D CO thg mo phon gM trong khong gian dg{n) vdi mot h^ng so
274 Tinhnan gia
D i n h nghia 4.2.2
P"^ la Idp cdc ngon ngU dxiOc khdng dinh bdi may Turing tat
dinh vdi ho phan tu van ve Ạ
NP"^ la Idp cdc ngon ngU duac khdng dinh bdi may Turing
khdng tat dinh vdi bg phan tu van ve Ạ
V i d u 4.2.2 N h i l da neu tren, may Turing tat dinh v6i bo phan t u van ve SAT c6 t h i khang dinh moi ngon ngii cua 16p N P trong thdi gian da thiic. Noi each khac, N P C P'^^^. Them nfla,
c o - N P C P ^ ^ ^ bdi VI P ^ ^ ^ la Idp phiic tap tat dinh nen bat bi6n
dudi tac dong ciia phep lay phan biị •
V i d u 4.2.3 Giong nhu p-^-^^ chiia nhiing ngon ngu: ma ta t i n la khdng thudc P , Idp NP'^'^^ cung chiia nhung ngon ngu: m^ t a t i n la khong thuoc N P .
That vay, ta nhac lai r^ng hai cong thiic Boole 0 va •0 vdi cac
bien x i , X 2 , . . . , Xfc la tuang ditang neu chung nhan cung mot gia t r i doi vdi bat cii mot phep gan t r i nao cho cac bien. Ta noi them rang mot cong thijfc la toi tiiu neu khdng mot cong thiic nho hdn nao tuong duong vdi nd. Gia sii
NONMIN-FORMULA = {(0) 10 khong la cong thiJc Boole tdi tigu}. Ngon ngii NONMIN-FORMULA dirdng nhu khdng thudc N P (tuy khdng biet lieu nd cd thuc sU thudc N P hay khdng). Tuy nhien,
NONMIN-FORMULA thudc N P ^ ^ ^ , bdi vi may Turing khdng tat
dinh thdi gian da thiic vdi bd phan t u van v6 SAT cd the kidm tra
xem lieu 0 cd la toi tigu hay khdng. Viec kigm t r a duoc tien hanh
nhu saụ Dau tien, t a nhan thay rang bai toan khdng tUOng dUOng
doi vdi hai cdng thilc Boole la bai toan thuoc N P , bdi v i may Turing
khong tat dinh cd kha nang chon duoc phep gan t r i de hai cdng thiic nhan nhiing gia t r i khac nhau va do đ bai toan tUOng duong thuQt^
c o - N P . Ti6p theo, dg khang dinh ngon ngii NONMIN-FORMULẠ
4.2 Phuang phdp quan he hog va van de P = NP 2751
tren dau vao (0) may Turing chon mot each khong t a t dinh mot
cdng thiic nhd hon rdi kiem tra xem lieu nd cd tUOng dudng vdi </> hay khong, bang each sijt dung t u van ve SAT, vh chap nhan neu
nhir la tuong duong. • N h u vay, thong qua su t u van ta cd the higu sau hon vg kha
nang cua phuong phap dudng cheo, cu thg la ve nhiing han chg cua phuong phap naỵ
4.2.2 H a n che c u a phifcfng p h a p di:f5ng cheo
Dinh ly sau day cho ta thay r i n g cd nhiing t u van ve A vh ve D
dg P ^ to ra khac NP"^ va dg P ^ to ra bang N P ^ . Hai t u van nay
la quan trong bdi v i sU ton t a i cija chung bao hieu rang t a khdng
chic se giai quyet duac van de P = N P bang each sii dung phuong
phap dudng cheọ
Vg ban chat, phuong phap dudng cheo la md phdng mot may Turing nay bdi mot may Turing khac. Viec md phdng duoc thuc hien sao cho may md phdng cd t h l xac dinh dupe hanh v i ciia may kia va khi ay hanh đng ngupc laị Gia sii ca hai may Turing nay dupe g i n ket vdi nhiing bp phan t u van giong nhaụ K h i đ, moi khi may dupe md phdng chat van bp phan t u van, may md phdng
Cling cd thg lam nhu vay va v i the viec md phdng cd t h i dign ra nhu trudc. Do vay, bat cii dinh ly nao lap luan ve may Turing chi bing each silt dung phuong phap dudng cheo se van giii nguygn gia tri neu ca hai may dupe trang bi ciing mot bp phan t u van.
Noi rigng, neu ta can chilng minh r i n g P va N P la khac nhau
bing phuong phap dudng cheo, ta cd thg kgt luan r i n g chiing khac
nhau nhd moi quan he vdi bat cii t u vkn nao thich hpp. Nhung P ^
yk N P ^ b i n g nhau, bdi the ket luan nhu vay la saị Do đ phuong
Phap dudng cheo la khdng du dg phan biet hai Idp P va N P . Tuong
khong mot chiing cut xac dang nao vg mot phep mo phdng don Sian cd the chiing to r i n g hai Idp nay la nhu nhau, bdi v i digu đ
276 T i n h nan giai se chilng to rang chung b^ng hhau nhd moi quan vdi mpt ttr van bat ky, nhiing su that la va N P ^ khac nhaụ
Dinh ly 4.2.3
(i) Co mgt tu van ve A ma nhd do P ^ 7^ N P " * .
(ii) Co mot tic van ve B ma nhd do P ^ = N P ^ .
ChUng minh Viec b i i u lo tir van B la kha dg dang. Vko v i t r i B ta
CO thg chon bat cii ngon ngiJ PS-day du nao, ching han nhil ngon ngii TQBF ma ta da xac dinh trong Muc 3.2.2.
Ta trirng bay A bang each xay dung. Viec xay difng A can dam bao r^ng mot ngon ngii nko do L{A) eua N P ' ^ ch^c han đi hoi su tim t o i vdi mpt no lue phi thirdng va nhd do L{A) khong thuoc P'^.
Khi xay dung can phai xem xet hoat dong eua tiing may Turing thdi gian da thilc vdi t u van dg dam bao rang moi may ay deu that bai trong viec k h i n g dinh ngon ngfl L{A).
Dau tien, doi vdi B = TQBF ta c6 cac bao ham thiJc sau: