V sao cho khong mot sinh vign nao thi hai mon trong ciing mpt • bu6ị Noi dung bai toan bao gom viec xac dinh xem lieu c
2. Thuc hien cdc budc lap tiep theo (dg tiet kiem trang thai,
khi bat dau moi budc lap tiep theo, may cung 6 trang thai ban dau qo y nhir k h i may bat dau tien h^nh budc lap thiJf nhat. Ta l u u y them r^ng, k h i d trang t h a i 92, may thuc hien viec cOng them mot don v i trong he n h i phan vko ket qua tilc t h d i ) :
<^(93,0) = ( 9 o , 0, R ) 5(gi,tt) = (g2,ti,R)
(5(92, l ) = ( g 2 , 0, R ) 5{q2,0) = {qs,l,L) I
% 3, 0 ) = ( g 3, 0, L ) .
31 Do phitc tap khong gian cua cdc may Turing 213'
3. Ket thuc:
3ạ Dao nguqc sang bgn trai dau tJ
S{qo,^) = {q,,lR) S{qlx) = {qlx,L) 6{qlx) = {qlx,L), v6ix = 0, l, t t , ® S{ql 0 ) = (96, 0 , R) 5{ql 0 ) = {qe, 1 , R) ( 5 ( 9 6 , 0 ) = (96, 0, R ) 6{qe,l) = {qe,l,R) 6{qeA) = {q4,lR) ^(94, ®) = (^4, ®, R); 3b. Chuyen den hinh thai chap nhan q^n
1 ^ ( 5 ( 9 4 , 0 ) = ( 9 7 , 0 , L) 5{q7,0) = (97, 0 , L)
H &{q7, ft) = (97, 0 , L) S{qr, 0 ) = {q,, 0 , L)
( 5 ( 9 7 , 1 ) = ( 9 7 , 1 , L) 5 ( 9 7 , 0 ) = ( 9 v > 0, R ) -
^ ^ N h a n thay r^ng sau m5i bitdc lap, so o ma may B sii dung khong tang them va thudng tifng budc giam d i 1, t i t cQ n o xuong con khoang 2 + log^n, so 6 diing de ghi dau ngan each tl v a kgt qua tGc thdi n ^ . Do do, may B c6 do phiic tap khong gian S B { n ) = 0[n .
Cu thg hdn, dg xac dinh chinh xac S B ( n ) , t a quan sat ky qua t r i n h
thuc hien budc lap tM nhat v a nhan ra r^ng vao thdi digm ket *^huc budc lap nay, may sii dung dung n + 2 0 , bdi v i k h i ay ngay ^au tiit vao da dUdc ghi thgm tjl va dau doc-ghi dang 6 dau tien trgn ^S.ng, t u y la o trong nhitng theo quy dinh v i n ditgc t i n h . Vao thdi ^i^m ay, may B cd hinh t h a i 9 3 0 I . • • I J t l vdi n - 1 ky t u 1 trudc
2l4 Do phijfc tap khong gian
V i d u 3.1.3 Trong Chvrang 2 ta da xem xet bai toan SAT.
Day \h mot bai toan NP-day du, cho nen no kho c6 t h i giai diroc
bdi thuat toan thdi gian da thutc va do do cang kho giai duoc bdi thuat toan t h d i gian tuyen tinh. Tuy nhien, sau day t a chutng to r^ng bai toan SAT giai duac bcii thuat toan khong gian tuygn tinh. Co duac ket qua nay la do khong gian diing cho t i n h toan c6 thg diidc sii dung lai, c5n thdi gian t h i khong. Trong v i d u viia neu,
viec sii dung lai cac o tren bang da dudc d i l n ra 6 Bxidc 3a ciia may Turing B. Nhxl vay, khong gian diing cho t i n h toan xem ra
linh hoat hon thdi gian.
Bay gid, dg khang dinh ngon ngii SAT trong khong gian tuyen t i n h , t a xay dung may Turing M nhtr sau (luu y rang, tap cac gia
t r i chan ly { D U N G , SAI} duoc thay the bdi tap { 1 , 0 } ) .
M = " T r g n m 5 i tilt vao (0), trong do 0 la mot cong thiic Boole:
1. Doi vdi m6i each gan t r i chan ly a 6 { 0 , 1 } ' " cho cac bien
xi,... ,Xm cua cong thutc 0:
l a . Tien hanh gan t r i a cho cac bien cua (f);
l b . T i n h gia t r i cua 0 trgn ạ ' 2. Ngu 0 vao mot luc nao do nhan gia t r i 1, t h i chap nhan;
ngu khong, bdc bọ"
R6 rang may Af t i n h toan trong khong gian tuyen t i n h , bdi vi
khi lap lai viec t i n h toan gia t r i cua ham 0 vdi mot phep gan t r i
chan ly nao do, may A/ sii dung lai chinh nhiJng o tren bang ma no da diing doi vdi phep gan t r i trirdc. Trong qua t r i n h t i n h toan,
ngoai khoang khong gian l u u triJ dfl: lieu dau vao (0), may chi ckn
them mOt luong khong gian 0 [ m ] diing dg ghi nhd t a m thdi nhiing
gia t r i chan ly ma may tign hanh gan cho cac bien x i , . . . , x ^ - Do rn khong thg vuot qua n , do dai cua dau vao, cho ngn may M t h i f c '
hien viec t i n h toan trong khong gian 0 [ n ] . 0
3.1 Do phvcc tap khong gian cua cac may Turing 215
3.1.2 D o phiJc tap khong gian c u a m a y T u r i n g tat d i n h n h i e u bang tat d i n h n h i e u bang
N h u thirdng le, do phiic tap khong gian cua may T u r i n g t a t dinh nhigu bang cQng duac xac dinh trong "trudng hdp xau nhat". Doi vdi may Turing nhigu bang, ngoai "dau vao xau nhat" va " t h d i digm xau nhat", t a con phai xet den ca "bang xau nhat", tiic bang ma tren do may sijt dung nhieu o nhat.
D i n h nghia 3.1.2 Cho M la mot may Turing nhieu bdng diing.
Dp phUCc tap khong gian cua may Turing nhieu bang {space
complexity of multitape Turing machine) M la ham s : N —> N, trong do gia tri s{n) la so toi da cac o tren mdi bang ma may M sxt dung tai mdi thdi diem trong qua trlnh tinh toan cua may tren bat cti tit vao nao dO dai n.
T h i du, t a hay xac dinh do phiic tap khong gian ciia may Turing
hai bang Ấ^\c mo t a trong Muc 2.1.2 n h ^ m k h i n g d i n h ngon
ngfl L = {ÓV I i = 0 , 1 , 2 , . . . } . Qua t r i n h t i n h toan cua may Â^)
trgn moi tiit vao duoc tien hanh nhu sau: Trudc tien, danh dau dau
bang hai bdi ky t u ft chang han va tirng budc sao chgp nhfing ky t u
0 cua t\t who d bang mot Ign bang hai cho t d i k h i gap ky t u 1, hkng each d i chuygn ca i i a i dau doc-ghi tir trdi qua phaị Tiep den, quay
dau doc-ghi bang hai ve hudng t r a i , Ian luot so sanh ti^fng ky t u 1
trgn bang mot vdi ttrng ky t u 0 tren bang hai va dong t h d i loQ,i bo nhiing til v&o ma sau ky t u 1 c6n xuat hien ky t u 0. Neu so nhiing
ky t u 1 tren bang mot triing vdi so nhiing ky t u 0 tren bang hai,
t h i chap nhan; ngUdc lai, bdc bọ
% Ro rang, doi vdi t i i vao 0" va vao t h d i digm k h i toan bo cdc ky t u 0 dudc sao chep Ign bang hai, so 6 ma may sii dung trgn bang hai la nhieu nhat. K h i ay, bang hai cd ndi dung fiO" va dau dpc ghi
bang hai dang d 6 ngay sau ky t u 0 cuoi cung. Vay s^(2) (n) = n + 2,
216 Do phutc tap khong gian Bay gid ta hay quay trci lai vdi su tuong duong giOa may Turing mOt bang va may Turing nhieu bang d6 xem xet mot each t i mi moi tuong quan giiia do phiic tap khong gian cua nhiing may aỵ Doi vdi m6i may Turing k bang M^''\g Dinh ly 1.2.1 ta da xay dung may Turing mot bang M tuong duong vk trong Dinh ly 2.1.3 ta da churng to rSng, neu may M^'') c6 do phutc tap thdi gian t{n), t{n) > n, thi may M c6 do phiJc tap thdi gian 0[t'^{n)]. Can luu y
r^ng, bang cua may M duoc chia thanh k doan sao cho tren m6i
doan the hien duqc noi dung bang tuong ling cua may M^''\o do, neu M^*^) CO do phiic tap khong gian s(n) thi do phiic tap khong
gian cua M khong vugt qua A; s(n). Vay ta c6 thg kgt luan nhu saụ
Dinh ly 3.1.3 Mdi may Turing tat dinh nhieu hang khong gian
s(n),s(n) > n, diu c6 may Turing tat dinh mqt hang khong gian
0[s{n)\ duang. •
• May Turing vdi bang chuyen doc. Nhan ihky r^ng, do phiic
tap thdi gian va do phiic tap khong gian cua cac may Turing quen
thuoc deu khong thg nho hon mot ranh gidi tuyen tinh. Chinh xac hon, ham f{n) duoc sii dung de xac dinh do phiic tap luon thô
man /(n) > n. Vay thi, lieu c6 thg tign hknh khao sat vdi mot ranh
gidi dudi tiiyen tinh hay khong? R6 rang, doi vdi cac may Turing
quen thuoc, do thdi gian dudi tuyen tinh la khong du de may truy cap dfl: ligu dau vao, ngn ta khong can phai xem xet dp phiic tai thdi gian dudi tuyen tinh. Tuong tu, ta cung khong can phai khao sdt do phiic tap khong gian dudi tuyen tinh doi vdi may Turing nC
chung, bdi vi trong pham vi khong gian nhu vay cung khong du d§
may luu tru: chi rigng du: lieu dau vaọ
Tuy nhign, doi vdi may Turing nhieu bang ta c6 thg khao s^t
do phiic t9,p khong gian dudi tuyen tinh, neu nhu dfl lieu dau ' duqc luu trfl rigng biet tren mdt bang nao do \h viec tinh toan du^c
thuc hien trgn cac bang khac; dong thdi, ta can cd mot quan m&^'
thuc te hon ve khong gian tinh toan cua maỵ
3.1 Do phac tap khong gian cua cac may Turing 217 Y tudng thiet thuc nay giup ta bo tri mot each thich hop khong Y tudng thiet thuc nay giup ta bo tri mot each thich hop khong
gian luu trfl vk khong gian tinh toan doi vdi may Turing nhieu bang, va nhd do ta c6 duoc mot may Turing kigu mdị Trong may Turing nhieu bang ta quy dinh bang thfl nhat la bang luu trfl va con lai la cac bang thao tac. Bang luu trfl diing de chfla dfl Ueu dau vao ma trong qua trinh tinh toan chi duoc phep khai thac chfl khong duoc
thay đi, vi the no con duqc gqi la hang chi duac dqc thdi (read-only tape) hay van t^t la hUng chuyen dgc. Viec tinh toan cua may duqc tien hanh tren cac bang thao tdc (work tapes). Dau dqc-ghi bang luu trfl, sau khi khai thac du nhflng thong tin can thiet, se dflng d
6 chfla ky tU dau tign hoac d 6 chfla ky tU cuoi cung cua dfl lỉu dau vaọ Trong qua trinh hoat dqng cua may, nhflng thong tin thu
thap duqc tfl bang luu trfl cd thg qua mot c5ng doan xii ly nho đ
va duqc chuygn sang bang thao tac. Qua trinh tinh toan ti^p theo duqc thuc hien trgn bang naỵ Bdi vay, thuc chat chi nhflng 6 duqc sii dung tren bang thao tac mdi tgo ngn khong gian tinh toan cua loai may Turing naỵ
Vdi quan niem nhu vay ve khong gian tinh toan ta c6 may Turing loai mdi, do la may Turing nhieu bang vdi bang chuyen doc
{multitape Turing machine with read-only tape), vk mot khai nigm mang tinh thuc tien ve dp phflc tap khong gian cua loai may nkỵ