ti Gii thiu v l thuyt tr chi
Thut ton Min-Max&Alpha-Beta v ng dng
trong tr chi c Caro.Mc lc
Li m u3I.Gii thiu v l thuyt tr chi v ng dng4II.Gii thiu tr chi i
khng v lch s cc chng trnh c52.1.Tr chi i khng52.2. Lch s cc chng
trnh c62.3.Gii thiu v tr chi C caro (Gomoku)8III.Phn tch bi
ton123.1. Biu din bi ton di dng cy tr chi (Game Tree)123.2.Chin lc
tm kim133.2.1 Thut ton vt cn liu c c s dng?133.2.2.Khng gian tm kim
nc i & chin lc tm kim trong c Caro14IV. Thut ton144.1.Thut ton
Min-Max14 4.2.Thut ton ct ta Alpha-Beta19Gii thiu sn phm22Kt
Lun25Ti Liu Tham Kho25
Li m u
L thuyt tr chi l mt nhnh ca ton hc, n s dng cc m hnh nghin cu cc
tnh hung chin thut, trong cc i th c gng lm ti a kt qu thu c ca
mnh.Trong thi i cng ngh thng tin pht trin mnh nh hin nay th L thuyt
tr chi thu ht c rt nhiu s ch ca cc nh khoa hc my tnh do ng dng ca n
trong Tr tu nhn to v iu khin hc
Trong bo co ny, em s trnh by mt trong nhng ng dng ca L thuyt tr
chi, l gii thut tm kim Min-Max, Alpha-Beta v ng dng trong vic xy
dng 1 chng trnh tr chi i khng, c th l tr chi c caro .
I. Gii thiu v l thuyt tr chi v ng dngTheo mt s ti liu th ln u
tin l thuyt tr chi xut hin trong mt l th vit bi James Waldegrave nm
1713, trong l th th tc gi a ra li gii chin thut hn hp Minimax cho
mt tr nh bi 2 ngi Leher.Tuy nhin th L thuyt tr chi ch thc s tn ti l
mt ngnh khi John von Neumann xut bn mt lot cc bi bo nm 1828.John
von Neumann cng l ngi u tin hnh thc ha L thuyt tr chi trong thi k
trc v trong chin tranh lnh, ch yu do p dng ca n trong chin lc qun
s, ni ting l khi nim m bo ph hy ln nhau (mutual assured
destruction).
Sau nhiu nm pht trin th hin nay L thuyt tr chi c s dng rng ri
trong nhiu ngnh khc nhau nh : Kinh t v kinh doanh, sinh hc, chnh tr
hc, trit hc, khoa hc my tnh v logic, vin thng, mt s show game trn
truyn hnh
Trong thi i Cng ngh thng tin pht trin nh hin nay th L thuyt tr
chi ng mt vai tr ht sc quan trng, c bit trong logic v khoa hc my
tnh.Mt s l thuyt logic c c s trong ng ngha tr chi.Thm vo nhng khoa
hc gia my tnh s dng tr chi m phng nhng tnh ton tng tc vi nhau.
Mt s thut ton trong L thuyt tr chi gip xy dng, pht trin nhng tr
chi hay, nh : thit k tr chi Nim; thit k kiu tr chi c nhn, c tnh i
xng; thut ton lin quan n chin lc tm kim,Bi bo co cp n thut ton tm
kim MinMax v thut ton ct ta Alpha-Beta trong vic xy dng chng trnh
tr chi c caro (Gomoku).
II. Gii thiu tr chi i khng v lch s cc chng trnh c2.1.Tr chi i
khng
Tr chi i khng din ra gia 2 i th.Nhn chung th tr chi thng c c im
:
Mi tr chi u c 1 lut chi m cc u th u phi c gng ginh phn thng v
mnh.Trn u phi c kt thc ha hoc phn nh thng thua ch khng ko di v
tn.
Mi u th c i mt nc i khi ti lt ca mnh.
Cc u th u bit thng tin v tnh trng ca trn u. Mt s tr chi i khng
nh : Tictactoe, C caro (Gomoku), c vua, c tng,nh hnh di ---- C vua
---- ----C tng---- ----Tictactoe---- ----Gomoku----2.2. Lch s cc
chng trnh cVo nm 1950, Alan Turing - mt nh nghin cu ngi Anh i tin
phong trong lnh vc my tnh s, vit chng trnh chi c u tin. Vo lc ,
Turing phi vit v chy chng trnh ca ng bng bt ch v giy. Chng trnh ,
cng nh ch nhn ca n, chi c rt ti, nhng t c mc ch: cho thy my tnh c
th chi c c. Cng vo nm , Claude Shannon vch ra mt chin lc cho my tnh
chi c tt. Nhng vo nhng nm 1950 tc my tnh rt chm nn khng ai dm tin
on liu my tnh c th thng con ngi c khng, d trong cc tr chi n gin nh
tr Checker.
Nm 1958, mt chng trnh chi c ln u tin h c i phng l con ngi. Ngi
thua l mt c th k ca chnh i lp trnh ra n, c cha bao gi chi c trc v c
dy chi c ch mt gi trc cuc u. i vi ngy nay chin cng ny tht nh nhoi,
nhng n cho thy tri thc c th c a vo trong mt chng trnh chi c. Lng
tri thc ny c o chnh xc bng mt gi hc chi.
Sau chin thng , mt s ngi trong nhm lp trnh c u tin tin on rng vo
nhng nm 60 s c chng trnh chi c c lit vo hng ng kin tng th gii. Vo
nhng nm cui ca thp k 60, Spassky tr thnh kin tng c th gii v cc chng
trnh chi c chim c nhng th hng cao trong hng ng nhng ngi chi cao cp.
Nhng nhiu ngi cho rng my tnh s khng bao gi c th gii quyt c nhng
nhim v thng minh, khng th t c chc V ch c th gii.
Li tin on ny c nhc li mt ln na vo nhng nm 70, lin quan n mt cuc
nh cc gia David Levy, mt kin tng quc t ngi Anh (theo phn loi ca Lin
on c quc t cc ng cp cao bao gm: Kin tng quc t, i kin tng v V ch th
gii) v John McCarthy, mt nh nghin cu trong lnh vc tr tu nhn to. Li
thch u c a ra vo nm 1978. Trn u c din ra v chng trnh c tt nht thi ,
CHESS 4.7 b Levy h trong trn u c nm vn ti Toronto vi thnh tch ba vn
ngi thng, mt ho v mt my thng. Levy khng ch chin thng m cn t ti s
tin nh cc 1000 bng.
Nu nh mc ch ca cuc nh cc l lm cho nhng nh nghin cu phi ngh k trc
khi tin on n ngy thng li, th ln nh cc ny cho thy: mc d tin on sai
trong nhng nm 1958-1968 v 1968-1978, cc chuyn gia chng trnh c li
tip tc tin on tip rng my tnh s t n v ch c th gii trong thp k tip
theo. Nhng mt ln na, vo nm 1988, V ch c th gii vn l con ngi.
Trong nm tip theo, Deep Thought, mt chng trnh c mnh nht t xa n
nay chin thng mt cch d dng Kin tng Quc t Levy. B no ca Deep Thought
c 250 chip v hai b x l trong mt bng mch n, n c kh nng xt 750.000 th
c trong mt giy v tm trc c n 10 nc. Cng trong nm , n l my tnh u tin
h c mt i kin tng (Bent Larsen). Deep Thought tr thnh mt trong mt
trm ngi chi c mnh nht th gii. Nhng trong trn u din ra vo nm 1989
gia nh V ch th gii Garry Kasparov v Deep Thought th n b nh v ch
bp.
Cc li tin on li n nh cc ln trc. ba ln cc nh nghin cu tin on:
'trong thp k ti'. Nhng ln ny h li sa li l: 'trong 3 nm ti'...
Trong nm 1993, Deep Thought h Judit Polgar - lc l i kin tng tr
nht trong lch s v l ngi ph n chi hay nht th gii, trong trn u 2
vn.
Trong nm 1996, Deep Blue (tn mi ca Deep Thought v lc ny n thuc
hng IBM) l mt my tnh song song c 32 b x l vi 256 mch tch hp c ln,
kh nng xt t 2 n 400 triu nc i mi giy) thng Gary Kasparov trong vn u
tin ca trn u 6 vn, nhng li thua trong ton trn (vi t s my thng 1, ho
2 v thua 3).
Cui cng ch m mi ngi ch i ti, nhng sau 9 nm t li tin on cui v 39
nm t lc c chng trnh chi c u tin, Deep Blue chin thng nh ng kim V ch
th gii Garry Kasparov vo thng 5/1997 trong mt cuc chin di y kh khn,
vi t s st nt 2 thng, 1 thua v 3 ho.
2.3.Gii thiu v tr chi C caro (Gomoku)
C caro chnh l mn c logic lu i v c xa nht trn Tri t. C caro c sng
to t nhiu nn vn minh khc nhau mt cch c lp. N bt u xut hin t nm 2000
trc CN sng Hong H, Trung Quc. Mt s nh khoa hc tm thy bng chng chng
minh Caro c pht minh Hy lp c i v Chu M trc thi Colombo. Mn c c ca
Trung Quc l Wutzu. C Caro du nhp t Trung Quc vo Nht Bn t khong nm
270 trc CN. N thng c gi l Gomoku nhng cng c cc tn gi khc tu theo
thi gian v a phng nh Kakugo, gomoku-narabe, Itsutsu-ishi... Ngi ta
tm thy mt tr chi c t mt di tch Nht nm 100 sau CN v thy n l mt bin
th ca Caro. N lan truyn nhanh chng vi ci tn Kakugo (tr 5 qun). Cc
nh s hc ni rng vo cc th k 17 v 18, mi ngi u chi tr ny-ngi gi cng nh
ngi tr. Nm 1858, khi quyn sch u tin v tr chi ny c xut bn, n c gi l
Kakugo. N tip tc c chi, c gi vi nhiu tn khc nhau nh Goren, Goseki,
ri Gomokunarabe, Gomoku v pht trin cho n ngy nay thnh th loi phc tp
nht trong h hng ng c ca n, l Renju (chui ngc trai).
Lut chi ca Gomoku c nh sau:
Bn c 15 x 15, qun en i trc.
Ai to c 5 qun lin nhau trc th thngKhi trnh cc k th Gomoku c nng
cao, h nhn ra rng nu ch chi n gin nh trong Gomoku th s l mt li th
qu ln cho bn tin tc bn en (thc t chnh l u th thng). Sau mt s nh ton
hc chng minh c rng nu chi vi lut Gomoku trn bn c bng hoc rng hn
15x15 th en chc chn thng (sure win), v sau cch i c th cng c tm ra,
h thng v phn loi.V chnh v vy, t Gomoku lm vo mt giai on khng hong.
Kh nng nh thng 100 phn trm ca en lm tr chi ny mt i ngha ca n. C
nhiu ci tin c xut, mt s b b qua nhanh chng, s khc lm xut hin cc bin
th mi ca Gomoku. tng chung ca cc ci tin l ra mt s hn ch cho en, nhm
cn bng u th i tin. Di y l mt s bin th ph bin: Gomoku. Hin nay c chi
chnh thc vi bn 13x13. Khng c ho. Nu ht t th Trng thng. Cha tm c
chng minh no cho thy en chc chn thng. Tuy nhin en vn c u th rt ln.
ProGomoku. Chi trn bn 15x15. Nc u ca en t sn trung tm. Nc th ba (nc
th hai ca en) phi t ngoi hnh vung cm. Hnh vung cm l hnh vung trung
tm kch thc 5x5. Khng c hn ch cho Trng. c chng minh en chc chn thng
trong bin th ny. Pente. Bin th ny khng cn ging Gomoku. Lut b sung l
c th n qun i phng. Nc n qun c thc hin bng cch chn hai u mt nc hai
qun i phng v n hai qun . Ai to c nc nm hoc n c 5 cp qun trc th
thng. Rt ph bin M. Chi trn bn 19x19.III. Phn tch bi ton
3.1.Biu din bi ton di dng cy tr chi (Game Tree) Tr chi c th c
biu din nh mt cy gm gc, nhng nt, nhng l v nhng nhnh
Gc l trng thi ban u ca tr chi.Vi mi tr chi c th th trng thi ( mi
thi im) li c c trng bi nhng thng s ring
Cc nt (Node) ca cy th hin tnh trng hin ti ca tr chi, gm nt cha
(Parent Node) v nt con (Children Node)
Cc nhnh ni gia cc nt th hin nc i, tc l cho bit t mt tnh hung ca
tr chi chuyn sang tnh hung khc thng qua ch mt nc i no .
Cc l (leave) hay cn gi l nt l (leave node), th hin thi im kt thc
khi m kt qu ca tr chi r rng.
Ngoi ra th cn mt thng s ca cy na l su (Fly) hay cn gi l mc ca
cy, s tng ca cy.
Thng th mi v tr kt thc ca tr chi (nt l) s gn mt trng s, chng hn
gn 1 cho chin thng, 0 cho ha v -1 cho thua trn.Ti mi nt cng c mt
trng s tng ng c xc nh bng mt cch no .Da vo cy tr chi ny, ngi ta c
th tm ra nc i tt ginh phn thng cho mnh (nu c th), bng cch tm kim
trn cy tm ra nc i tt nht.Di y l v d v cy tr chi qua tr chi bc si Gi
thit c 3 hp bi, s lng bi trong mi hp l (1,2,2). Mi lt chi ngi chi
ch c bc trong 1 hp bi, vi s lng ty .Ngi chi no bc bi cui cng s l
ngi thua cuc.
Da vo nh gi cy tr chi di, ta thy c nhng nt l m c trng s l 1, tc
l i theo nhng nhnh no m cui cng n c nhng l y th ngi chi Max s ginh
thng li.
3.2. Chin lc tm kimNh vy vi mt tr chi i khng, khi m ta biu din c
tr chi di dng mt cy tr chi, th vn t ra l phi tm c chin thut i trn
cy tr chi chim li th.Tc l phi c chin lc tm kim tt m bo ng i ca mnh
l tt
3.2.1Thut ton vt cn liu c c s dng?Nu nh thut ton vt cn thc s dng
c tm kim trn cy tr chi th ta ch cn chn nhnh cy dn ti nt chin thng
i, v nh vy cc tr chi khng cn s hp dn thng c.V thc t l, trong cc tr
chi i khng th sau mt vi lt i th li sinh ra rt nhiu kh nng nh tip
theo (bng n t hp), ch c mt s t cc trng hp l c th tm kim theo kiu vt
cn ht cc kh nng ny.Do khng dng thut ton vt cn cho chin lc tm kim
c.
3.2.2. Khng gian tm kim nc i & chin lc tm kim trong c CaroNh
chng ta bit, trong c caro th c sau mi nc i s trng s gim.V vy vic tm
kim nc i tip theo l vic tm kim trong khng gian cc trng cn li, sau
mi lt i th khng gian tm kim s gim dn
Chin lc thng c c ngi ln my dng l phn tch th c ch sau mt nc i no
ca c 2 bn.Tc l trn cy tr chi, vic tm kim nc i l chn 1 nt trn cy sao
cho nc i l tt .V nh gi c nt th thng phi nhn xa, lin quan n su ca cy
(tng ng vi vic ngi chi phi nhn xa xem bn c c nhng kh nng bin i no
sau mt s nc, t nh gi c tt xu ca th c hin ti) Vi my tnh th th c ny c
nh gi tt hn th c kia nh so snh im ca th c do b lng gi tr li.V khng
gian tm kim l qu ln nn chng ta gii hn cho my tnh ch tm kim mt su
nht nh, v tt nhin su cng ln th chng trnh cng thng minh nhng tr gi v
mt thi gian
IV.Thut ton
4.1.Thut ton Min-Max
Trong 2 ngi chi th mt ngi gi l ngi chi cc i (Max) v i th ca h l
ngi chi cc tiu (Min).C 2 u th u c gng i nhng nc th no im tuyt i ca
mnh ln hn hay cao nht c th.Tc l ngi chi Max s tm cch lm im ca mnh
cao hn v lm im ca i th bt m hn (gim v tr s) .Trong khi ngi chi Min
th ngc li, s c gng lm cho im ca mnh m hn v lm cho im ca i th
gim.
Gii thut tm kim Min-Max c s dng xc nh tt c nhng din bin tip theo
ca tr chi cho n tng c yu cu.im s ban u c gn cho l, sau bng cch lng
gi cc nc i, im s c gn cho cc tng trn qua gii thut Min Max, thut gii
thc hin mt lt ct cho trc v tnh im trn .
tng c bn ca thut gii Min-Max theo quy
Nu mc ang xt l ngi chi cc tiu th p dng thut ton Min-Max cho cc
con ca n.Lu kt qu l gi tr nh nht.
Nu mc ang xt l ngi chi cc i th p dng thut ton Min-Max cho cc con
ca n.Lu kt qu l gi tr ln nht.
Nu mc ang xt l l (tng cui cng ca cy tm kim), tnh gi tr tnh ca th
c hin ti ng vi ngi chi .Ghi nh kt qu.M 1MinMax(x){ // x l nt mun
tnh im
If x is a leaf
// x l nt l
Return score of x;
Else
If x in a minNode
// ngi chi min
For allChildren of x : v1,,vnReturn min
{MinMax(v1),,Min-Max(vn)}
Else
For allChildren of x : v1,,vnReturn max
{Min-Max(v1),,Min-Max(vn)}
}Tuy nhin trn mt cy c kch thc ln th ta khng th tm ht tt c cc nt
m ta ch gii hn trong mt s tng ca cy v xem nh y l m phng gn ng ca mt
cy Min-Max (cha bit) bng cch gn trng s cho cc l ca n.Trng s y l
trng s khng cn chnh xc tuyt i m l c lng.Trng s nhn c theo cch ny gi
l c tnh ton vi s gip ca hm lng gi, hm ny c xy dng vi ngi dng da trn
s hiu bit v kinh nghim.M 2function MinMax (pos, depth):
integer;
{
if depth = 0 then //t n gii hn
MinMax = Eval (pos) //Tnh gi tr th c pos
else
{
Gen (pos); // Sinh ra mi nc i t th c pos
while cn ly c mt nc i m do
{
pos = Tnh th c mi nh i m;
value = MinMax (pos, depth-1); // Tnh im ca pos
}
}
}Tham s depth su tm kim gip ta bit phi tm n u, tham s pos cho
bit th c hin ti t bit cch tnh tip. Gi tr tr v ca hm chnh l im ca th
c pos. Hm lng gi Eval s nh gi c cht lng ca th c pos hin ti. Cc th c
con pos' l cc th c c to ra t pos bng cch i mt nc i hp l x no . Do
ta phi c cc lnh thc hin i qun n cc th c mi. bit t th c pos c th i c
nhng nc no, ta dng mt th tc Gen c tham s l th c cha pos. Th tc ny s
ct cc th c con pos' vo b nh (dng danh sch). Vic tip theo l ta ly
tng th c ra v p dng tip th tc MinMax cho n tnh im value ca n.M
3function MinMax (pos, depth): integer;
{ if depth = 0 then MinMax = Eval (pos) // Tnh gi tr th c
pos
else
{best = -INFINITY; // m v cng
Gen (pos); // Sinh ra mi nc i t th c pos
while cn ly c mt nc i m
do
{
pos = Tnh th c mi nh i m;
value = -Minimax (pos, depth - 1);
if value > best then best = value;
}
MinMax = best; //Tr v gi tr tt nht
}
}Thng thng, bn c c biu din bng cc bin ton cc. Do thay cho truyn
tham s l mt bn c mi pos vo th thc MinMax th ngi ta bin i lun bin
ton cc ny nh thc hin nc i "th" (nc i dn n bn c mi pos). Sau khi
MinMax thc hin vic tnh ton da vo bn c lu bin ton cc th thut ton s
dng mt s th tc loi b nc i ny. Nh vy MinMax b cc tham s pos nh
sau:
M 4function MinMax (depth): integer;
{
if depth = 0 then MinMax = Eval // Tnh th c pos trong bin ton
cc
else
{
best = -INFINITY;
Gen; // Sinh ra mi nc i t th c pos
while cn ly c mt nc i m do
{
thc hin nc i m;
value = -MinMax (depth - 1);
b thc hin nc i m;
if value > best then best = value;
}
MinMax = best;
}
} nh gi thut ton :Gi s h s nhnh trung bnh ca cy l a , xt su b th
s nt y phi lng gi l ab .Thc t s nhnh kh ln nn ch cn xt su nh (c nh
hn 10) th s nt cn xt cng rt ln.
Hnh v v d vi s nhnh l 5
DepthNode Count
01
15
8390625
n5n
4.2.Thut ton ct ta Alpha-Beta
Thut ton ct ta Alpha Beta l ci tin ca thut ton Min Max vi t tng
Nu thy mt vic lm l t th khng nn mt thi gian xem n t n mc no .
Thut ton lm gim s nt cn thit ca vic tm kim khng lng ph thi gian
tm kim nhng nc i bt li r cho ngi chi.Gii thut Alpha Beta ci tin so
vi Min Max bng cch thm vo 2 tham s l alpha v beta.Chng cho bit cc
gi tr nm ngoi khng [alpha, beta] l cc im khng cn xem xt na.Th tc
Alpha Beta c bt u ti nt gc vi gi tr ca alpha l - infinity v beta l
+ infinity .Th tc s t gi quy chnh n vi khong cch gia cc gi tr alpha
v beta ngy cng hp dn.M 1
evalutemin(x, B) // x l nt Min{ Alpha=+infinity; if x = leaf
return the score; else for all children v of u { Val =
evalutemax(v, B); alpha= Min{alpha, Val}; if Alpha= Beta then exit
loop; } return Alpha; }
M 2function AlphaBeta(alpha, beta, depth): integer;
{
if depth = 0 then
AlphaBeta = Eval // Tnh gi tr th c pos
else
{
best = -INFINITY;
Gen; //Sinh ra mi nc i t v tr pos
while (cn ly c mt nc i m) and (best < beta) do
{
if best > alpha then
alpha = best;
thc hin nc i m;
value = -AlphaBeta(-beta, -alpha, depth-1);
b thc hin nc i m;
if value > best then best = value;
}
AlphaBeta = best;
}
} nh gi thut ton :Ngi ta tnh ton c l, trong iu kin l tng th thut
ton Alpha Beta ch phi xt s nt theo cng thc
+ 2.ab/2 - 1 nu b chn
+a(b+1)/2 + ab/2 - 1 nu b l
Trong a l s nhnh trung bnh ca cy, b l su ca cy
Qua cng thc trn th ta thy c thut ton Alpha Beta phi xt s nt t hn
thut ton Min Max kh nhiu. Chng hn ly a = 30, b=6 th s nt phi xt vi
thut ton Alpha Beta l 53999 trong khi s nt cn xt vi thut ton MinMax
l xp x 2.2 x 1023
Gii thiu sn phm
Chng trnh c lp trnh bng ngn ng C#, code v chng trnh c ghi trong
CD i km.
Di y l mt s hnh nh ca chng trnh khi chy thc t
Khi bn chin thng
Khi my chin thng
Kt Lun
Nh vy qua bo co ta thy c tm quan trng ca L thuyt tr chi, mt
trong rt nhiu l thuyt ca n l gii thut tm kim Min Max, gii thut ct
ta Alpha Beta v ng dng vo trong tr chi i khng.
Ti Liu Tham Kho
1. http://vi.wikipedia.org/wiki/L_thuyt_tr_chi 2.
http://www.ocf.berkeley.edu/~yosenl/extras/alphabeta/alphabeta.html3.
Tr tu nhn to
4. Mt s chng trnh Gomoku m ngun m
[email protected]
[email protected] 25
