20100930 proof complexity_hirsch_lecture03
TRANSCRIPT
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Ñëîæíîñòü ïðîïîçèöèîíàëüíûõ äîêàçàòåëüñòâ
Ýäóàðä Àëåêñååâè÷ Ãèðø
http://logic.pdmi.ras.ru/~hirsch
ÏÎÌÈ ÐÀÍ
30 ñåíòÿáðÿ 2010 ã.
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÏðåäñòàâëåíèå ëèíåéíûõ íåðàâåíñòâ
I ïîáèòíîå êîäèðîâàíèå ÷èñåë ôîðìóëàìè;I íåðàâåíñòâà âèäà
∑t ctyt ≥ c ,
ãäå ct , c ≥ 0, yt ∈ {0, 1} (yt = xt èëè yt = ¬xt).
I ct · yt � ýòî (Y0, . . . ,Yk),ãäå Yi = yt , åñëè (ct)i = 1; èíà÷å Yi = False.
I íàäî âû÷èñëèòü ñóììó∑
t ctyt äëÿ yt ∈ {0, 1} è ñðàâíèòü ñ c .I Add((F0, . . . ,Fk), (G0, . . . ,Gk))i =
Fi ⊕ Gi ⊕∨0≤j<i
(Fj ∧ Gj ∧∧
j<l<i
(Fl ⊕ Gl )).
I SAdd(~F , ~G , ~H)i = Fi ⊕ Gi ⊕ Hi .I CAdd(~F , ~G , ~H)i+1 = (Fi ∧ Gi ) ∨ (Fi ∧ Hi ) ∨ (Gi ∧ Hi ).I SUM(c1y1, . . . , cnyn): ñêëàäûâàåì SAdd, CAdd, ïîñëåäíèå � Add.
I ~F > ~G ïðåäñòàâëÿåòñÿ êàê∨i
(Fi ∧ ¬Gi ∧∧j>i
(Fj ≡ Gj)).
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÏðåäñòàâëåíèå ëèíåéíûõ íåðàâåíñòâ
I ïîáèòíîå êîäèðîâàíèå ÷èñåë ôîðìóëàìè;I íåðàâåíñòâà âèäà
∑t ctyt ≥ c ,
ãäå ct , c ≥ 0, yt ∈ {0, 1} (yt = xt èëè yt = ¬xt).I ct · yt � ýòî (Y0, . . . ,Yk),
ãäå Yi = yt , åñëè (ct)i = 1; èíà÷å Yi = False.
I íàäî âû÷èñëèòü ñóììó∑
t ctyt äëÿ yt ∈ {0, 1} è ñðàâíèòü ñ c .I Add((F0, . . . ,Fk), (G0, . . . ,Gk))i =
Fi ⊕ Gi ⊕∨0≤j<i
(Fj ∧ Gj ∧∧
j<l<i
(Fl ⊕ Gl )).
I SAdd(~F , ~G , ~H)i = Fi ⊕ Gi ⊕ Hi .I CAdd(~F , ~G , ~H)i+1 = (Fi ∧ Gi ) ∨ (Fi ∧ Hi ) ∨ (Gi ∧ Hi ).I SUM(c1y1, . . . , cnyn): ñêëàäûâàåì SAdd, CAdd, ïîñëåäíèå � Add.
I ~F > ~G ïðåäñòàâëÿåòñÿ êàê∨i
(Fi ∧ ¬Gi ∧∧j>i
(Fj ≡ Gj)).
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÏðåäñòàâëåíèå ëèíåéíûõ íåðàâåíñòâ
I ïîáèòíîå êîäèðîâàíèå ÷èñåë ôîðìóëàìè;I íåðàâåíñòâà âèäà
∑t ctyt ≥ c ,
ãäå ct , c ≥ 0, yt ∈ {0, 1} (yt = xt èëè yt = ¬xt).I ct · yt � ýòî (Y0, . . . ,Yk),
ãäå Yi = yt , åñëè (ct)i = 1; èíà÷å Yi = False.I íàäî âû÷èñëèòü ñóììó
∑t ctyt äëÿ yt ∈ {0, 1} è ñðàâíèòü ñ c .
I Add((F0, . . . ,Fk), (G0, . . . ,Gk))i =
Fi ⊕ Gi ⊕∨0≤j<i
(Fj ∧ Gj ∧∧
j<l<i
(Fl ⊕ Gl )).
I SAdd(~F , ~G , ~H)i = Fi ⊕ Gi ⊕ Hi .I CAdd(~F , ~G , ~H)i+1 = (Fi ∧ Gi ) ∨ (Fi ∧ Hi ) ∨ (Gi ∧ Hi ).I SUM(c1y1, . . . , cnyn): ñêëàäûâàåì SAdd, CAdd, ïîñëåäíèå � Add.
I ~F > ~G ïðåäñòàâëÿåòñÿ êàê∨i
(Fi ∧ ¬Gi ∧∧j>i
(Fj ≡ Gj)).
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÏðåäñòàâëåíèå ëèíåéíûõ íåðàâåíñòâ
I ïîáèòíîå êîäèðîâàíèå ÷èñåë ôîðìóëàìè;I íåðàâåíñòâà âèäà
∑t ctyt ≥ c ,
ãäå ct , c ≥ 0, yt ∈ {0, 1} (yt = xt èëè yt = ¬xt).I ct · yt � ýòî (Y0, . . . ,Yk),
ãäå Yi = yt , åñëè (ct)i = 1; èíà÷å Yi = False.I íàäî âû÷èñëèòü ñóììó
∑t ctyt äëÿ yt ∈ {0, 1} è ñðàâíèòü ñ c .
I Add((F0, . . . ,Fk), (G0, . . . ,Gk))i =
Fi ⊕ Gi ⊕∨0≤j<i
(Fj ∧ Gj ∧∧
j<l<i
(Fl ⊕ Gl )).
I SAdd(~F , ~G , ~H)i = Fi ⊕ Gi ⊕ Hi .I CAdd(~F , ~G , ~H)i+1 = (Fi ∧ Gi ) ∨ (Fi ∧ Hi ) ∨ (Gi ∧ Hi ).I SUM(c1y1, . . . , cnyn): ñêëàäûâàåì SAdd, CAdd, ïîñëåäíèå � Add.
I ~F > ~G ïðåäñòàâëÿåòñÿ êàê∨i
(Fi ∧ ¬Gi ∧∧j>i
(Fj ≡ Gj)).
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÏðåäñòàâëåíèå ëèíåéíûõ íåðàâåíñòâ
I ïîáèòíîå êîäèðîâàíèå ÷èñåë ôîðìóëàìè;I íåðàâåíñòâà âèäà
∑t ctyt ≥ c ,
ãäå ct , c ≥ 0, yt ∈ {0, 1} (yt = xt èëè yt = ¬xt).I ct · yt � ýòî (Y0, . . . ,Yk),
ãäå Yi = yt , åñëè (ct)i = 1; èíà÷å Yi = False.I íàäî âû÷èñëèòü ñóììó
∑t ctyt äëÿ yt ∈ {0, 1} è ñðàâíèòü ñ c .
I Add((F0, . . . ,Fk), (G0, . . . ,Gk))i =
Fi ⊕ Gi ⊕∨0≤j<i
(Fj ∧ Gj ∧∧
j<l<i
(Fl ⊕ Gl )).
I SAdd(~F , ~G , ~H)i = Fi ⊕ Gi ⊕ Hi .I CAdd(~F , ~G , ~H)i+1 = (Fi ∧ Gi ) ∨ (Fi ∧ Hi ) ∨ (Gi ∧ Hi ).
I SUM(c1y1, . . . , cnyn): ñêëàäûâàåì SAdd, CAdd, ïîñëåäíèå � Add.
I ~F > ~G ïðåäñòàâëÿåòñÿ êàê∨i
(Fi ∧ ¬Gi ∧∧j>i
(Fj ≡ Gj)).
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÏðåäñòàâëåíèå ëèíåéíûõ íåðàâåíñòâ
I ïîáèòíîå êîäèðîâàíèå ÷èñåë ôîðìóëàìè;I íåðàâåíñòâà âèäà
∑t ctyt ≥ c ,
ãäå ct , c ≥ 0, yt ∈ {0, 1} (yt = xt èëè yt = ¬xt).I ct · yt � ýòî (Y0, . . . ,Yk),
ãäå Yi = yt , åñëè (ct)i = 1; èíà÷å Yi = False.I íàäî âû÷èñëèòü ñóììó
∑t ctyt äëÿ yt ∈ {0, 1} è ñðàâíèòü ñ c .
I Add((F0, . . . ,Fk), (G0, . . . ,Gk))i =
Fi ⊕ Gi ⊕∨0≤j<i
(Fj ∧ Gj ∧∧
j<l<i
(Fl ⊕ Gl )).
I SAdd(~F , ~G , ~H)i = Fi ⊕ Gi ⊕ Hi .I CAdd(~F , ~G , ~H)i+1 = (Fi ∧ Gi ) ∨ (Fi ∧ Hi ) ∨ (Gi ∧ Hi ).I SUM(c1y1, . . . , cnyn): ñêëàäûâàåì SAdd, CAdd, ïîñëåäíèå � Add.
I ~F > ~G ïðåäñòàâëÿåòñÿ êàê∨i
(Fi ∧ ¬Gi ∧∧j>i
(Fj ≡ Gj)).
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÏðåäñòàâëåíèå ëèíåéíûõ íåðàâåíñòâ
I ïîáèòíîå êîäèðîâàíèå ÷èñåë ôîðìóëàìè;I íåðàâåíñòâà âèäà
∑t ctyt ≥ c ,
ãäå ct , c ≥ 0, yt ∈ {0, 1} (yt = xt èëè yt = ¬xt).I ct · yt � ýòî (Y0, . . . ,Yk),
ãäå Yi = yt , åñëè (ct)i = 1; èíà÷å Yi = False.I íàäî âû÷èñëèòü ñóììó
∑t ctyt äëÿ yt ∈ {0, 1} è ñðàâíèòü ñ c .
I Add((F0, . . . ,Fk), (G0, . . . ,Gk))i =
Fi ⊕ Gi ⊕∨0≤j<i
(Fj ∧ Gj ∧∧
j<l<i
(Fl ⊕ Gl )).
I SAdd(~F , ~G , ~H)i = Fi ⊕ Gi ⊕ Hi .I CAdd(~F , ~G , ~H)i+1 = (Fi ∧ Gi ) ∨ (Fi ∧ Hi ) ∨ (Gi ∧ Hi ).I SUM(c1y1, . . . , cnyn): ñêëàäûâàåì SAdd, CAdd, ïîñëåäíèå � Add.
I ~F > ~G ïðåäñòàâëÿåòñÿ êàê∨i
(Fi ∧ ¬Gi ∧∧j>i
(Fj ≡ Gj)).
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÌîäåëèðîâàíèå ïðàâèë
I ïðîñóììèðóåì∑
ctyt ≥ c è∑
dtyt ≥ d :
I äîêàæåì ïî èíäóêöèè, ÷òîAdd(SUM(. . . , ctyt , . . .), SUM(. . . , dtyt , . . .)) ≡ SUM(. . . , (ct + dt)yt , . . .),
I ðàâåíñòâî Add(ctyt , dtyt)i ≡ (ct + dt)iyt � ðàçáîð ñëó÷àåâ.I yt + ¬yt � àíàëîãè÷íî.
I äîêàæåì ~F ≥ ~G ∧ ~F ′ ≥ ~G ′ ⊃ Add(~F , ~F ′) ≥ Add(~G , ~G ′).I óìíîæåíèå (äåëåíèå) íà êîíñòàíòó. . .I îêðóãëåíèå ∑
(act)yt ≥ ac + r∑ctyt ≥ c + 1
(r < a)
I ñâîéñòâà íóëÿ: Add(~F , 0)i ≡ Fi è 0 < 1.I SUM(0y1, . . . , 0yn) ≥ 1, î÷åâèäíî, ëîæíî.
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÌîäåëèðîâàíèå ïðàâèë
I ïðîñóììèðóåì∑
ctyt ≥ c è∑
dtyt ≥ d :I äîêàæåì ïî èíäóêöèè, ÷òîAdd(SUM(. . . , ctyt , . . .), SUM(. . . , dtyt , . . .)) ≡ SUM(. . . , (ct + dt)yt , . . .),
I ðàâåíñòâî Add(ctyt , dtyt)i ≡ (ct + dt)iyt � ðàçáîð ñëó÷àåâ.I yt + ¬yt � àíàëîãè÷íî.
I äîêàæåì ~F ≥ ~G ∧ ~F ′ ≥ ~G ′ ⊃ Add(~F , ~F ′) ≥ Add(~G , ~G ′).I óìíîæåíèå (äåëåíèå) íà êîíñòàíòó. . .I îêðóãëåíèå ∑
(act)yt ≥ ac + r∑ctyt ≥ c + 1
(r < a)
I ñâîéñòâà íóëÿ: Add(~F , 0)i ≡ Fi è 0 < 1.I SUM(0y1, . . . , 0yn) ≥ 1, î÷åâèäíî, ëîæíî.
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÌîäåëèðîâàíèå ïðàâèë
I ïðîñóììèðóåì∑
ctyt ≥ c è∑
dtyt ≥ d :I äîêàæåì ïî èíäóêöèè, ÷òîAdd(SUM(. . . , ctyt , . . .), SUM(. . . , dtyt , . . .)) ≡ SUM(. . . , (ct + dt)yt , . . .),
I ðàâåíñòâî Add(ctyt , dtyt)i ≡ (ct + dt)iyt � ðàçáîð ñëó÷àåâ.
I yt + ¬yt � àíàëîãè÷íî.
I äîêàæåì ~F ≥ ~G ∧ ~F ′ ≥ ~G ′ ⊃ Add(~F , ~F ′) ≥ Add(~G , ~G ′).I óìíîæåíèå (äåëåíèå) íà êîíñòàíòó. . .I îêðóãëåíèå ∑
(act)yt ≥ ac + r∑ctyt ≥ c + 1
(r < a)
I ñâîéñòâà íóëÿ: Add(~F , 0)i ≡ Fi è 0 < 1.I SUM(0y1, . . . , 0yn) ≥ 1, î÷åâèäíî, ëîæíî.
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÌîäåëèðîâàíèå ïðàâèë
I ïðîñóììèðóåì∑
ctyt ≥ c è∑
dtyt ≥ d :I äîêàæåì ïî èíäóêöèè, ÷òîAdd(SUM(. . . , ctyt , . . .), SUM(. . . , dtyt , . . .)) ≡ SUM(. . . , (ct + dt)yt , . . .),
I ðàâåíñòâî Add(ctyt , dtyt)i ≡ (ct + dt)iyt � ðàçáîð ñëó÷àåâ.I yt + ¬yt � àíàëîãè÷íî.
I äîêàæåì ~F ≥ ~G ∧ ~F ′ ≥ ~G ′ ⊃ Add(~F , ~F ′) ≥ Add(~G , ~G ′).I óìíîæåíèå (äåëåíèå) íà êîíñòàíòó. . .I îêðóãëåíèå ∑
(act)yt ≥ ac + r∑ctyt ≥ c + 1
(r < a)
I ñâîéñòâà íóëÿ: Add(~F , 0)i ≡ Fi è 0 < 1.I SUM(0y1, . . . , 0yn) ≥ 1, î÷åâèäíî, ëîæíî.
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÌîäåëèðîâàíèå ïðàâèë
I ïðîñóììèðóåì∑
ctyt ≥ c è∑
dtyt ≥ d :I äîêàæåì ïî èíäóêöèè, ÷òîAdd(SUM(. . . , ctyt , . . .), SUM(. . . , dtyt , . . .)) ≡ SUM(. . . , (ct + dt)yt , . . .),
I ðàâåíñòâî Add(ctyt , dtyt)i ≡ (ct + dt)iyt � ðàçáîð ñëó÷àåâ.I yt + ¬yt � àíàëîãè÷íî.
I äîêàæåì ~F ≥ ~G ∧ ~F ′ ≥ ~G ′ ⊃ Add(~F , ~F ′) ≥ Add(~G , ~G ′).
I óìíîæåíèå (äåëåíèå) íà êîíñòàíòó. . .I îêðóãëåíèå ∑
(act)yt ≥ ac + r∑ctyt ≥ c + 1
(r < a)
I ñâîéñòâà íóëÿ: Add(~F , 0)i ≡ Fi è 0 < 1.I SUM(0y1, . . . , 0yn) ≥ 1, î÷åâèäíî, ëîæíî.
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÌîäåëèðîâàíèå ïðàâèë
I ïðîñóììèðóåì∑
ctyt ≥ c è∑
dtyt ≥ d :I äîêàæåì ïî èíäóêöèè, ÷òîAdd(SUM(. . . , ctyt , . . .), SUM(. . . , dtyt , . . .)) ≡ SUM(. . . , (ct + dt)yt , . . .),
I ðàâåíñòâî Add(ctyt , dtyt)i ≡ (ct + dt)iyt � ðàçáîð ñëó÷àåâ.I yt + ¬yt � àíàëîãè÷íî.
I äîêàæåì ~F ≥ ~G ∧ ~F ′ ≥ ~G ′ ⊃ Add(~F , ~F ′) ≥ Add(~G , ~G ′).I óìíîæåíèå (äåëåíèå) íà êîíñòàíòó. . .
I îêðóãëåíèå ∑(act)yt ≥ ac + r∑ctyt ≥ c + 1
(r < a)
I ñâîéñòâà íóëÿ: Add(~F , 0)i ≡ Fi è 0 < 1.I SUM(0y1, . . . , 0yn) ≥ 1, î÷åâèäíî, ëîæíî.
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÌîäåëèðîâàíèå ïðàâèë
I ïðîñóììèðóåì∑
ctyt ≥ c è∑
dtyt ≥ d :I äîêàæåì ïî èíäóêöèè, ÷òîAdd(SUM(. . . , ctyt , . . .), SUM(. . . , dtyt , . . .)) ≡ SUM(. . . , (ct + dt)yt , . . .),
I ðàâåíñòâî Add(ctyt , dtyt)i ≡ (ct + dt)iyt � ðàçáîð ñëó÷àåâ.I yt + ¬yt � àíàëîãè÷íî.
I äîêàæåì ~F ≥ ~G ∧ ~F ′ ≥ ~G ′ ⊃ Add(~F , ~F ′) ≥ Add(~G , ~G ′).I óìíîæåíèå (äåëåíèå) íà êîíñòàíòó. . .I îêðóãëåíèå ∑
(act)yt ≥ ac + r∑ctyt ≥ c + 1
(r < a)
I ñâîéñòâà íóëÿ: Add(~F , 0)i ≡ Fi è 0 < 1.I SUM(0y1, . . . , 0yn) ≥ 1, î÷åâèäíî, ëîæíî.
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÌîäåëèðîâàíèå ïðàâèë
I ïðîñóììèðóåì∑
ctyt ≥ c è∑
dtyt ≥ d :I äîêàæåì ïî èíäóêöèè, ÷òîAdd(SUM(. . . , ctyt , . . .), SUM(. . . , dtyt , . . .)) ≡ SUM(. . . , (ct + dt)yt , . . .),
I ðàâåíñòâî Add(ctyt , dtyt)i ≡ (ct + dt)iyt � ðàçáîð ñëó÷àåâ.I yt + ¬yt � àíàëîãè÷íî.
I äîêàæåì ~F ≥ ~G ∧ ~F ′ ≥ ~G ′ ⊃ Add(~F , ~F ′) ≥ Add(~G , ~G ′).I óìíîæåíèå (äåëåíèå) íà êîíñòàíòó. . .I îêðóãëåíèå ∑
(act)yt ≥ ac + r∑ctyt ≥ c + 1
(r < a)
� ðàçáîð ñëó÷àåâ (ò.å. äîê-âî îò ïðîòèâíîãî):
SUM(. . . , ctyt , . . .) ≥ c + 1 ∨ ¬(SUM(. . . , ctyt , . . .) ≥ c + 1),
èç âòîðîãî ñëåäóåò ≤ c , óìíîæèì îáðàòíî íà a. . . .
I ñâîéñòâà íóëÿ: Add(~F , 0)i ≡ Fi è 0 < 1.I SUM(0y1, . . . , 0yn) ≥ 1, î÷åâèäíî, ëîæíî.
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Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ ÔðåãåÌîäåëèðîâàíèå ïðàâèë
I ïðîñóììèðóåì∑
ctyt ≥ c è∑
dtyt ≥ d :I äîêàæåì ïî èíäóêöèè, ÷òîAdd(SUM(. . . , ctyt , . . .), SUM(. . . , dtyt , . . .)) ≡ SUM(. . . , (ct + dt)yt , . . .),
I ðàâåíñòâî Add(ctyt , dtyt)i ≡ (ct + dt)iyt � ðàçáîð ñëó÷àåâ.I yt + ¬yt � àíàëîãè÷íî.
I äîêàæåì ~F ≥ ~G ∧ ~F ′ ≥ ~G ′ ⊃ Add(~F , ~F ′) ≥ Add(~G , ~G ′).I óìíîæåíèå (äåëåíèå) íà êîíñòàíòó. . .I îêðóãëåíèå ∑
(act)yt ≥ ac + r∑ctyt ≥ c + 1
(r < a)
� ðàçáîð ñëó÷àåâ (ò.å. äîê-âî îò ïðîòèâíîãî):
SUM(. . . , ctyt , . . .) ≥ c + 1 ∨ ¬(SUM(. . . , ctyt , . . .) ≥ c + 1),
èç âòîðîãî ñëåäóåò ≤ c , óìíîæèì îáðàòíî íà a. . . .I ñâîéñòâà íóëÿ: Add(~F , 0)i ≡ Fi è 0 < 1.I SUM(0y1, . . . , 0yn) ≥ 1, î÷åâèäíî, ëîæíî.
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Îïòèìàëüíûå ïîëóàëãîðèòìû
Îïðåäåëåíèå
A � îïòèìàëüíûé ïîëóàëãîðèòì äëÿ L ⇐⇒äëÿ âñÿêîãî A′ èìååòñÿ ïîëèíîì p, ò.÷. ∀x ∈ L
timeA(x) ≤ p(timeA′(x) + |x |).
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Îïòèìàëüíûå ïîëóàëãîðèòìû
Îïðåäåëåíèå
A � îïòèìàëüíûé ïîëóàëãîðèòì äëÿ L ⇐⇒äëÿ âñÿêîãî A′ èìååòñÿ ïîëèíîì p, ò.÷. ∀x ∈ L
timeA(x) ≤ p(timeA′(x) + |x |).
Ëåâèíñêèé îïòèìàëüíûé àëãîðèòì äëÿ ðåøåíèÿ çàäà÷è ïîèñêà SAT:
çàïóñòèòü �ïàðàëëåëüíî� âñå âîçìîæíûå àëãîðèòìû, ïðîâåðèòü
âûäàííûé �âûïîëíÿþùèé� íàáîð, åñëè âåðåí � âûäàòü.
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Îïòèìàëüíûå ïîëóàëãîðèòìû
Îïðåäåëåíèå
A � îïòèìàëüíûé ïîëóàëãîðèòì äëÿ L ⇐⇒äëÿ âñÿêîãî A′ èìååòñÿ ïîëèíîì p, ò.÷. ∀x ∈ L
timeA(x) ≤ p(timeA′(x) + |x |).
Ëåâèíñêèé îïòèìàëüíûé àëãîðèòì äëÿ ðåøåíèÿ çàäà÷è ïîèñêà SAT:
çàïóñòèòü �ïàðàëëåëüíî� âñå âîçìîæíûå àëãîðèòìû, ïðîâåðèòü
âûäàííûé �âûïîëíÿþùèé� íàáîð, åñëè âåðåí � âûäàòü.
Çàìå÷àíèå
Ëåâèíñêèé àëãîðèòì íå äëÿ ÿçûêà TAUT.
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Îïòèìàëüíûå ïîëóàëãîðèòìû
Îïðåäåëåíèå
A � îïòèìàëüíûé ïîëóàëãîðèòì äëÿ L ⇐⇒äëÿ âñÿêîãî A′ èìååòñÿ ïîëèíîì p, ò.÷. ∀x ∈ L
timeA(x) ≤ p(timeA′(x) + |x |).
Ëåâèíñêèé îïòèìàëüíûé àëãîðèòì äëÿ ðåøåíèÿ çàäà÷è ïîèñêà SAT:
çàïóñòèòü �ïàðàëëåëüíî� âñå âîçìîæíûå àëãîðèòìû, ïðîâåðèòü
âûäàííûé �âûïîëíÿþùèé� íàáîð, åñëè âåðåí � âûäàòü.
Çàìå÷àíèå
Ëåâèíñêèé àëãîðèòì íå äëÿ ÿçûêà TAUT.
. . . è íå äëÿ ÿçûêà SAT.
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Îïòèìàëüíûå ïîëóàëãîðèòìû vs ñèñòåìû äîêàçàòåëüñòâ
Òåîðåìà (Kraj���cek, Pudl�ak, 1989)
∃ p-îïòèìàëüíàÿ ñèñòåìà äîê-â ⇐⇒∃ îïòèìàëüíûé ïîëóàëãîðèòì äëÿ TAUT.
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Îïòèìàëüíûå ïîëóàëãîðèòìû vs ñèñòåìû äîêàçàòåëüñòâ
Òåîðåìà (Kraj���cek, Pudl�ak, 1989)
∃ p-îïòèìàëüíàÿ ñèñòåìà äîê-â ⇐⇒∃ îïòèìàëüíûé ïîëóàëãîðèòì äëÿ TAUT.
⇐=:
I Îïòèìàëüíûé ïîëóàëãîðèòì O ðàáîòàåò ïîëèíîìèàëüíîå âðåìÿ
íà ëþáîì ïîäìíîæåñòâå òàâòîëîãèé èç P.I Äëÿ ëþáîé ñèñòåìû äîêàçàòåëüñòâ Π, ëåãêî (çà ïîëèíîìèàëüíîå
âðåìÿ) çàïèñàòü òàâòîëîãèþ ConΠ,n, îçíà÷àþùóþ �Π êîððåêòíà
äëÿ ôîðìóë ðàçìåðà n�.I Çíà÷èò, O ïîëèíîìèàëåí íà CΠ = {ConΠ,n}n∈N.
I Îïòèìàëüíîå äîê-âî ôîðìóëû F ðàçìåðà n:I Íîìåð ñèñòåìû Π;
I Ïðîòîêîë ðàáîòû O íà ConΠ,n;
I Π-äîêàçàòåëüñòâî ôîðìóëû F .
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Îïòèìàëüíûå ïîëóàëãîðèòìû vs ñèñòåìû äîêàçàòåëüñòâ
Òåîðåìà (Kraj���cek, Pudl�ak, 1989)
∃ p-îïòèìàëüíàÿ ñèñòåìà äîê-â ⇐⇒∃ îïòèìàëüíûé ïîëóàëãîðèòì äëÿ TAUT.
⇐=:I Îïòèìàëüíûé ïîëóàëãîðèòì O ðàáîòàåò ïîëèíîìèàëüíîå âðåìÿ
íà ëþáîì ïîäìíîæåñòâå òàâòîëîãèé èç P.
I Äëÿ ëþáîé ñèñòåìû äîêàçàòåëüñòâ Π, ëåãêî (çà ïîëèíîìèàëüíîå
âðåìÿ) çàïèñàòü òàâòîëîãèþ ConΠ,n, îçíà÷àþùóþ �Π êîððåêòíà
äëÿ ôîðìóë ðàçìåðà n�.I Çíà÷èò, O ïîëèíîìèàëåí íà CΠ = {ConΠ,n}n∈N.I Îïòèìàëüíîå äîê-âî ôîðìóëû F ðàçìåðà n:
I Íîìåð ñèñòåìû Π;
I Ïðîòîêîë ðàáîòû O íà ConΠ,n;
I Π-äîêàçàòåëüñòâî ôîðìóëû F .
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Îïòèìàëüíûå ïîëóàëãîðèòìû vs ñèñòåìû äîêàçàòåëüñòâ
Òåîðåìà (Kraj���cek, Pudl�ak, 1989)
∃ p-îïòèìàëüíàÿ ñèñòåìà äîê-â ⇐⇒∃ îïòèìàëüíûé ïîëóàëãîðèòì äëÿ TAUT.
⇐=:I Îïòèìàëüíûé ïîëóàëãîðèòì O ðàáîòàåò ïîëèíîìèàëüíîå âðåìÿ
íà ëþáîì ïîäìíîæåñòâå òàâòîëîãèé èç P.I Äëÿ ëþáîé ñèñòåìû äîêàçàòåëüñòâ Π, ëåãêî (çà ïîëèíîìèàëüíîå
âðåìÿ) çàïèñàòü òàâòîëîãèþ ConΠ,n, îçíà÷àþùóþ �Π êîððåêòíà
äëÿ ôîðìóë ðàçìåðà n�.
I Çíà÷èò, O ïîëèíîìèàëåí íà CΠ = {ConΠ,n}n∈N.I Îïòèìàëüíîå äîê-âî ôîðìóëû F ðàçìåðà n:
I Íîìåð ñèñòåìû Π;
I Ïðîòîêîë ðàáîòû O íà ConΠ,n;
I Π-äîêàçàòåëüñòâî ôîðìóëû F .
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Îïòèìàëüíûå ïîëóàëãîðèòìû vs ñèñòåìû äîêàçàòåëüñòâ
Òåîðåìà (Kraj���cek, Pudl�ak, 1989)
∃ p-îïòèìàëüíàÿ ñèñòåìà äîê-â ⇐⇒∃ îïòèìàëüíûé ïîëóàëãîðèòì äëÿ TAUT.
⇐=:I Îïòèìàëüíûé ïîëóàëãîðèòì O ðàáîòàåò ïîëèíîìèàëüíîå âðåìÿ
íà ëþáîì ïîäìíîæåñòâå òàâòîëîãèé èç P.I Äëÿ ëþáîé ñèñòåìû äîêàçàòåëüñòâ Π, ëåãêî (çà ïîëèíîìèàëüíîå
âðåìÿ) çàïèñàòü òàâòîëîãèþ ConΠ,n, îçíà÷àþùóþ �Π êîððåêòíà
äëÿ ôîðìóë ðàçìåðà n�.I Çíà÷èò, O ïîëèíîìèàëåí íà CΠ = {ConΠ,n}n∈N.
I Îïòèìàëüíîå äîê-âî ôîðìóëû F ðàçìåðà n:I Íîìåð ñèñòåìû Π;
I Ïðîòîêîë ðàáîòû O íà ConΠ,n;
I Π-äîêàçàòåëüñòâî ôîðìóëû F .
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Îïòèìàëüíûå ïîëóàëãîðèòìû vs ñèñòåìû äîêàçàòåëüñòâ
Òåîðåìà (Kraj���cek, Pudl�ak, 1989)
∃ p-îïòèìàëüíàÿ ñèñòåìà äîê-â ⇐⇒∃ îïòèìàëüíûé ïîëóàëãîðèòì äëÿ TAUT.
⇐=:I Îïòèìàëüíûé ïîëóàëãîðèòì O ðàáîòàåò ïîëèíîìèàëüíîå âðåìÿ
íà ëþáîì ïîäìíîæåñòâå òàâòîëîãèé èç P.I Äëÿ ëþáîé ñèñòåìû äîêàçàòåëüñòâ Π, ëåãêî (çà ïîëèíîìèàëüíîå
âðåìÿ) çàïèñàòü òàâòîëîãèþ ConΠ,n, îçíà÷àþùóþ �Π êîððåêòíà
äëÿ ôîðìóë ðàçìåðà n�.I Çíà÷èò, O ïîëèíîìèàëåí íà CΠ = {ConΠ,n}n∈N.I Îïòèìàëüíîå äîê-âî ôîðìóëû F ðàçìåðà n:
I Íîìåð ñèñòåìû Π;I Ïðîòîêîë ðàáîòû O íà ConΠ,n;I Π-äîêàçàòåëüñòâî ôîðìóëû F .
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Îïòèìàëüíûå ïîëóàëãîðèòìû vs ñèñòåìû äîêàçàòåëüñòâ
Òåîðåìà (Kraj���cek, Pudl�ak, 1989)
∃ p-îïòèìàëüíàÿ ñèñòåìà äîê-â ⇐⇒∃ îïòèìàëüíûé ïîëóàëãîðèòì äëÿ TAUT.
=⇒:I Ïóñòü Π � p-îïòèìàëüíàÿ.
I Îïòèìàëüíûé ïîëóàëãîðèòì: �ïàðàëëåëüíûé� çàïóñê âñåõ Oi ,
ïðåòåíäóþùèõ íà âûäà÷ó Π-äîêàçàòåëüñòâ.I Âûäàííîå Oi �äîê-âî� ïðîâåðÿåòñÿ Π;
åñëè ïðàâèëüíîå � âåðíóòü �1�.
I Ïî p-îïòèìàëüíîñòè Π äëÿ ëþáîãî àëãîðèòìà A åãî ïðîòîêîë
ìîæåò áûòü çà ïîëèíîìèàëüíîå âðåìÿ ïðåîáðàçîâàí â Π-äîê-âî
íåêîòîðûì f . Êîìïîçèöèÿ A è f èìååòñÿ â {Oi}i .
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Îïòèìàëüíûå ïîëóàëãîðèòìû vs ñèñòåìû äîêàçàòåëüñòâ
Òåîðåìà (Kraj���cek, Pudl�ak, 1989)
∃ p-îïòèìàëüíàÿ ñèñòåìà äîê-â ⇐⇒∃ îïòèìàëüíûé ïîëóàëãîðèòì äëÿ TAUT.
=⇒:I Ïóñòü Π � p-îïòèìàëüíàÿ.I Îïòèìàëüíûé ïîëóàëãîðèòì: �ïàðàëëåëüíûé� çàïóñê âñåõ Oi ,
ïðåòåíäóþùèõ íà âûäà÷ó Π-äîêàçàòåëüñòâ.I Âûäàííîå Oi �äîê-âî� ïðîâåðÿåòñÿ Π;
åñëè ïðàâèëüíîå � âåðíóòü �1�.
I Ïî p-îïòèìàëüíîñòè Π äëÿ ëþáîãî àëãîðèòìà A åãî ïðîòîêîë
ìîæåò áûòü çà ïîëèíîìèàëüíîå âðåìÿ ïðåîáðàçîâàí â Π-äîê-âî
íåêîòîðûì f . Êîìïîçèöèÿ A è f èìååòñÿ â {Oi}i .
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Îïòèìàëüíûå ïîëóàëãîðèòìû vs ñèñòåìû äîêàçàòåëüñòâ
Òåîðåìà (Kraj���cek, Pudl�ak, 1989)
∃ p-îïòèìàëüíàÿ ñèñòåìà äîê-â ⇐⇒∃ îïòèìàëüíûé ïîëóàëãîðèòì äëÿ TAUT.
=⇒:I Ïóñòü Π � p-îïòèìàëüíàÿ.I Îïòèìàëüíûé ïîëóàëãîðèòì: �ïàðàëëåëüíûé� çàïóñê âñåõ Oi ,
ïðåòåíäóþùèõ íà âûäà÷ó Π-äîêàçàòåëüñòâ.I Âûäàííîå Oi �äîê-âî� ïðîâåðÿåòñÿ Π;
åñëè ïðàâèëüíîå � âåðíóòü �1�.
I Ïî p-îïòèìàëüíîñòè Π äëÿ ëþáîãî àëãîðèòìà A åãî ïðîòîêîë
ìîæåò áûòü çà ïîëèíîìèàëüíîå âðåìÿ ïðåîáðàçîâàí â Π-äîê-âî
íåêîòîðûì f . Êîìïîçèöèÿ A è f èìååòñÿ â {Oi}i .
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p-Optimal proof system from optimal acceptorfor any paddable language [Messner, 99]
De�nition
L is paddable if there is an injective non-length-decreasing polynomial-time
padding function padL : {0, 1}∗ × {0, 1}∗ → {0, 1}∗ that is polynomial-time
invertible on its image and such that ∀x ,w (x ∈ L ⇐⇒ padL(x ,w) ∈ L).
Optimal proof:
I description of proof system Π;I Π-proof π of F ;I 1t (for how long can we work?).
Veri�cation:
I run optimal acceptor on padL(x , π);I for a correct proof, it accepts in a polynomial time because for a
correct system Π, the set {padL(x , π) | x ∈ L, Π(x , π) = 1} ⊆ L can
be accepted in a polynomial time.6 / 6