D CO thg mo phon gM trong khong gian dg{n) vdi mot h^ng so
j^pTQBFg NPS CP SC pTQBF^
Bao ham thilc t h i l nhat la dung dan, bdi v i ta c6 thg chuygn d5i may Turing khong tat dinh thdi gian da thiic vdi ttt van ve TQBF thanh may Turing khong tat dinh khong gian da thiic ma c6 thg tra Idi nhiJng cau hoi lign quan den TQBF hkng each tinh todn triTc tigp, khong can den bo phan t\i van. Bao ham thiic thi3 hai dirdc suy ra tilt Dinh ly Savitch. Bao ham thute t h i i ba thoa man, bdi vi
TQBFlh PS-day dụ Do do, ta k i t luan rang P^<5BF ^ j^pTQBF^
Tiep theo, ta trinh bay each xay diTng tir van Ạ Doi vdi mot A bat ky, gia sii L{A) \h tap tat ca eac tCt vdi do dai nhu cua mot tir nao do CO trong A, tiic
L{A) = {w\3xe A:\x\ \w\}. • *
Ro rang, doi vdi bat eii A, ngon ngfl L{A) thuoc N P ^ . ^
4.2 Phucang phdp quan he hoa va vdn de P=NP 277:
D i chiing to khong thuoc P ^ , ta xay dung A nhu saụ Gia
sijf A/i, . . . la danh sach tat ca cac may Turing vdi t u van. Dg don gian, ta c6 the gia thiet ra^g Mi hoat dong trong thdi gian n\
Viec xay dung duoe dign ra theo cac budc. 6 budc i , mOt phan ciia
A dUdc xay dung dg dam bao rang Mi khong khing dinh L(/l). Ta xay dung A bang each khai bao r^ng nhiing ttt nao do thuOc A va nhiing t\i khac khong thuoc Ạ Mdi budc xac dinh "than phan" cua chi mot so hiiu han cac tif. Liic dau ta khong c6 thong t i n gi ve Ạ
Bay gid t a dign ta cac budc, kg tir Budc 1.
Bifdrc ị Cho den thdi diem nay, mot s6 hflu han cac tilt duoc khai bao la thuoc hoac khong thuoc Ạ Ta chon n Idn hdn do dai cua bat
cut t i i nao trong so ay va dii Idn den miJc de 2" Idn hon n\i gian
hoat dong cua A/j. Ta trinh bay each cung cap thgm thong t i n ve
A sao cho Mi chap nhan tCt 1" mdi khi t i l nay khong thuoc L{A).
Ta cho Mi tinh toan trgn dau vao 1" va dap iJng dg bo phan tu van cua no tra Idi c h i t van nhu saụ Neu Mi chat van ve than phan cua y da duoc xac dinh trude do nhu the nao, ta dap ling mot each thich hop. Neu than phan cua y ehua duoe xac dinh, ta tra Idi
KHONG va khai bao r^ng y khong thuoc Ạ Ta tiep tuc mo phong
Mi cho den khi no diing.
Bay gid ta xem xet mot tinh huong t\l boi canh cua Afj. Neu no t i m thay mot tilt do dai n trong / I , no se chap nhan bdi v i no biet rang 1" c6 trong IJ{A). Neu hh xac dinh r^ng tat ca cac t\i do
dai n khong cd trong yl, no se bac bo bdi vi no biet rang 1" khong
CO trong L{A). Song, no khong cd du thdi gian dg chat van ve tat ca cac tir do dai n vạ ta t r a Idi KHONG đi vdi tiTng chat van aỵ Do do, khi Mi dirng va phai quyet dinh chap nhan hoac bac bo, no khong cd du thong t i n dg cam thay chac chSn rang quyet dinh cua no la chuin xac.
Muc tieu cua ta la chiing to r^ng quygt dinh ciia Mi la khong chudn xdc. Ta thue hien dieu nay hhxg each quan sat quyet dinh
278 Tinh nan giai Mot each cu the, n^u Mi chap nhan 1", ta khai bao rang tat ca cac tiit con lai do dai n deu khong thuoc A va bdi the xac dinh rSng 1" khong CO trong L { A ) . Ngu Mi bac bo 1", ta tim duoc mot til do dai n mh Mi bo qua khi chat van va ta khai bao ring ttr nay thuoc
A , d l dam bao rang 1" c6 trong L{A). Til nhxl vay chac chan ton tai bdi vi Mi hoat dong trong thfii gian n\g thdi gian kha it so vdi 2", so tat cac cac tit do dai n. Nhu vay, ta tin chac r^ng Mf-
khong thg khang dinh L { A ) . Budc i duoc thuc hien vh ta tien hknh Budc i + 1 tiep theọ
Sau khi ket thuc tat ca cac budc, ta hoan thanh viec xay dung A
bkng each manh dan khai bao rang bat cil tit nao ma than phan cua no c6n chiia duoc xac dinh trong nhUng budc ay la khong thuoc A .
Vay la khong mot may Turing nao vdi tu van A c6 t h i khing dinh
L{A) trong thdi gian da thuic. Dinh ly dudc chiing minh. • Tom lai, phuong phap quan he hoa bay to cho ta each thilc giai
quyet v§,n de P = N P . Tuy nhien, ta can phai tien hanh phan tick
cac qua trinh tinh toan mot each t i mi, chijf khong chi don thuSn thuc hien viec mo phong. Trong phan tiep theo, ta se gidi thieu m5t each tiep can c6 the dan den viec phan tich nhu vaỵ ^
4.3 Mach Boole doi vdi van de P = N P
May tinh dUdc lap rap tit nhiing thiet bi dien tijf, duoc ghep noi lai vdi nhau theo ban thiet kg duoc goi la mach chU so {digital circuit).
Ta cung c6 t h i mo phong nhieu mo hinh ly thuyet, ching han nhU cdc may Turing, b^ng nhUng mach chU so ma duoc goi la mach Boole {Boolean circuit). Hai muc dich sau day duoc dap ting b&ng each thiet lap moi lien quan gifla may Turing va mach Boolẹ Thii nhdt, nhung chuyen gia nghien ciiu trong linh vuc khoa hoe tinh toan tin r^ng mach Boole cung cap mot mo hinh tinh toan thich i hop cho viec tan cong van de P = N P va cac van dg lign quan-
4.3 Mach Boole doi vdi van di P^NP
279
Thii hai, mach Boole dua den cho ta mOt each chUng minh khdc doi vdi Dinh ly Cook-Levin ve tinh NP-dSy du cua bai toan thoa duoc. Day la hai chu de ma ta de cap den trong phan naỵ
4.3.1 Mach Boole doan nhan ngon nguf
Ta bat dau t i i khai niem mach Boolẹ
D i n h nghia 4.3.1 Mach Boole la mot hg cac cdng {gates)
vd cac loi vao {inputs) duoc ket nSi vdi nhau b&ng cac day dan
{wires). Co ha dang cdng ca sd: cong AND, cdng OR va cdng NOT,
duac hiiu thi bang nhiing sd do tren hinh ve sau daỵ
loi vho \ loi vao 0 loi ra OR loi I6i ra NOT H i n h 4.1 Cong AND, cdng OR va cdng NOT
CAc bien dau v^o cua mach Boole duoc ky hieu la x i , ... Cac day d i n cua mach Boole truyen tai nhiing gid tri Boole 1 va 0. Cac pQng la cac thiet bi xii ly don gian de tinh nhiing ham Boole
AND, OR va NOT. Ham AND cho ket qua 1 neu ca hai dau vao cua no deu la 1 va cho ket qua 0 trong cac trudng hop nguoc laị Ha,m
OR cho ket qua 0 neu ca hai dau vao cua no deu la 0 vh cho ket qua