11. Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ Ôðåãå
Ìîäåëèðîâàíèå ïðàâèë
ïðîñóììèðóåì ct yt ≥ c è dt yt ≥ d :
äîêàæåì ïî èíäóêöèè, ÷òî
Add(SUM(. . . , ct yt , . . .), SUM(. . . , dt yt , . . .)) ≡ SUM(. . . , (ct + dt )yt , . . .
ðàâåíñòâî Add(c y , d y )
t t t t i ≡ (c + d ) y ðàçáîð ñëó÷àåâ.
t t i t
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12. Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ Ôðåãå
Ìîäåëèðîâàíèå ïðàâèë
ïðîñóììèðóåì ct yt ≥ c è dt yt ≥ d :
äîêàæåì ïî èíäóêöèè, ÷òî
Add(SUM(. . . , ct yt , . . .), SUM(. . . , dt yt , . . .)) ≡ SUM(. . . , (ct + dt )yt , . . .
ðàâåíñòâî Add(c y , d y ) ≡ (c + d ) y ðàçáîð ñëó÷àåâ.
t t t t i t t i t
y + ¬y àíàëîãè÷íî.
t t
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13. Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ Ôðåãå
Ìîäåëèðîâàíèå ïðàâèë
ïðîñóììèðóåì ct yt ≥ c è dt yt ≥ d :
äîêàæåì ïî èíäóêöèè, ÷òî
Add(SUM(. . . , ct yt , . . .), SUM(. . . , dt yt , . . .)) ≡ SUM(. . . , (ct + dt )yt , . . .
ðàâåíñòâî Add(c y , d y ) ≡ (c + d ) y ðàçáîð ñëó÷àåâ.
t t t t i t t i t
y + ¬y àíàëîãè÷íî.
t t
äîêàæåì F ≥G ∧ F ≥G ⊃ Add(F , F ) ≥ Add(G , G ).
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14. Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ Ôðåãå
Ìîäåëèðîâàíèå ïðàâèë
ïðîñóììèðóåì ct yt ≥ c è dt yt ≥ d :
äîêàæåì ïî èíäóêöèè, ÷òî
Add(SUM(. . . , ct yt , . . .), SUM(. . . , dt yt , . . .)) ≡ SUM(. . . , (ct + dt )yt , . . .
ðàâåíñòâî Add(c y , d y ) ≡ (c + d ) y ðàçáîð ñëó÷àåâ.
t t t t i t t i t
y + ¬y àíàëîãè÷íî.
t t
äîêàæåì F ≥G ∧ F ≥G ⊃ Add(F , F ) ≥ Add(G , G ).
óìíîæåíèå (äåëåíèå) íà êîíñòàíòó. . .
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15. Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ Ôðåãå
Ìîäåëèðîâàíèå ïðàâèë
ïðîñóììèðóåì ct yt ≥ c è dt yt ≥ d :
äîêàæåì ïî èíäóêöèè, ÷òî
Add(SUM(. . . , ct yt , . . .), SUM(. . . , dt yt , . . .)) ≡ SUM(. . . , (ct + dt )yt , . . .
ðàâåíñòâî Add(c y , d y ) ≡ (c + d ) y ðàçáîð ñëó÷àåâ.
t t t t i t t i t
y + ¬y àíàëîãè÷íî.
t t
äîêàæåì F ≥G ∧ F ≥G ⊃ Add(F , F ) ≥ Add(G , G ).
óìíîæåíèå (äåëåíèå) íà êîíñòàíòó. . .
îêðóãëåíèå
(act )yt ≥ ac + r
(r a)
ct yt ≥ c + 1
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16. Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ Ôðåãå
Ìîäåëèðîâàíèå ïðàâèë
ïðîñóììèðóåì ct yt ≥ c è dt yt ≥ d :
äîêàæåì ïî èíäóêöèè, ÷òî
Add(SUM(. . . , ct yt , . . .), SUM(. . . , dt yt , . . .)) ≡ SUM(. . . , (ct + dt )yt , . . .
ðàâåíñòâî Add(c y , d y ) ≡ (c + d ) y ðàçáîð ñëó÷àåâ.
t t t t i t t i t
y + ¬y àíàëîãè÷íî.
t t
äîêàæåì F ≥G ∧ F ≥G ⊃ Add(F , F ) ≥ Add(G , G ) .
óìíîæåíèå (äåëåíèå) íà êîíñòàíòó. . .
îêðóãëåíèå
(act )yt ≥ ac + r
(r a)
ct yt ≥ c + 1
ðàçáîð ñëó÷àåâ (ò.å. äîê-âî îò ïðîòèâíîãî):
SUM(. . . , ct yt , . . .) ≥ c + 1 ∨ ¬(SUM(. . . , ct yt , . . .) ≥ c + 1),
èç âòîðîãî ñëåäóåò ≤ c, óìíîæèì îáðàòíî íà a. . . .
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17. Ìîäåëèðîâàíèå ñåêóùèõ ïëîñêîñòåé â ñèñòåìàõ Ôðåãå
Ìîäåëèðîâàíèå ïðàâèë
ïðîñóììèðóåì ct yt ≥ c è dt yt ≥ d :
äîêàæåì ïî èíäóêöèè, ÷òî
Add(SUM(. . . , ct yt , . . .), SUM(. . . , dt yt , . . .)) ≡ SUM(. . . , (ct + dt )yt , . . .
ðàâåíñòâî Add(c y , d y ) ≡ (c + d ) y ðàçáîð ñëó÷àåâ.
t t t t i t t i t
y + ¬y àíàëîãè÷íî.
t t
äîêàæåì F ≥G ∧ F ≥G ⊃ Add(F , F ) ≥ Add(G , G ) .
óìíîæåíèå (äåëåíèå) íà êîíñòàíòó. . .
îêðóãëåíèå
(act )yt ≥ ac + r
(r a)
ct yt ≥ c + 1
ðàçáîð ñëó÷àåâ (ò.å. äîê-âî îò ïðîòèâíîãî):
SUM(. . . , ct yt , . . .) ≥ c + 1 ∨ ¬(SUM(. . . , ct yt , . . .) ≥ c + 1),
èç âòîðîãî ñëåäóåò ≤ c, óìíîæèì îáðàòíî íà a. . . .
ñâîéñòâà íóëÿ: Add(F , 0)i ≡ Fi è 0 1.
SUM(0y1 , . . . , 0yn ) ≥ 1, î÷åâèäíî, ëîæíî.
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18. 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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30. Îïòèìàëüíûå ïîëóàëãîðèòìû vs ñèñòåìû äîêàçàòåëüñòâ
Òåîðåìà @ur—j¡™ekD €udl¡kD IWVWA
%§ —
∃ p-îïòèìàëüíàÿ ñèñòåìà äîê-â ⇐⇒
∃ îïòèìàëüíûé ïîëóàëãîðèòì äëÿ TAUT.
=⇒:
Ïóñòü Π p-îïòèìàëüíàÿ.
Îïòèìàëüíûé ïîëóàëãîðèòì: ïàðàëëåëüíûé çàïóñê âñåõ Oi ,
ïðåòåíäóþùèõ íà âûäà÷ó Π-äîêàçàòåëüñòâ.
Âûäàííîå Oi äîê-âî ïðîâåðÿåòñÿΠ;
åñëè ïðàâèëüíîå âåðíóòü 1.
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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31. p -Optimal proof system from optimal acceptor
for any paddable language [Messner, 99]
he(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:
description of proof system Π;
Π-proof π of F ;
t
1 (for how long can we work?).
Verication:
run optimal acceptor on padL (x , π);
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.
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