåñòâî ÿâëÿåòñÿ äèîôàíòîâûì.
Îïðåäåëåíèå. Ìíîæåñòâî M íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R äëÿðåãèñòðîâîé ìàøèíû, òàêóþ ÷òî
R-a -îñòàíîâêà, åñëè a ∈M
âå÷íàÿ ðàáîòà â ïðîòèâíîì ñëó÷àå
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.
Ãèïîòåçà Martin'a Davis'à (=DPRM-òåîðåìà)
Ãèïîòåçà M. Davis'à (DPRM-òåîðåìà). Êàæäîåïåðå÷èñëèìîå ìíîæåñòâî ÿâëÿåòñÿ äèîôàíòîâûì.
Îïðåäåëåíèå. Ìíîæåñòâî M íàòóðàëüíûõ ÷èñåë íàçûâàåòñÿïåðå÷èñëèìûì, åñëè ìîæíî íàïèñàòü ïðîãðàììó R äëÿðåãèñòðîâîé ìàøèíû, òàêóþ ÷òî
R-a -îñòàíîâêà, åñëè a ∈M
âå÷íàÿ ðàáîòà â ïðîòèâíîì ñëó÷àå
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
......
......
......
...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` =
∑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
......
......
......
...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` =
∑qt=0
r`,tbt
......
......
......
...
......
......
......
...Z` z`,q . . . z`,t+1 z`,t . . . z`,0 = z` =
∑qt=0
z`,tbt
......
......
......
...
Íîâûå çíà÷åíèÿ ðåãèñòðîâ
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
Íîâûå çíà÷åíèÿ ðåãèñòðîâ
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
Íîâûå çíà÷åíèÿ ðåãèñòðîâ
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
Íîâûå çíà÷åíèÿ ðåãèñòðîâ
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
Íîâûå çíà÷åíèÿ ðåãèñòðîâ
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
Íîâûå çíà÷åíèÿ ðåãèñòðîâ
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
Íîâûå çíà÷åíèÿ ðåãèñòðîâ
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
Íîâûå çíà÷åíèÿ ðåãèñòðîâ
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
Íîâûå çíà÷åíèÿ ðåãèñòðîâ
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
Íîâûå ñîñòîÿíèÿ
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
Íîâûå ñîñòîÿíèÿ
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
Íîâûå ñîñòîÿíèÿ
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
Íîâûå ñîñòîÿíèÿ
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
Íîâûå ñîñòîÿíèÿ
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
Íîâûå ñîñòîÿíèÿ
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
Íîâûå ñîñòîÿíèÿ
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
Íîâûå ñîñòîÿíèÿ
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
Íîâûå ñîñòîÿíèÿ
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
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
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
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
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
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`,tbt
2c f ∧ ((2c − 1)f + r`) = 2cz`
f =
q∑t=0
1 · bt+1 =bq+1 − 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`,tbt
2c f ∧ ((2c − 1)f + r`) = 2cz` f =
q∑t=0
1 · bt+1 =bq+1 − 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`,tbt
2c f ∧ ((2c − 1)f + r`) = 2cz` f =
q∑t=0
1 · bt+1 =bq+1 − 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`,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 =
{01 . . . 1, åñëè r`,t = 01 ∗ · · · ∗ â ïðîòèâíîì ñëó÷àå
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 =
{01 . . . 1, åñëè r`,t = 01 ∗ · · · ∗ â ïðîòèâíîì ñëó÷àå
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 =
{01 . . . 1, åñëè r`,t = 01 ∗ · · · ∗ â ïðîòèâíîì ñëó÷àå
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 =
{01 . . . 1, åñëè r`,t = 01 ∗ · · · ∗ â ïðîòèâíîì ñëó÷àå
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 =
{01 . . . 1, åñëè r`,t = 01 ∗ · · · ∗ â ïðîòèâíîì ñëó÷àå
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 =
{01 . . . 1, åñëè r`,t = 01 ∗ · · · ∗ â ïðîòèâíîì ñëó÷àå
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 =
{01 . . . 1, åñëè r`,t = 01 ∗ · · · ∗ â ïðîòèâíîì ñëó÷àå
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 =
{01 . . . 1, åñëè r`,t = 01 ∗ · · · ∗ â ïðîòèâíîì ñëó÷àå
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 =
{01 . . . 1, åñëè r`,t = 01 ∗ · · · ∗ â ïðîòèâíîì ñëó÷àå
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
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
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
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
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
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
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
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
Ïðîòîêîë
q . . . t + 1 t . . . 0
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Sk sk,q . . . sk,t+1 sk,t . . . sk,0 = sk =∑Q
t=0sk,tb
t
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R` r`,q . . . r`,t+1 r`,t . . . r`,0 = r` =∑Q
t=0r`,tb
t
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Z` z`,q . . . z`,t+1 z`,t . . . z`,0 = z` =∑Q
t=0z`,tb
t
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