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
220 Do phiic tap khong gian m^y mot bang, tirong t u nhir trong V i du 1.1.2, duy chi c6 diem ^
kh^c nhau la 6 day diiac b^t dau t\l cac ky t\t cua t i t thuf hai fc^.)
D6i vdi moi x e { 0 , 1 } , t a c6:
5{ql,(2),x) = {ql,<2),lS,L) . , 0 , *) = (94, 0> S, R) (5(9f,0,y) = (9f,0,?/,S,L), v6ij/e {o,i,tt} S{ql(2),x) = ( g 5 , 0 , * , S , R ) <5(95,0,2;) = (g5,0,2:,S,R) (^(g6,0j) = (g6,0,fi,S,R) % 6 , 0 , 0 ) = ( g 7, 0 , 0 , S , L ) 5(97, 0 , * ) = ( 9 8, 0 , * , S , R ) 5(98,0,11) = (9Y,0,tt,S,S). W , 0 , a : ' ) = ( g ^ , 0 , x ' , S , S ) , vdix' € {0,1,} vhx' 5(94",0,tt) = ( g N , 0 , « , s , s ) 5(95,0,1}) = ( 9 6 ,0 , « , S , R ) 5(96,0,:r) = ( 9 f, 0 , « , S , L) 5( 9 7 ,0 , y ) = ( 9 7, 0 , y , S , L ) 5(98,0,3:) = ( 9N , 0 , X , S , S )
Nhtr vay, k h i fc^ = / i ^ do do fc = /i, tutc k = h, may B^^^ chuyen t i t hinh t h a i ban dau [q,, (£,0H'=), (£,0)) den hinh t h a i chap
nhan (9v, (ó=l',0), (*'"'',«'=+')), nghia Ih mdy TXiring B^'^ khSng dinh ngon ngQ L.
D l danh gia do phiic tap khong gian cua mdy B^'^\a can luu
y r^ng trong qud t r i n h t i n h todn ciia mdy, moi ky t u cua tif vdo tren bang mot khong he thay d6i va noi dung bang hai vao thdi
diem c^n nhieu o nhat, chang han nhu */i^tlfc^, c6 dp dai khSng
vuot qud 21og2n. Dieu nay chifng to r^ng B^^^ Id mdy Turing vdi bang chuyen doc (cu thg Id bang mot) vd c6 do phiic tap khSng
gian 0 [ l o g n ] ^ Vay Id mdy Turing vdi bang chuyen doc B^^^ khdng dinh ngon ngii L trong khong gian logạ
^Di don gi^n, logj a thil6ng duqc vi6t Ik log ạ
I
3.1 Do phiic tQ,p khong gian cua cac mdy Turing 221
Cuoi cung, hodn todn tuong t u nhu D i n h ly 3.1.3, doi vdi mdy
Turing nhieu bdng vdi bang chuyen doc t a c6 k^t luan sau daỵ
D i n h l y 3.1.5 M5i mdy Turing tat dinh nhieu bdng vdi bdng
chuyen doc vdvdi dO phiic tap khong gian s{n), s{n) > l o g n , deu c6 may Turing tdt dinh hai bdng vdi bdng chuyen doc vd vdi do phiic
tQ,p khong gian 0 [ s ( n ) ] tuang duang. •
N h u vay, theo quan d i l m thong thudng ve do phiic tap khong
gian cua may Turing, Dinh ly 3.1.3 chiing to r^ng may Turing
mOt bang vd mdy Turing nhi^u bdng hodn todn tuong duong nhau va tuong dudng 6 miic dO tuy^n tính. Hon niia, trgn quan diem tdch biet "khong gian l u u t r f l " vdi "khQng gian thao tdc" cua mdy,
Dinh ly 3.1.5 cung cho ta dieu khdng dinh tudng t u doi vdi cdc may
Turing nhieu bdng vdi bdng chuygn doc. Do do, k h i khao sdt kha ndng doan nhan ngdn ngii cua mdy trong pham v i khong gian vdi
ranh gidi tuyen tinh, t a c6 the han che viec xem xet doi vdi mdy Turing mot bdng; con trong pham v i vdi ranh gidi dudi tuyen tinh,
\a chi can xem xet doi vdi mdy Turing hai hang, trong do bdng
tha nhdt Id bdng chuySn dpc vd bdng thU hai la bdng thao tdc, t a goi
tSt Id mdy Turing vdi bdng chuyen doc (Turing machine with
read-only tape), de phan biet vdi mdy Turing nhieu bdng vdi bdng \chuyen doc c6 nhieu han mOt bang thao tdc.
Nhdn t h i t a dinh nghia khdi niem rieng ve hinh thdi cua mdy
\ vdi bdng chuyen dgc vd c6 doi dieu nhan xet cdn thiet.
Trudc tien, t a hay quay t r d lai vdi quan niem thong thudng ve
hinh thdi cua mdy Turing k bdng noi chung, A; > 1, t a i mSt thdi
Ldilm ndo do trong qua trinh tính todn ciia may tren tiit vdo w cho
rudẹ Do Id mot bo (9, (uuvi),..., {ui,Vi),{uk;uk)) vdi nghia Id:
t?ii thdi diem xem xet, mdy d trang thdi 9, bdng t h i i i {I < i < k) •d noi dung mvi vd dau doc-ghi bdng ndy dang d o chiia ky tu dau
222 Do phtic tap khong gian
Thii nhat la t h i hien kit qua tinh todn titc thdi cua may, thong qua noi dung cua cac bSng. Thit hai la t h i hien cdc yeu to can thiit cho
viec thuc hien phep hien d6i co ban thich hdp tiep theo (nhu trang
thai va cac ky t u ma may doc duoc, tiic ky ttr dau tien cua Vi). Vdi
nhiJng muc dich n h u vay, trong hinh-thai [q, {ui,vi), {u2,V2)) cua
may Turing vdi bSng chuyen doc, thanh phan t h i i hai lien quan den bang chuyen doc c6 thg dUdc thay the bdi ky t u ma dau
doc-ghi doc duoc tren bang chuyen doc, tiic ky t u dSu tien cua v i ,
hay bdi vi tri cua dau doc-ghi tren bang chuyen doc.
D i n h nghia 3.1.6 Hlnh thdi cua may Turing vdi bang
chuyen doc tai mot thdi diem ndo do trong qud trinh tinh todn
cua may tren tic vdo cho trvcdc w = uị. .Oị. .Unld day {q, ai, {u;u))
vdi noi dung: tai thdi diim xem xet, mdy d trang thdi q, dau doc-ghi
bang chuyen doc dang doc ky tiC thii i cua tit vdo, tren hang thao tdc CO kit qud titc thdi uv vd dau doc-ghi hang ndy dang d 6 chica
ky tu dau tien cua v. Noi cdch khdc, hlnh thdi cua mdy Turing vdi bang chuyen doc duoc xdc dinh bdi trang thdi cua mdy, noi dung cua bang thao tdc vd vi tri cua cdc dau dgc-ghi tren bang tuang iing tai thdi diim xem xet. .
Vdi dinh nghia n h u vay va theo cac dinh nghia ve do phutc tap khong gian, t a c6 nhan xet chung sau day: Doi vdi bat cil may Turing nao vdi do phutc tap khong gian s(n),s(n) > l o g n , moi hinh t h a i cua may tren t i t vao do dai n deu c6 do dai bieu di§n cQ 0 [ s ( n ) ] . Theo dinh nghia tiep theo vg do phiic tap khong gian khong tat dinh, nhan xet nay cung dung doi vdi may Turing khong tat dinh va may Turing khong tat dinh vdi bang chuyen doc.
3.1.3 D o phiJc t a p khong gian cua may T u r i n g k h o n g t a t d i n h k h o n g t a t d i n h
N h u da biet, toan bo qua t r i n h tinh toan cua mdy Turing khong I
tat dinh diing tren mot i\l vao cho trudc thudng duoc dign t a bdi ^
3.1 Do phvCc tap khong gian cua cdc mdy Turing 223
cay t i n h toan va mdi qua t r i n h tinh toan khong t a t dinh cua may di§n ra theo mot nhanh nao do (xuat phat tijt goc den la) duqc lua chon mot each khong t a t dinh. Do phiic tap khong gian cua may Turing khong t a t dinh duqc dinh nghia n h u saụ
D i n h nghia 3.1.7 Cho N Id mdy Turing khong tat dinh dicng.
Do phiic tap khong gian cua mdy Turing khong tdt dinh N
Id ham s: N —> N, trong do s{n) la so toi da cdc 6 tren bang duac siC dung tai moi thdi diem trong qud trinh tinh todn theo moi nhdnh thuoc cay tinh todn cua mdy N tren bat cit tit vdo ndo dg dai n.
N h u vay, do phiic tap khong gian cua may Turing khong tkt
dinh cung duoc xac dinh trong "trudng hdp xau nhat". Doi vdi loai may Turing nay, ngoai yeu to "dau vao xau nhat" va " t h d i d i i m xau nhat", t a con phai xet den ca "nhanh t i n h toan xau nhat", tufc nhanh trong cay t i n h toan ma may sijf dung nhieu o jnhat v^o thdi
d i i m nao dọ •
| k V i d u 3.1.5 Ta hay xem xet do phiic tap khdng gian ciia mdy
H Turing khong t a t dinh N dUdc xay dung trong Dinh ly 2.3.3, dg H giai bai toan D I H A M P A T H , bai toan ve dudng d i Hamilton til