Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Àëåêñàíäð Ðûáàëîâ ÎÔ ÈÌ ÑÎ ÐÀÍ, Îìñê Íîÿáðü, 2012 Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ïîäõîäû ê àëãîðèòìè÷åñêèì ïðîáëåìàì 1 Êëàññè÷åñêèé ïîäõîä ê âû÷èñëèìîñòè (Òüþðèíã, ×åð÷, Ìàðêîâ, Åðøîâ, Ãîí÷àðîâ è äð.). 2 Êëàññè÷åñêèé ïîäõîä ê ñëîæíîñòè âû÷èñëåíèé ñëîæíîñòü â õóäøåì ñëó÷àå (Êóê, Ëåâèí, Êàðï è äð.) Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ñëîæíîñòü â ñðåäíåì 1 Èíòóèòèâíî (íî íå òî÷íî) ñëîæíîñòü â ñðåäíåì ñðåäíåå ÷èñëî øàãîâ àëãîðèòìà ïî âñåì âõîäàì. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ñëîæíîñòü â ñðåäíåì 1 Èíòóèòèâíî (íî íå òî÷íî) ñëîæíîñòü â ñðåäíåì ñðåäíåå ÷èñëî øàãîâ àëãîðèòìà ïî âñåì âõîäàì. 2 Ëåâèí è Ãóðåâè÷ ââåë èíâàðèàíòíîå ïîíÿòèå ñëîæíîñòè â ñðåäíåì. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ñëîæíîñòü â ñðåäíåì 1 Èíòóèòèâíî (íî íå òî÷íî) ñëîæíîñòü â ñðåäíåì ñðåäíåå ÷èñëî øàãîâ àëãîðèòìà ïî âñåì âõîäàì. 2 Ëåâèí è Ãóðåâè÷ ââåë èíâàðèàíòíîå ïîíÿòèå ñëîæíîñòè â ñðåäíåì. 3 Òèïè÷íîå çàáëóæäåíèå: ñëîæíîñòü â ñðåäíåì = ÷èñëî øàãîâ ðàáîòû àëãîðèòìà íà òèïè÷íîì (ñëó÷àéíîì) âõîäå. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ñëîæíîñòü â ñðåäíåì 1 Èíòóèòèâíî (íî íå òî÷íî) ñëîæíîñòü â ñðåäíåì ñðåäíåå ÷èñëî øàãîâ àëãîðèòìà ïî âñåì âõîäàì. 2 Ëåâèí è Ãóðåâè÷ ââåë èíâàðèàíòíîå ïîíÿòèå ñëîæíîñòè â ñðåäíåì. 3 Òèïè÷íîå çàáëóæäåíèå: ñëîæíîñòü â ñðåäíåì = ÷èñëî øàãîâ ðàáîòû àëãîðèòìà íà òèïè÷íîì (ñëó÷àéíîì) âõîäå. 4 Ñëîæíîñòü â ñðåäíåì íå ïðèãîäíà äëÿ êðèïòîãðàôèè: åñòü àëãîðèòìû, êîòîðûå íà ïî÷òè âñåõ âõîäàõ ðàáîòàþò çà ëèíåéíîå âðåìÿ, à èõ ñëîæíîñòü ýêñïîíåíöèàëüíà. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ãåíåðè÷åñêàÿ ñëîæíîñòü 1 Ãåíåðè÷åñêèé ïîäõîä (Ðåìåñëåííèêîâ, Áîðîâèê, Ìÿñíèêîâ, Øóïï, Êàïîâè÷, Ðûáàëîâ, Äæîêóø). Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ãåíåðè÷åñêàÿ ñëîæíîñòü 1 Ãåíåðè÷åñêèé ïîäõîä (Ðåìåñëåííèêîâ, Áîðîâèê, Ìÿñíèêîâ, Øóïï, Êàïîâè÷, Ðûáàëîâ, Äæîêóø). 2 Ãåíåðè÷åñêàÿ ñëîæíîñòü = êëàññè÷åñêàÿ ñëîæíîñòü íà ìíîæåñòâå ïî÷òè âñåõ âõîäîâ. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ãåíåðè÷åñêàÿ ñëîæíîñòü 1 Ãåíåðè÷åñêèé ïîäõîä (Ðåìåñëåííèêîâ, Áîðîâèê, Ìÿñíèêîâ, Øóïï, Êàïîâè÷, Ðûáàëîâ, Äæîêóø). 2 Ãåíåðè÷åñêàÿ ñëîæíîñòü = êëàññè÷åñêàÿ ñëîæíîñòü íà ìíîæåñòâå ïî÷òè âñåõ âõîäîâ. 3 Ïðîáëåìà ãåíåðè÷åñêè ïîëèíîìèàëüíà, åñëè ñóùåñòâóåò ÷àñòè÷íûé àëãîðèòì äëÿ åå ðåøåíèÿ, êîòîðûé îñòàíàâëèâàåòñÿ íà ïî÷òè âñåõ âõîäàõ. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ãåíåðè÷åñêàÿ ñëîæíîñòü 1 Ãåíåðè÷åñêèé ïîäõîä (Ðåìåñëåííèêîâ, Áîðîâèê, Ìÿñíèêîâ, Øóïï, Êàïîâè÷, Ðûáàëîâ, Äæîêóø). 2 Ãåíåðè÷åñêàÿ ñëîæíîñòü = êëàññè÷åñêàÿ ñëîæíîñòü íà ìíîæåñòâå ïî÷òè âñåõ âõîäîâ. 3 Ïðîáëåìà ãåíåðè÷åñêè ïîëèíîìèàëüíà, åñëè ñóùåñòâóåò ÷àñòè÷íûé àëãîðèòì äëÿ åå ðåøåíèÿ, êîòîðûé îñòàíàâëèâàåòñÿ íà ïî÷òè âñåõ âõîäàõ. 4 Êàê è â ñëîæíîñòè â ñðåäíåì, ïðîáëåìû ðàññìàòðèâàþòñÿ ñ çàäàííîé ìåðîé íà ìíîæåñòâå âõîäîâ. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ãåíåðè÷åñêàÿ ñëîæíîñòü 1 Ãåíåðè÷åñêèé ïîäõîä (Ðåìåñëåííèêîâ, Áîðîâèê, Ìÿñíèêîâ, Øóïï, Êàïîâè÷, Ðûáàëîâ, Äæîêóø). 2 Ãåíåðè÷åñêàÿ ñëîæíîñòü = êëàññè÷åñêàÿ ñëîæíîñòü íà ìíîæåñòâå ïî÷òè âñåõ âõîäîâ. 3 Ïðîáëåìà ãåíåðè÷åñêè ïîëèíîìèàëüíà, åñëè ñóùåñòâóåò ÷àñòè÷íûé àëãîðèòì äëÿ åå ðåøåíèÿ, êîòîðûé îñòàíàâëèâàåòñÿ íà ïî÷òè âñåõ âõîäàõ. 4 Êàê è â ñëîæíîñòè â ñðåäíåì, ïðîáëåìû ðàññìàòðèâàþòñÿ ñ çàäàííîé ìåðîé íà ìíîæåñòâå âõîäîâ. 5 Ìîæíî è èíòåðåñíî ðàññìàòðèâàòü ãåíåðè÷åñêóþ ñëîæíîñòü íåðàçðåøèìûõ ïðîáëåì. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Êëàññè÷åñêèé ïîäõîä Àëãîðèòì (áûñòðî) ðàáîòàåò íà âñåõ âõîäàõ. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ñëîæíîñòü â ñðåäíåì Àëãîðèòì áûñòðî ðàáîòàåò íà ïî÷òè âñåõ âõîäàõ, ìåäëåííî íà ïëîõèõ, íî M(Tn ) ≤ p(n). Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ãåíåðè÷åñêèé ïîäõîä Àëãîðèòì áûñòðî ðàáîòàåò íà ïî÷òè âñåõ âõîäàõ è èãíîðèðóåò ïëîõèå. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ñðàâíåíèå ïîäõîäîâ Ðàçðåøèìîñòü (çà ïîëèíîìèàëüíîå âðåìÿ): Êëàññè÷åñêè ðàçðåøèìà ⇒  ñðåäíåì ðàçðåøèìà ⇒ Ãåíåðè÷åñêè ðàçðåøèìà. Êëàññè÷åñêè ðàçðåøèìà :  ñðåäíåì ðàçðåøèìà : Ãåíåðè÷åñêè ðàçðåøèìà. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Èñòîðèÿ ïîÿâëåíèÿ ãåíåðè÷åñêîãî ïîäõîäà 1 I. Kapovich, A. Myasnikov, P. Schupp, V. Shpilrain. Generic-case complexity, decision problems in group theory and random walks. J. Algebra, vol. 264, no. 2 (2003), pp. 665694. 2 A.Borovik, A.Myasnikov, V.Remeslennikov Multiplicative measures on free groups // Int. Journal of Algebra and Computations, V.13, n.6 (2003) pp. 705731. 3 I.Kapovich, A.Myasnikov, P.Schupp, V.Shpilrain Avrage-case complexity of the word and membership problems in group theory // Advances in Mathematics, V.190, 2005, n.2,pp. 343359. 4 A. V. Borovik, A. G. Myasnikov, V. N. Remeslennikov, The conjugacy problem in amalgamated products I: Regular elements and black holes // Intern. J. of Algebra and Computation, 17:7 (2007), 1299-1333. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Àñèìïòîòè÷åñêàÿ ïëîòíîñòü Îïðåäåëåíèå Ïóñòü I âñå âõîäû, In âñå âõîäû ðàçìåðà n. Àñèìïòîòè÷åñêàÿ ïëîòíîñòü ìíîæåñòâà S ⊆ I µ(S) = lim n→∞ |S ∩ In | . |In | Çàìå÷àíèå |S∩In | |In | âåðîÿòíîñòü ïîëó÷èòü âõîä èç S åñëè ìû ãåíåðèðóåì âõîäû ðàçìåðà n ñëó÷àéíî è ðàâíîìåðíî. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ãåíåðè÷åñêèå ìíîæåñòâà Îïðåäåëåíèå Ìíîæåñòâî âõîäîâ S ⊆ I íàçûâàåòñÿ ãåíåðè÷åñêèì åñëè µ(S) = 1 ïðåíåáðåæèìûì åñëè µ(S) = 0 Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ïðèìåðû ãåíåðè÷åñêèõ è ïðåíåáðåæèìûõ ìíîæåñòâ I = N ìíîæåñòâî íàòóðàëüíûõ ÷èñåë, ïðåäñòàâëåííûõ äâîè÷íûìè ñòðîêàìè, ðàçìåð n äëèíà ñòðîêè n. Ìíîæåñòâî ïðîñòûõ ÷èñåë P ïðåíåáðåæèìî, ò.ê. |P ∩ In | π(2n ) = =O |In | 2n 1 . n Ìíîæåñòâî ñîñòàâíûõ ÷èñåë C ãåíåðè÷åñêîå, ò.ê. C = N \ P. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ãåíåðè÷åñêàÿ ðàçðåøèìîñòü Îïðåäåëåíèå S ⊆ I ãåíåðè÷åñêè ðàçðåøèìî (çà ïîëèíîìèàëüíîå âðåìÿ) åñëè ñóùåñòâóåò ìíîæåñòâî G òàêîå, ÷òî: 1 G ãåíåðè÷åñêîå 2 G ðàçðåøèìî (çà ïîëèíîìèàëüíîå âðåìÿ) 3 S ∩ G ðàçðåøèìî (çà ïîëèíîìèàëüíîå âðåìÿ) Ãåíåðè÷åñêèé àëãîðèòì A äëÿ S ðàáîòàåò íà âõîäå a òàê: 1 2 3 ïðîâåðÿåò a ∈ G ? åñëè a ∈ / G , âûäàåò "ÍÅ ÇÍÀÞ!" åñëè a ∈ G ðåøàåò a ∈ S ∩ G ? Îïðåäåëåíèå Ìíîæåñòâî BH(A) = I \ G íàçûâàåòñÿ ÷åðíîé äûðîé àëãîðèòìà A. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ïðèìåðû 1 Ïðîáëåìà îñòàíîâêè äëÿ ìàøèí Òüþðèíãà ñ îäíîíàïðàâëåííîé ëåíòîé (Õýìêèíñ, Ìÿñíèêîâ, 2006) 2 Ïðîáëåìà ðàâåíñòâà ñëîâ äëÿ ïîëóãðóïï Ìàðêîâà, Ïîñòà, Öåéòèíà è Ìàòèÿñåâè÷à (Ìÿñíèêîâ, Óøàêîâ, Äîíã Âóê Âîí 2005). 3 Çàäà÷à ëèíåéíîãî ïðîãðàììèðîâàíèÿ. Ñèìïëåêñ-ìåòîä ãåíåðè÷åñêè ýôôåêòèâåí (Ñìåéë, Âåðøèê, 1983). 4 Ïðîáëåìà èçîìîðôèçìà ãðàôîâ (Ýðäåø, Êàðï, Ëèïòîí, 1970-å). 5 Ìíîãî÷èñëåííûå NP -ïîëíûå ïðîáëåìû. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ïðîáëåìà èçîìîðôèçìà ãðàôîâ Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ïðîáëåìà èçîìîðôèçìà ãðàôîâ ÂÕÎÄ: Ãðàôû G1 è G2 ñ îäèíàêîâûì ÷èñëîì âåðøèí, çàäàííûå ìàòðèöàìè ñìåæíîñòè M(G1 ) è M(G2 ). ÂÛÕÎÄ: ÄÀ, åñëè ãðàôû G1 è G2 èçîìîðôíû, ÍÅÒ, èíà÷å. Ðàçìåð âõîäà = ÷èñëî âåðøèí â ãðàôàõ. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ïðîáëåìà èçîìîðôèçìà ãðàôîâ Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ãåíåðè÷åñêàÿ ëåãêîðàçðåøèìîñòü Èà ÀËÃÎÐÈÒÌ: ñðàâíèòü ÷èñëî ðåáåð G1 è G2 (÷èñëî 1 â ìàòðèöàõ ñìåæíîñòè). Åñëè îíî ðàçíîå, òî âûäàòü ÍÅÒ, èíà÷å ÍÅ ÇÍÀÞ. Äîëÿ ïàð ìàòðèö ñ îäèíàêîâûì ÷èñëîì 1: Pn2 i i i=0 Cn2 Cn2 2 2n 2 = √ n C2n 2 2 n2 = (2n2 )! ∼ 2n2 (n2 !)2 2 4πn2 (2n2 /e)2n 1 √ ∼ n2 → 0. 2 = 2 2n πn 2 · 2π(n /e) Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ãåíåðè÷åñêîé ðàçðåøèìîñòè ìàëî! Ïóñòü G ãåíåðè÷åñêîå ïîëèíîìèàëüíîå ìíîæåñòâî òàêîå, ÷òî n−1 |G ∩ In | = . |In | n Áûñòðûé âåðîÿòíîñòíûé àëãîðèòì äëÿ ãåíåðàöèè âõîäîâ x ∈ / G: Øàã 1. Ñãåíåðèðîâàòü ñëó÷àéíûé âõîä x ðàçìåðà n. Øàã 2. Åñëè x ∈ G , òî ïîâòîðèòü øàã 1, èíà÷å çàêîí÷èòü. Âåðîÿòíîñòü ïîëó÷àòü âõîäû òîëüêî èç G çà n2 ðàóíäîâ: n−1 n n 2 = Àëåêñàíäð Ðûáàëîâ 1 1− n n n → e −n Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ñèëüíî ãåíåðè÷åñêàÿ ðàçðåøèìîñòü Îïðåäåëåíèå Ìíîæåñòâî âõîäîâ S ⊆ I íàçûâàåòñÿ ñèëüíî ãåíåðè÷åñêèì åñëè ïîñëåäîâàòåëüíîñòü ρn = |S ∩ In | , n = 1, 2, 3, . . . |In | ýêñïîíåíöèàëüíî áûñòðî ñòðåìèòñÿ ê 1. Îïðåäåëåíèå Ïðîáëåìà S ⊆ I ñèëüíî ãåíåðè÷åñêè ðàçðåøèìà (çà ïîëèíîìèàëüíîå âðåìÿ) åñëè ñóùåñòâóåò ìíîæåñòâî G òàêîå, ÷òî 1 G ñèëüíî ãåíåðè÷åñêîå 2 G ðàçðåøèìî (çà ïîëèíîìèàëüíîå âðåìÿ) 3 S ∩ G ðàçðåøèìî (çà ïîëèíîìèàëüíîå âðåìÿ) Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Íåðàçðåøèìîñòü íà ãåíåðè÷åñêèõ ìíîæåñòâàõ Îïðåäåëåíèå Ìíîæåñòâî S ⊆ I íàçûâàåòñÿ ñèëüíî íåðàçðåøèìûì åñëè S íå ñèëüíî ãåíåðè÷åñêè ðàçðåøèìî, ò.å. íåðàçðåøèìî íà ëþáîì ðåêóðñèâíîì ñèëüíî ãåíåðè÷åñêîì ìíîæåñòâå ñóïåð íåðàçðåøèìûì åñëè S íå ãåíåðè÷åñêè ðàçðåøèìî, ò.å. íåðàçðåøèìî íà ëþáîì ðåêóðñèâíîì ãåíåðè÷åñêîì ìíîæåñòâå àáñîëþòíî íåðàçðåøèìûì åñëè S íåðàçðåøèìî íà ëþáîì ðåêóðñèâíîì íå ïðåíåáðåæèìîì ìíîæåñòâå Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ãåíåðè÷åñêàÿ àìïëèôèêàöèÿ Îïðåäåëåíèå Ïóñòü I è J ìíîæåñòâà. Êëîíèðîâàíèåì I â J íàçûâàåòñÿ ôóíêöèÿ C : I → P(J) òàêàÿ, ÷òî 1 ∀x, y ∈ I x 6= y → C (x) ∩ C (y ) = ∅ 2 Åñòü àëãîðèòì E : I × N → J òàêîé, ÷òî äëÿ âñåõ x ∈ I C (x) = {E (x, 0), E (x, 1), . . . , } Îïðåäåëåíèå Êëîíèðîâàíèå S ⊆ I ýòî C (S) = S x∈S C (x). Îïðåäåëåíèå Êëîíèðîâàíèå C ñóùåñòâåííîå åñëè µ(C (x)) > 0 äëÿ âñåõ x ∈ I è ñèëüíî ñóùåñòâåííî åñëè äëÿ âñåõ x ∈ I ìíîæåñòâî C (x) íå ñèëüíî ïðåíåáðåæèìî. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ãåíåðè÷åñêàÿ àìïëèôèêàöèÿ Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ãåíåðè÷åñêàÿ àìïëèôèêàöèÿ Òåîðåìà (Ìÿñíèêîâ, Ðûáàëîâ, 2008) Ïóñòü C : I → P(J) êëîíèðîâàíèå è S ⊆ I íåðàçðåøèìî. Òîãäà 1 Åñëè C ñóùåñòâåííî, òî C (S) ñóïåð íåðàçðåøèìî. 2 Åñëè C ñèëüíî ñóùåñòâåííîå, òî C (S) ñèëüíî íåðàçðåøèìî. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ïðîáëåìà îñòàíîâêè Òåîðåìà (Ðûáàëîâ, 2007) Ïðîáëåìà îñòàíîâêè ñèëüíî íåðàçðåøèìà. P ìíîæåñòâî ïðîãðàìì âñåõ ìàøèí Òüþðèíãà. Êëîíèðîâàíèå C : P → P(P) äîáàâëÿåò íîâûå ñîñòîÿíèÿ è ê ïðîãðàììå (q1 , 0) → (qj1 , t1 , D1 ), (q1 , 1) → (qj2 , t2 , D2 ), ... (qn , 1) → (qj2n , t2n , D2n ) ïðîèçâîëüíûå êîìàíäû äëÿ íîâûõ ñîñòîÿíèé (qn+1 , 0) → (qk1 , u1 , E1 ), (qn+1 , 1) → (qk2 , u2 , E2 ), ... (qn+m , 1) → (qk2m , u2m , E2m ) Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ïðîáëåìà ðàâåíñòâà â ïîëóãðóïïàõ S = ha1 , . . . , am |Ri ïîëóãðóïïà ñ íåðàçðåøèìîé ïðîáëåìîé ðàâåíñòâà. Òåîðåìà (Ìÿñíèêîâ, Ðûáàëîâ, 2008)  ïîëóãðóïïå S + (x) = hA, x|R, xai = x, xx = xi ïðîáëåìà ðàâåíñòâà ñóïåð íåðàçðåøèìà. Ïóñòü A = {a1 , . . . , an } è Ax = A ∪ {x}. Êëîíèðîâàíèå îòîáðàæåíèå èç A∗ × A∗ â A∗x × A∗x : C (w1 , w2 ) = {(w1 xv , w2 xu) : v , u ∈ A∗x }. Òåîðåìà (Ãèëìàí, Ìÿñíèêîâ, Îñèí, 2010) Ñóùåñòâóåò êîíå÷íî îïðåäåëåííàÿ ãðóïïà ñ ñèëüíî íåðàçðåøèìîé ïðîáëåìîé ðàâåíñòâà. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ãåíåðè÷åñêàÿ ñëîæíîñòü òåîðèé ïåðâîãî ïîðÿäêà 1 Ïóñòü Th(A) íåðàçðåøèìàÿ òåîðèÿ. Áóäåò ëè Th(A) ãåíåðè÷åñêè ðàçðåøèìîé? 2 Ïóñòü Th(A) ðàçðåøèìàÿ òåîðèÿ ñ âûñîêîé âû÷èñëèòåëüíîé ñëîæíîñòüþ. Áóäåò ëè Th(A) ãåíåðè÷åñêè ðàçðåøèìîé çà ïîëèíîìèàëüíîå âðåìÿ? Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ïðåäñòàâëåíèå ôîðìóë ∀x1 ∀x2 ∃x3 (((x1 = x2 ) ∨ (x2 = x1 + x3 ))&(x1 = x2 x3 )) ∨ (x3 6= x2 x1 ) Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ãåíåðè÷åñêàÿ ñëîæíîñòü òåîðèé 1 Åñëè òåîðèÿ íåðàçðåøèìà, òî îíà ñèëüíî íåðàçðåøèìà (Ðûáàëîâ, Ìÿñíèêîâ, 2008). 2 Àðèôìåòèêà Ïðåñáóðãåðà ñèëüíî íåðàçðåøèìà çà ýêñïîíåíöèàëüíîå âðåìÿ (Ðûáàëîâ, 2010). 3 Ïðè óñëîâèè P = BPP òåîðèÿ óïîðÿäî÷åííîãî ïîëÿ R ñèëüíî íåðàçðåøèìà çà ïîëèíîìèàëüíîå âðåìÿ (Ðûáàëîâ, Ôåäîñîâ, 2011). 4 Äåñÿòàÿ ïðîáëåìà Ãèëüáåðòà ñèëüíî íåðàçðåøèìà (Ðûáàëîâ 2011). 5 Ïðè óñëîâèè P = BPP è P 6= NP ïðîáëåìà âûïîëíèìîñòè áóëåâûõ ôîðìóë ñèëüíî íåðàçðåøèìà çà ïîëèíîìèàëüíîå âðåìÿ (Ðûáàëîâ, 2012). 6 Ïðè óñëîâèè P = BPP è P 6= PSPACE ïðîáëåìà èñòèííîñòè áóëåâûõ ôîðìóë ñèëüíî íåðàçðåøèìà çà ïîëèíîìèàëüíîå âðåìÿ (Ðûáàëîâ, 2012). Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Ýôôåêòèâíàÿ ãåíåðàöèÿ ïëîõèõ âõîäîâ Òåîðåìà (Ðûáàëîâ, 2012) Äëÿ ëþáîãî ïîëèíîìèàëüíîãî ãåíåðè÷åñêîãî àëãîðèòìà A, ðåøàþùåãî Äåñÿòóþ ïðîáëåìó Ãèëüáåðòà. Ïðîáëåìó îñòàíîâêè äëÿ ìàøèí Òüþðèíãà. Íåðàçðåøèìóþ òåîðèþ ïåðâîãî ïîðÿäêà. Àðèôìåòèêó Ïðåñáóðãåðà. ñóùåñòâóåò ïîëèíîìèàëüíûé âåðîÿòíîñòíûé àëãîðèòì, êîòîðûé ïî ÷èñëó n ãåíåðèðóåò âõîä èç BH(A) ñ âåðîÿòíîñòüþ > 1 − e1n . Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Îòêðûòûå ïðîáëåìû Áóäóò ëè ãåíåðè÷åñêè ðàçðåøèìû (çà ïîëèíîìèàëüíîå âðåìÿ) ñëåäóþùèå ïðîáëåìû: 1) Ôîðìàëüíàÿ àðèôìåòèêà? 2) Àðèôìåòèêà Ïðåñáóðãåðà? 3) Óïîðÿäî÷åííîå ïîëå âåùåñòâåííûõ ÷èñåë? 4) Ïîëå p -àäè÷åñêèõ ÷èñåë? 5) Ïðîáåìà èñòèííîñòè áóëåâûõ ôîðìóë? 6) Ïðîáëåìà âûïîëíèìîñòè áóëåâûõ ôîðìóë? 7) Ïðîáëåìà îñòàíîâêè äëÿ ìàøèí ñ äâóíàïðàâëåííîé ëåíòîé? 8) Äåñÿòàÿ ïðîáëåìà Ãèëüáåðòà? 9) Ïðèìåð ê.ï. ãðóïïû ñ ñóïåð íåðàçðåøèìîé ïðîáëåìîé ðàâåíñòâà. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì ×òî äàåò ãåíåðè÷åñêèé ïîäõîä? 1 Âîçíèêëè íîâûå ïîñòàíîâêè àëãîðèòìè÷åñêèõ ïðîáëåì, ñâÿçàííûå ñ êëàññè÷åñêèìè àëãîðèòìè÷åñêèìè ïðîáëåìàìè. 2 Íîâûå ìåòîäû îöåíêè õîðîøèõ ÷àñòåé àëãîðèòìè÷åñêèõ ïðîáëåì, ÷åðíûõ äûð àëãîðèòìîâ (êîìáèíàòîðíûå è òåîðåòèêî-âåðîÿòíîñòíûå ìåòîäû, öåïè Ìàðêîâà è ò.ä.). 3 Ãåíåðè÷åñêàÿ òåîðèÿ ðåêóðñèè (ðàáîòû Äæîêóøà, ãåíåðè÷åñêàÿ èåðàðõèÿ Åðøîâà è äð.). 4 Íîâûå ìåòîäû îöåíêè ñòîéêîñòè êðèïòîãðàôè÷åñêèõ ñèñòåì (ñèñòåìû, îñíîâàííûå íà íåêîììóòàòèâíûõ ãðóïïàõ). Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì Áèáëèîãðàôèÿ 1 2 3 4 5 6 I. Kapovich, A. Myasnikov, P. Schupp, V. Shpilrain. Generic-case complexity, decision problems in group theory and random walks. J. Algebra, vol. 264, no. 2 (2003), pp. 665694. A. Rybalov. On the strongly generic undecidability of the Halting Problem. Theoretical Computer Science, vol. 377 (2007), pp. 268270. A. Myasnikov, A. Rybalov. Generic complexity of undecidable problems. Journal of Symbolic Logic, vol. 73, no. 2 (2008), pp. 656673. A. Rybalov. Generic Complexity of Presburger Arithmetic. Theory of Computing Systems, vol. 46, no. 1, (2010), pp. 28. À. Ðûáàëîâ, Â. Ôåäîñîâ. Î ãåíåðè÷åñêîé ñëîæíîñòè àëãåáðû Òàðñêîãî // Âåñòíèê Îìñêîãî óíèâåðñèòåòà, 2, 2011, Ñ. 39-43. À. Ðûáàëîâ. Î ãåíåðè÷åñêîé íåðàçðåøèìîñòè Äåñÿòîé ïðîáëåìû Ãèëüáåðòà // Âåñòíèê Îìñêîãî óíèâåðñèòåòà, 4, 2011, Ñ. 19-22. Àëåêñàíäð Ðûáàëîâ Ãåíåðè÷åñêèé ïîäõîä ê àëãîðèòìè÷åñêèì ïðîáëåìàì