D CO thg mo phon gM trong khong gian dg{n) vdi mot h^ng so
d nao ọ Do g{n) la o[/(rt)], nen ton tai mot h^ng so no sao cho
(ioíi^) < fin) doi vdi moi n > nọ V i the, qua trinh ma D mo phong
M se thuc hien day du ngay ca doi vdi nhflng dau vao do dai no hoac Idn hon. Ta hay xem dieu gi xay ra khi D tinh toan tren dau vao (y\/)10"ọ Dau vao nay c6 do dai Idn hon no, cho ngn qua t r i n h mo phong trong Budc 4 se duoc thuc hien day dụ Bdi vay, D se thuc hien nguoc lai vdi M trgn ciing mot dau vaọ Do do M khong thg khang dinh L, trai vdi gia thigt cua tạ N h u vay L khong duoc
khang dinh trong khong gian o [ / ( n ) ] . •
He qua 4.1.3 Ddi vdi bat ky hai ham / i , /2 : N —> N ma / i ( n )
o[/2(n)] vd f2 Id ham khong gian kien thiet dxiac,
S P A C E ( / i ( n ) ) C SPACE(/2(n)). •
He qua nay cho phep ta phan biet cac Idp phiic tap khong gian. T h i du, ta c6 thg chiJng to rang ham ri*^ la khong gian kien thiet duoc ^6i vdi bat ky so t u nhien A,'. Do d6, doi vdi bat ky hai s6 t u nhien ^1 < A;2, t a dg dang k h i n g dinh rang SpACE(n'=i) C SpACE(n'=2).
266 Tinh nan giai
Bang each phan tieh t i mi ta c6 t h i chilng to rang n'^ la hkm khong | gian ki6n thiet duae doi vdi bat ky so hiiu t i c > 0 va nh5 do bao
ham thilc viJta neu v l n dung d6i vdi bat ky hai so hiiu t i 0 < ci < d.
Nhan thay rang giiia hai so thuc bat ky r i < ton tai hai so hau
ti ci va C2, sao cho r i < ci < C2 < r-ị Vay la ta thu duoe them mot
he qua biiu thi trat tu tinh te trong pham vi Idp phiic tap P S . He qua 4.1.4 Doi vdi bat ky hai so thicc 0<n <r2,
S P A C E K ' ) $ SPACECÚ"^). •
Ta cung c6 thg sijt dung dinh ly ve trat tu khong gian dg phan biet hai Idp phtrc tap khong gian quan trong ma d cuoi Chilong 3 ta da de cap den.
He qua 4.1.5
N L S C P S .
Chitng minh Dinh ly Savitch chirng to rang N L S C SpACE(log^ n)
va dinh ly ve trat tu khong gian (Dinh ly 4.1.2) chiing to rang
SpACE(log^ n) C SPACE(n). Ttt day ta suy ra he quạ
Nhu da thay d cuoi Muc 3.4.2, su khac nhau nay cho phep ta kgt luan rkng TQBF ^ N L S , bdi vi TQBFlh PS-day du theo quy
d i n khong gian logạ • Bay gid ta tien tdi muc tieu cua chuong nay: Chiing minh sii ton
tai cac hai todn ma vi nguyen tdc la gidi duoc nhung tren thuc te thi khong, tiic cdc bdi todn gidi duac nhung rat gian nan. Doi vdi bat ky so tu nhien k, trtng Idp SpACE(n'') chila trong 16p SPACE(n'°^"). Tiep theo, Idp SpACE(n'°s") thuc sir chiia trong Idp SPACE(2"), nghia la SpACE(n'°s") C SPACE(2"). Bdi vay, ta thu dildc them he qua nh^m phan biet Idp P S vdi Idp E S = SPACE(2"'').
He qua 4.1.6
P S C E S
4.1 Trat tu cua cdc Idp phiic tap 267
4.1.2 Trat tu" thdi gian
Tren day ta vvia chi5ng to su t6n tai cac bai toan nan giai v^ mat khong gian tinh toan, tire nhiing bai toan giai dugc v^ mat ly thuygt nhung vdi mot luong khong gian (bo nhd) tinh toan Idn den mile ma thuc tg khong thg nao dap ling n6ị Cu t h i , theo He qua 4.1.6, ton tai ngon ngfl ma dugc khing dinh trong khong gian ham mu, nhung khong thg duoc khang dinh trong khong gian da thiic. Tuy vay, su thg hien ngon ngii ay cd phan gugng gao, chi tien Iqi doi vdi muc dich phan biet cac Idp phiic tap. 6 day, ta cung se sii dung chinh ngon ngu ay dg chiing minh tinh nan giai cua cac ngon ngQ khac, tu nhign hon, sau khi trinh bay dinh ly ve trat tu thdi gian.
Dinh nghIa 4.1.7 Ham t : N —> N, md t{n) > 0 [ n log
duac goi Id thdi gian kiin thiet diidc {time constructible), neu ton tai mot ham tinh ducfc trong thdi gian 0[t{n)] sao cho mdi tic
1" duac biin doi thdnh biiu diin nhi phan cua t{n).
Ndi each khac, t la ham thdi gian kign thigt dugc neu ton tai mot may Turing tat dinh đng M vdi do phiic tap thdi gian 0[t{n)
sao cho, khi tinh toan trgn m5i tir vao 1 " , may M cho ket qua la bieu dign nhi phan cua t{n).
Thi du, ta cd thg chiing minh r^ng mgi ham thudng gap ma it
nhat bang 0['n logn], nhu nlogsn, ny/n, n? v a 2", deu la thdi gian
kien thiet dugc. Chang han, doi vdi cac ham n logj n va n^, ta nhan thay r^ng nhiing may Turing dugc xay dung trong V i du 4.1.1, nham bien doi t i i vao 1 " thanh cac bigu dien nhi phan cua n l o g j n va n\ do phiic tap thdi gian 0[n log2n] va 0[n^] tiiong ling.
Doi vdi do phiic tap thdi gian ta cd dinh ly ve trat tu thdi gian tugng tu nhu dinh ly ve trat tu khong gian, Dinh ly 4.1.2. The nhung, vi ly do ky thuat se xuat hien trong chiing minh dinh ly, nen dan den mot trat tu thdi gian long leo hon khong dang kg so vdi trat tu khong gian ma ta da thu dugc. Trong khi bdt cd mot
268 Tinh nangiai