Analysis of the Alpha-beta Pruning Algorithm

7/27/2019 Analysis of the Alpha-beta Pruning Algorithm

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Carnegie Mellon University

Research Showcase

Computer Science Department School of Computer Science

1-1-1973

Analysis of the alpha-beta pruning algorithmSamuel H. FullerCarnegie Mellon University

John G. Gaschnig

Gillogly

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Recommended CitationFuller, Samuel H.; Gaschnig, John G.; and Gillogly, "Analysis of the alpha-beta pruning algorithm" (1973).Computer ScienceDepartment. Paper 1701.hp://repository.cmu.edu/compsci/1701
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ANALYSI S OF THE ALPHA-BETA PRUNI NG ALGORI THM

S. H. Ful l er , J . G. Gaschni g and J. J . Gi l l ogl y

Department of Computer Sci enceCarnegi e- Mel l on Uni versi ty

Pi t t sburgh, Pennsyl vani a 15213

J ul y, 1973

Thi s work was supported by the Advanced Research Proj ects Agencyof the Of f i ce of the Secretary of Def ense ( F44620- 73- C- 0074) andi s moni tored by t he Ai r Force Of f i ce of Sci ent i f i c Research.


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ABSTRACT

Many game-pl ayi ng programs must search very l arge game trees. Use

of the al pha- beta pruni ng al gor i thm i nstead of the si mpl e mi ni max search

reduces by a l arge f actor the number of bot tomposi t i ons whi ch must be

exami ned i n the search. An anal yti cal expressi on f or the expected number

of bot tom posi t i ons exami ned i n a game t ree usi ng al pha- beta pruni ng i s

deri ved, subj ect to the assumpt i ons that the branchi ng f actor N and the

depth D of the tree are arbi t rary but f i xed, and the bot tomposi t i ons

are a random permutat i on of N uni que val ues. A si mpl e approxi mat i on t o the

growth rate of the expected number of bot tomposi t i ons exami ned i s suggested,

based on a Mont e Car l o si mul ati on for l arge val ues of N and D. The behavi or

of the model i s compared wi t h t he behavi or of the al pha- beta al gor i thm i n a

chess pl ayi ng program and t he ef f ects of correl at i on and non- uni que bot tom

posi t i on val ues i n real game t rees are exami ned.


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TABLE OF CONTENTS

Secti on Page

1. I nt roducti on 1

2. The Al pha- Beta Pruni ng Al gor i thm 2

3. A Probabi l i st i c Model of Game Trees and Some I ni t i al 14

Observat i ons

4. The Probabi l i ty of Eval uati ng a Node i n the Game Tree 18

5. The Expected Number of BottomPosi t i ons Eval uated 23

6. Appl i cati on of the Game Tree Model to Chess 37

7. Empi r i cal Observat i ons 42

8. Concl usi on 47

Appendi x: Notat i on 49

References 51

i i


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1. I NTRODUCTI ON

Searchi ng t rees of possi bl e al ternat i ves i s a task common to a wi de

range of programs. The ef f i ci ency wi th whi ch these t rees can be searched

i s of cri t i cal i mportance to such programs, si nce the t rees are typi cal l y

very l arge. Thi s paper i s concerned wi t h measur i ng the ef f i ci ency of a

part i cul ar t ree- searchi ng al gor i t hm, the mi ni max search of a game t ree

wi t h al pha- beta pruni ng.

The probabi l i st i c model used i n our study i s presented i n the next

sect i on and we der i ve an anal yti cal expressi on f or the expected number of

bot tomposi t i ons eval uated i n the search of a game tree usi ng al pha- beta

pruni ng. A reasonabl y accurate si mpl e approxi mat i on to the anal yt i cal

resul t based upon an empi r i cal anal ysi s i s suggested. Si nce our model i n

corporates several si mpl i f yi ng assumpt i ons, the rel evance of our model wi l l

be exami ned i n Sect i on 6 where we compare the behavi or of our model wi t h

the observed behavi or of the al pha- beta procedure as i t i s used i n a non-

t r i vi al exampl e, a chess pl ayi ng program.

I n thi s paper, the operat i on of the mi ni max search procedure and the

al pha- beta pruni ng procedure are i l l ust rated i n the context of game pl ay

i ng programs. We gi ve t he name Max to the pl ayer whose t urn i t i s to move

and t he name Mi n to hi s opponent . Max at tempts to maxi mi ze the ul t i mate

val ue of the game whi l e Mi n at tempts to mi ni mi ze the val ue. A number of

strategi es exi st to ai d a pl ayer i n determi ni ng hi s next move, but the

mi ni max procedure has recei ved the most attent i on i n programs whi ch pl ay

games of perf ect i nf ormat i on. The procedure i s most easi l y i l l ustrated


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wi t h the ai d of the si mpl e game tree of Fi gure 1. 1. The nodes of the t ree

are i nterpreted as posi t i ons, and the arcs f romeach node are the l egal

moves f rom that posi t i on. The square nodes i ndi cate i t i s Max' s turn to

move whi l e the ci rcl es i ndi cate i t i s Mi n' s turn. The stati c val ues

associ ated wi t h each of the ni ne bot tomposi t i ons are gi ven i ndependent l y

of the appl i cat i on of any search procedure. I ncreasi ng val ues are i nterpreted

as a measure of the "goodness11 of a board posi t i on, i . e. , the amount of ad

vantage to pl ayer Max. I n the mi ni max procedure t he backed- up val ue of a

Max posi t i on i s the maxi mum of the val ues of i ts i mmedi ate successors and

si mi l ar l y, the backed- up val ue of a Mi n posi t i on i s the mi ni mumof the

val ues of i t s i mmedi ate successors, i . e. , at each node t he pl ayer to move

wi l l choose t he move whi ch i s most f avorabl e t o hi msel f . The mi ni max pro

cedure recursi vel y appl i es these two rul es unt i l t he stat i c val ues at the

l eaf nodes have been used t o generate a backed- up val ue f or the root node.

For exampl e, i n Fi gure 1.1 the backed- up val ues of p (1 ) , p( 2) , and p( 3)

are 3, -2 and - 10, r especti vel y and t he backed- up val ue of p, the root node,

i s 3. For a more compl ete di scussi on of the mi ni max procedure see Shannon

[1950] or Ni l sson [1970] .

We wi l l f requent l y use t he game of chess i n thi s paper to i l l ustrate

some of the practi cal i mpl i cat i ons and l i mi tat i ons of our anal ysi s. The

cl assi c exampl e of the l i mi tat i on of the mi ni max procedure i s i ts appl i ca

t i on to chess. Consi der the game tree for chess where the posi t i on p i s

def i ned by t he l ocat i on and i dent i ty of each pi ece on the board, the i dent i ty

of the pl ayer whose t urn i t i s to move, and hi stor i cal i nf ormat i on rel at i ng

to cast l i ng, en passant capt ures, and draws by repet i t i on. Suppose we ex

tend the chess game tree unt i l every l eaf node i s a wi n, l oss, or draw.


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4

Fi gure 1. 1. A game tree wi t h branchi ng f actor 3and depth 2.


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Then the mi ni max procedure coul d be appl i ed to thi s t ree to f i nd the

opt i mal pl ayi ng st rategy. However , the exponent i al expl osi on of the

"l ook-ahead11 t ree makes t hi s i mpossi bl e i n pract i ce. ( I t i s est i mated

40

that there are about 10 possi bl e checkers games [ Samuel , 1959] and about

10^ possi bl e chess games [Shannon, 1950] , but l ess than 10^ mi croseconds

per century. ) Theref ore, the l ook- ahead process i s typi cal l y cont i nued

down to some non- termi nal (and possi bl y f i xed) depth at whi ch the posi t i on

i s eval uated wi t h a l ess accurate eval uat i on functi on. I f the branchi ng

f actor , N, and the dept h, D, are both f i xed, then N bot tom posi t i ons are

generated i n the mi ni max search. Even usi ng i ncompl ete ( non- termi nal )

trees, the l ook-ahead t rees f or most game pl ayi ng programs are st i l l very

l arge. I n chess, f or exampl e, a typi cal val ue f or the number of l egal

moves f roma mi ddl e-game posi t i on i s 35. I f = , then the

number of bot tom posi t i ons, ND, whi ch must be eval uated usi ng si mpl y mi ni

max search i s 1, 500, 625. For , N = 42, 521, 875. Chess

pl ayi ng programs are expected t o sat i sf y the t i me const rai nts of tournament

pl ay: they are al l owed t wo hours of computati on t i me to make 40 moves. For

a t ree of si ze =, thi s woul d mean that on the average about 220

mi croseconds woul d be avai l abl e f or eval uat i on of each bot tomposi t i on i f

the mi ni max al gor i thmwere used, i ncl udi ng the tree- searchi ng overhead i n

vol ved i n reachi ng that posi t i on. The need to ef f ecti vel y reduce the si ze

of the tree to be searched i s apparent .

I n the remai nder of thi s paper we wi l l rest r i ct our at tent i on to Max-

trees, i . e. , game trees that maxi mi ze at the top l evel . We can do thi s

wi t hout any l oss i n general i ty because of the obvi ous mappi ngs that exi st to


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t ransformMi n- t rees toMax- t rees. For exampl e, consi der thei somorphi sm:

cp(x) - x. Thenby t hedef i ni ti on of the mi n and maxoperatorswe see:

max( x1, x2, . . . , xn) = - mi n( - x- , - x2, . . . , - x^= cp(mi n(cptx1), cp(x2) , . . . , co(xn>)

and

mi n( x1, x2, . . . , xn) - - max( - x- , - x 2, . . . , -x r)

= cp(max(cptx1), co(x2) , . . . , co(xn) )

Si nce these i dent i t i es can be appl i ed r ecursi vel y, anarbi t rary Mi n- t ree

canbeanal yzed byanal yzi ng thecorrespondi ng Max- t ree created bycompl e

ment i ngal l theval ues i n theMi n- t r ee, repl aci ng al l mi n' sbymax' s, and

repl aci ng al l max' sbymi n' s. Theonl y di f f erence betweenaMi n- t reeand

i ts associ ated Max- t ree i sthat al l backed- up (andstat i c) val ues i n the

Max-t ree wi l l be thecompl ement of thecorrespondi ng val ues i n the Mi n- t ree

2. THEALPHA-BETA PRUNI NG ALGORI THM

The al pha- beta al gor i thm i sequi val ent to themi ni max al gor i thmi n

that they both f i nd thesame best move f romposi t i onp andbot h wi l l assi gn

the same val ueofexpected advantage t o i t . Al pha- beta i s faster thanmi ni

max because i t does not expl ore some branchesof the tree that wi l l not

af f ect thebacked-up val ue. Theal gor i thm can be i l l ustrated wi t h thetree

of depth three i nFi gure 2. 1. Assumi ng that thesearchi ng proceeds i n a

depth- f i rst f ashi on f rom l ef t tor i ght and that theroot node i s a Max

node, thesuccessors of Mi nnode p( l ) are f i rst exami ned and themaxi mum

val ue 3 i sbacked up top( 1, 1) . Theval ue3 nowbecomes anupper l i mi t

(beta val ue) f or thebacked- up val ue ofnode p( l ) . At thi s poi nt thef i nal

val ue p( 1) i sunknown, but si nce p( 1) i s a Mi nnodewe doknow that i t s

val ue must be at most 3.


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pO);v( 1) - 3

3 2 5 -8 1 -1

| +1: posi t i on not eval uated because ofacutof f s,

0 : posi t i on not eval uated because of |3 cutof f s.

Fi gure 2. 1. Exampl e of al pha and beta cut of f s.


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Next the procedure begi ns t o exami ne the successors of p( 1, 2) . When

p( l , 2, 2) i s eval uated t he l ower l i mi t (al pha val ue) f or the backed- up

val ue of the Max node p( l , 2) becomes 5. Si nce the al pha val ue of p(1, 2)

i s greater than the beta val ue of p( 1) ( =3) , p( 1, 2) cannot be t he l owest

val ued successor of p( l ) , and t hus there i s no need to eval uate t he re

mai ni ng successors of p( 1, 2) 0 That i s, Mi n wi l l not sel ect p(1, 2) because

Max can choose a branch l eadi ng to a hi gher val ue t han Mi n knows can be

achi eved wi t h p( l , l ) . Hence we have a beta cutof f at p( 1, 2, 2) . Addi t i onal

beta cutof f s occur at p( l , 3, l ) and p( 3, 2, 2) .

Af t er the bet a prunes at p( l , 2, 2) and p( l , 3, l ) occur , the val ue 3 i s

backed- up to p( l ) and becomes the l ower l i mi t (al pha val ue) f or the backed-

up val ue of node p. The procedure now begi ns to i nvest i gate the successors

of p(2). On eval uat i on of p( 2, l ) the beta val ue of p( 2) becomes 1. Si nce

thi s i s l ess than the al pha val ue (-3) of p, an al pha prune occurs at p( 2, 1)

Because of al pha cutof f s, nodes p( 2, 2) , p( 2, 3) , and p( 3, 4) , and thei r suc

cessors, are not eval uated. Note that onl y 15 bot tomposi t i ons are eval uate

by the al pha-beta procedure, whereas the mi ni max procedure woul d exami ne al l

28.

I n thi s exampl e t he al pha val ue used to obtai n the al pha cutof f s was

associ ated wi t h t he root node and t he cutof f s occurred near the bot tom l evel

of the tree. Not e that i f the tree i n the exampl e were one of greater

depth, the cutof f s at p( 2, l ) and p( 3, 3) woul d prune the potent i al l y vast

subt rees r ooted at p( 2, 2) , p( 2f3) , and p( 3, 4) . Fur t hermore, an al pha or

beta val ue may generate cutof f s at any node an even number of l evel s bel ow

i t . These are cal l ed deep cutof f s and a deep al pha cutof f i s i l l ustrated i n

Fi gure 2. 2.


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Backed- up val ues: 3

Stat i cval ues:

x x

x| posi t i on not eval uated because of deepa cutof f

x posi t i on not eval uated because of shal l owa cutof f

Fi gure 2. 2. Exampl e of deep al pha cutof f s.


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Beta cutof f s are anal ogous to al pha cut of f s, wi th t he rol es ofmi ni

mi zi ng and maxi mi zi ng reversed. The beta val ue speci f i es an upper l i mi t

f or the backed- up val ue of a Mi n node and i s used to generate cutof f s

among the successors of Max nodes at any l evel deeper i n the tree. Assum

i ng t hat the root node ( D O) i s a Max node, al pha cutof f s occur at even

l evel s and bet a cutof f s occur at odd l evel s.

I n order to formal l y def i ne the al pha- beta pruni ng al gor i thmdescri bed

above, we i nt roduce a few notat i onal conveni ences. Consi der the part i al

game t ree shown i n Fi gure 2. 3. We i dent i f y a node at depth d ^ D i n the

tree as p(T^), where i , someti mes denoted ( i ^ , i ^ ) , i s a vector of

l ength d whose components i.j , i 2, . . . j i i dent i f y the branch sel ected from

the nodes at successi ve depths i n the t ree al ong t he path f rom the root

node to p(T^). v(i^) i s the backed- up (f or an i ntermedi ate node) or stat i c

(f or a l eaf node) val ue associ ated wi t h node p( i ) .

To si mpl i f y subsequent subscri pts and summat i on ranges, we i nt roduce

the notat i on

, IK i fK i s evenLKL = 2L|J =. JK i fK i s oddLKJ = 2L SF I] + 1 =/ (2. 2) ] k-1 i f k i s even

Consi der the path f rom the root node to p( i ) . At l evel j , f or j and d eve

and 0 j < d, a maxi mi zi ng operati on i s i n progress and we have a l ower

bounda^(i^) on v( i ) , denoted the j - l evel al pha val ue, where


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depth

5 3 i 5 = (2, 3, 1, 4, 3)

Of ( l 5) = -1

0( i 5) =10

Fi gure 2. 3. A game t ree i l l ustrat i ng our notat i on f or the al pha- beta al gor i thm.


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a j ( i d ) - p m a x { v ( i 1 v( i 1 , . . . , 2) , . . . ,

v( i 1, . . . , i j , i j +1- 1 ) } f or > 1 (2. 3)

for i J + 1 = 1

Si mi l ar l y, at l evel j ,f or j and d odd and 1 j < d, a mi ni mi zi ng operat i o

i s i n progress and we have an upper bound b ( i ) on v( i ) , denoted the j -

l evel beta val ue where

b. . ( i d) ^ m i n j V C ^ , . . . , ^ , ! ) , v(i j , , i ^ , 2) , . . . , v( i1, . , i ^ , i ^+1-

f or i ^ > 1 (2. 4)

for i J + 1 = 1

Fi nal l y, def i ne the greatest al pha val ue, or si mpl y al pha val ue as

( i d) = m a x { a 0 ( i d ) , a 2 ( i d ) a


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v( l j :> p( i j and d odda a

a beta cutof f occurs. A cutof f at node p0- d) means that the remai nder of

the subt ree rooted at p( i , . . . >i d- 1) > i . e. , p( i d>! s parent node, i s not

exami ned i n the mi ni max search.

The above di scussi on of j - l evel al pha and bet a val ues proves the fol l ow

i ng f undamental l emma.

>

Al pha-Beta Lemma. Let v ( i g) be the backed- up val ue of a game tree usi ng

the al pha- beta pruni ng al gor i thmand l et v ^^g) be the backed- up val ue of

the same game tree usi ng the mi n-max al gor i thm. Then

I t shoul d be noted that there i s at l east one cl ass of r i sk- f ree pruni ng

al gor i thms that i s not subsumed by t he al pha- beta al gor i thm. For exampl e,

consi der the case where a top l evel move i s f ound to l ead to a wi n. Usi ng

the al pha-beta al gor i thm the next branch woul d have to be expl ored to some

extent bef ore bei ng pruned; but i t i s cl ear that al l other branches at the

top l evel coul d be pruned i mmedi atel y. Thi s coul d, of course, be appl i ed

at any poi nt i n the t ree where a wi n f or the pl ayer to move i s f ound.

The use of al pha- beta pruni ng i n the mi ni max search reduces by a l arge

f actor the number of bot tom posi t i ons whi ch need to be exami ned, typi cal l y

Some care must be taken i n tne i mpl ementat i on of thi s al gor i thm. I n theSecond Annual Computer Chess Champi onshi p (Chi cago, 1971) a chess programusi ng t hi s al gor i thm di scovered a mate i n two moves and t ermi nated i ts search.Af ter the opponent moved, the programbegan t he search agai n, di scover i ngf i rst a mate i n three. I t i mmedi atel y pruned and made the f i rst move of thi ssequence, mi ssi ng t he possi bl e mate on the move. I t cont i nued f i ndi ng matesi n more than one move unt i l due t o another bug i t f i nal l y l ost the game.


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by several orders of magni tude i n many game pl ayi ng programs. Previ ous

resul t s [Sl agl e and Di xon, 1969] have establ i shed the l ower l i mi t f or the

number of bot tom posi t i ons exami ned. The l ower l i mi t wi l l be achi eved i f

the stat i c val ues of the bot tom posi t i ons are i n "per f ect or der 11, i . e. ,

ordered such that every possi bl e al pha and beta cutof f occurs. I t can be

shown that i f per f ect order i s achi eved at every l evel , so that every pos

si bl e al pha or bet a cutof f occur s, then the number of posi t i ons at thebot

tomof the t ree of depth D and constant branchi ng f actor N i s:

D

NBP = 2N2 - 1 f or D even,po

D+1 D-1

NBP = N 2 + N 2 - 1 f or D odd.po

4

Thus f or = , NBP - 2449, whi ch di f f ers f rom35 - 1, 500, 625

by a f actor of 612.

Thi s very l arge rat i o of extremes i n per f ormance has i mportant i mpl i ca

t i ons f or searchi ng l arge game trees. The perf ormance of the al pha- beta

procedure may be f urther i mproved by the i ncorporat i on of heur i st i cs whi ch

reorder the nodes of the tree i nto a "more perf ect 11 arrangement . Var i ous

techni ques of f i xed and dynami c order i ng of nodes at i ntermedi ate l evel s

of the t ree are avai l abl e [e. g. , Sl agl e, 1963] . The rat i onal e f or t hese

types of heur i st i cs i s based on a correl at i on between the stat i c val ues of

nodes at i ntermedi ate l evel s of the tree and the f i nal backed- up val ues ob

tai ned f or these nodes. Thi s means that the nodes may be reordered before

eval uat i on of thei r subt rees to more cl osel y approxi mate per f ect order i ng an

thus obtai n a hi gher rate of pruni ng. The eval uat i on of the expected gai n


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over the si mpl e al pha- beta al gor i thm obtai ned by t he use of such heur i st i cs i s

compl i cated by t he f act t hat , whi l e the perf ect order i ng r esul ts provi de a great

est l ower bound f or the number of bot t om posi t i ons eval uated, the upper bound

of N i s unreal i st i c because i t i s great er , of ten by several orders of magni tude

than the number of bot tom posi t i ons eval uated wi t h the unmodi f i ed al pha- beta

al gor i thm.

Knowl edge of the expected val ue of the number of bottomposi t i ons

eval uated i n a l ook-ahead t ree usi ng al pha- beta pruni ng shoul d be usef ul

because the expected val ue provi des a much t i ghter upper bound f or the

average perf ormance of the t ree- searchi ng procedures than does the upper

bound gi ven by t he mi ni max al gor i thm. Thus, when eval uat i ng the ef f ect i ve

ness of heur i st i cs to be used i n conj uncti on wi t h t he al pha- beta al gor i thm

one mi ght determi ne not onl y how cl osel y the resul t i ng per f ormance approach

es the l i mi t under perf ect order i ng, but al so how much bet t er (or worse!)

the resul t i ng per f ormance i s compared wi t h t hat of the unmodi f i ed al pha- beta

al gor i thm.

3. A PROBABI LI STI C MODEL OF GAME TREES AND SOME I NI TI AL OBSERVATI ONS

I n order to draw some quant i tat i ve concl usi ons about the per f ormance of

the al pha- beta procedure i t i s necessary t o preci sel y model game t rees.

However, our purpose here i s to keep the model suf f i ci ent l y si mpl e so that

anal yti cal techni ques can be appl i ed t o our study of the perf ormance of the

al pha- beta procedure.

Our model i ncl udes three si mpl i f yi ng assumpt i ons.

1. Let us assume our game trees are compl ete t rees of depth D wi t h

constant branchi ng f actor N, e. g. , Fi gure 2. 3 where D = 5 and N = 4.


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- 15-

Note that there are al ways bot tom posi t i ons, and i n general

dN nodes at depth d i n the game t ree.

2. To study the probabi l i st i c propert i es of the game trees we must

provi de a model of the stat i c val ues assi gned to the bot tomposi

t i ons. A si mpl e yet appeal i ng assumpt i on to make i s. that the

val ues, v( i p) , of al l N bot tom posi t i ons are i ndependent , i den

t i cal l y di str i buted ( i i d) r andomvari abl es wi t h arbi t rary di s

t r i but i on f uncti on VD( x ) .

3. The onl y requi rement on VD( x ) , i n addi t i on to the standard prop

er t i es of a cumul at i ve di st r i but i on functi on [cf . Parzen, 1960] ,

i s t hat i t be cont i nuous. I n other wor ds, we requi re that the

probabi l i ty t hat the val ue of a l eaf node i s preci sel y x i s van-

i shi ngl y smal l , to el i mi nate t he possi bi l i t y of two or more nodes

havi ng the same val ue.

The second and thi rd assumpt i ons can be equi val ent l y restated by model

i ng the l eaf nodes as a randompermutat i on of the ordered l i st of val ues;

i . e. , each of the NI assi gnment of val ues to the nodes i s equal l y l i kel y.

Not e that the actual val ues of the N bot tom posi t i ons i s not of i nterest

when st udyi ng the behavLDr of mi ni max searchi ng, and the al pha-beta procedur

i n par t i cul ar , but onl y thei r rel at i ve order i ng. Our previ ous di scussi on o

the t ransf ormat i on of Mi n- t rees to Max-t rees i mpl i es that the probabi l i ty

of exami ni ng a part i cul ar bot tom posi t i on i n a Mi n- t ree wi t h cont i nuous

di st r i but i on Vp( x) i s equal to the probabi l i t y of exami ni ng the cor re

spondi ng bot tomposi t i on i n the associ ated Max- t ree wi t h di st r i but i on

Vp(-x). Thus, si nce the behavi or of the search i s i ndependent of the

speci f i c di str i but i on (as l ong as i t i s cont i nuous) , each of the subsequent

resul ts about Max-t rees wi l l be t rue of Mi n- t rees aswel l .


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We can now make t he obvi ous but i mportant observat i on that the val ue

of a node at any l evel i n the game t ree i s i ndependent of the val ues of the

other nodes at the same l evel * I n addi t i on, si nce the l eaf nodes are i i d

random var i abl es, i t f ol l ows f rom the structure of our game t rees, i . e. ,

uni f ormdepth at al l bot tom posi t i ons and constant branci ng f actors, that

al l the nodes at any l evel i n the t ree are i i d random var i abl es. I t i s

i nterest i ng to consi der the actual di st r i but i on of the val ues of the nodes

at an arbi tr ary l evel . I t f ol l ows f rom f i rst pri nci pl es i n order stati sti cs

that the di st r i but i on f uncti on of the maxi mum of n i i d random var i abl es wi t h

di str i but i on funct i on F( x) i s [F (x)] n and the di st r i but i on f uncti on of the

mi ni mumof n i i d random vari abl es wi th di st r i but i on f uncti on F( x) i s

1-[1-F(x)] n. Hence:

VQ( x) = [ V x) ] * ,

V x) = 1 - [1 -V2(x ) ]

N

,

V, ( x) = [ V2( x ) ]N;

and i n general :

Vf c(x) = V +1( x ) , f or k=0, 2, . . . , LD- 1J e ( 3. 1)

Vf c(x) ^+ 1 ( x ) , f or k=1, 3, . . . , LD- 1j Q (3. 2)

where F( x) denotes the survi vor f unct i on, i . e. , F(x) = 1- F( x).

To i l l ust rate the rel at i on of the di st r i but i on of the nodes f romone

l evel to the next, the di st r i but i on f uncti on at al l the l evel s i n the game

tree of Fi gure 2. 3 are shown i n Fi gure 3. 1. The val ue of the l eaf nodes

are assumed t o be uni f orml y di st r i buted over the uni t i nterval i n Fi gure 3. 1,

but thi s i s onl y f or i l l ustr at i ve purposes; as stated bef ore, V D (X ) c a n he

any cont i nuous di st r i but i on f uncti on.


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Fi gure 3. Cumul at i ve di st r i but i on f uncti on of val ues of nodes i ngame t ree wi t h .


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4. THE PROBABI LI TY OF EVALUATI NG A NODE I N THE GAME TREE

We are i nterested i n the stat i st i cs concerni ng t he number of bot tom

posi t i ons eval uated i n an al pha- beta search of a game tree. We wi l l start

by f i ndi ng the probabi l i t y t hat an arbi t rary node wi t h i ndi ces i ^ i s ex

ami ned by t he al pha- beta procedure; cal l thi s probabi l i ty of exami nat i on

Pr{i d}.

To f i nd Pr[ i } we f i rst consi der the path f rom the root node to p( i ) .At l evel j , f or j a non- negat i ve, even i nteger l ess t han d, a maxi mi zi ng

operat i on i s i n progress, we have a l ower bound on v ( i ) , i . e. , a i ) ,

and the di st r i but i on functi on for the j - l evel al pha val ue i s

A. (x) = [V ( x ) ] l j + 1 \ (4. 1)3 9 j +1 J

As i . approaches N, the f ormof A. . (x) approaches V . ( x ) .

Si mi l ar l y, at l evel j , f or j a posi t i ve, odd i nteger l ess t han D, a

-*

mi ni mi zi ng operat i on i s i n progress, we have an upper bound on v ( i ) , i . e. ,

b. ( i , ) and the survi vor f unct i on f or the j - l evel beta val ue i sJ d

B. . " (x) - [ V- . - C x ) ] 1 ^ 1 \ (4. 2)

Note that the j - l evel al pha and beta val ues associ ated wi t h i ^ are

i ndependent but not i dent i cal l y di st r i buted random vari abl es and thedi s

t r i but i on funct i on of cKi ) i s

Arf (x) = An (x) A (x). . . A (x) (4. 3)0 , 1 , 2, i 3 l d- 1^,1o


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and si mi l arl y the survi vor f uncti on ofP dd )i s

B-* (x) - B- . (x) B (x). . . B. . . (x) (4. 4)1* 12 3 j l 4 L d" 1 J o 1LdJ e

We can now prove several f undamental propert i es of the al pha- beta

pruni ng al gor i thm.

Theorem 1. Node p( i d) i s exami ned, i . e. , not pruned, by the cHS pruni ng

al gori thm i f and onl y i f

of(id) < P( i d) . (4. 5)

Proof . Fi r st , suppose

I n other wor ds, i f p( i d) i s not exami ned, an al pha or beta cutof f has

occurred; the candi dates f or p(i *) are shown i n Fi gure 2. 4. I f we con

si der the al pha cutof f case, Eqn. (4. 6) , we see


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Fi gure2. 4


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- 21-

si nce bj , (Tj) i s the mi ni mum of the i -1 successors of P(*-j ] ) Cl ear l y

PTFJ.,) , (1 (4. 9)

by def i ni t i on of the bet a val ue, Eqn. 2. 6. FromEquat i ons ( 4 . 6 ) , ( 4 . 8 ) ,

and ( 4. 9) i t f ol l ows that

ati^j* PTFJ-I> ( 4 - 1 0 )

and f romEqns. (2. 5) and (2. 6)

of(i d) *P( i d) (4. 11)

whi ch cont radi cts Eqn. (4. 5) . By a preci sel y anal ogous argument our second

case, Eqn. ( 4. 7) , al so l eads to Eqn. ( 4. 11) , and hence a cont radi ct i on.

Now i t remai ns to be shown that i f p(i ) i s exami ned, then Eqn. (4. 5)

must f ol l ow. Agai n proof by cont radi cti on provi des the si mpl est argument ,

i . e. , suppose

( i d) * B(i d) . ( 4. 12

I t f ol l ows f rom thi s i nequal i ty t hat there must exi st aj and ak such

that

bj (V * a k (V - ( 4 - 1 3 )

Supposek>j;then t here exi st s a nodep(T +,*) such that

v ( T k + I * >= V V - ( 4 - 1 4 )

However, the above t wo equati ons guarantee a beta cutof f no l ater than

p( i k+1*) and t hi s cont radi cts the assumpt i on of no cut of f .


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I f k < j , by a preci sel y anal ogous argument we get an al pha cutof f ,

agai n a cont radi cti on,

Now that Theorem 1 has been f ormal l y presented i t may be hel pf ul to

provi de an i ntui t i ve descr i pt i on. Theorem 1 says that a node i n a game

tree i s exami ned i f and onl y i f the associ ated upper bound (beta val ue) i s

greater than the associ ated l ower bound (al pha val ue) . Not e that i n thi s

paper we have def i ned al pha and beta val ues f or al l nodes i n the t ree, not

j ust those nodes exami ned by the al pha- beta procedure.

The next theorem i s the cent ral resul t of thi s sect i on: an expressi on

f or the probabi l i ty of eval uat i ng an arbi t rary node i n the game t ree.

Theorem2. Let A-+ (x) and (x) be the di st r i but i on f unct i ons of the al pha

J J -and bet a val ues, respecti vel y f or a node p( l ) at depth d i n a game t ree.

Then i f i . > 1 f or some j {2, 4, ,LdJ }:

(a) Pr{i d} = f Bj (z) dAj (z)-c o d d

and i f i j > 1 f or some j 6 { 1, 3 , . . . , L dJ q } :

(b) Pr{i , } = r A? (z) dBj (z)d J -c o^ d ~*d

and i f i . = 1 for al l j :J

(c) Pr{i d} =1 .

Proof . Fi r st , par t ( a) . FromTheorem 1 we know that the statement "posi

t i onp(i , ) i s not pruned11 i s equi val ent to the statement ) < B( i ) "

and so:


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Pr{i d} = Pr{(y(i d)

(Note: the condi t i on cKi ) = z i s def i ned onl y i f some el ement of i ^ wi th

an even i ndex i s greater than 1. )

-o o d d

-o o d d

The proof of par t (b) i s anal ogous to the above proof f or part ( a) . Part

(c) i s obvi ous, si nce the f i rst l eaf node must al ways be eval uated.

5. THE EXPECTED NUMBER OF BOTTOM POSI TI ONS

I n order to der i ve the expected number of bot tom posi t i ons E[NBP^ ]

eval uated i n a tree of depth D and branchi ng f actor N whi ch conf orms to

our model , we t ake advantage of the l i near i ty of the expected val ue operat or ,

i . e. , E[ Exi ] = SE[ xi ] . Hence E[ NBP D ] i s equal to the sumover the set of

al l bot tom posi t i ons of the probabi l i ty that the bot tom posi t i on i s eval uated,

i . e. ,

E[NBP ] - S E . . . S PRFL} (5. 1)w , u l t l i 2 N

and we may compute Pr i j usi ng Theorem2.

To i l l ust rate the method we wi l l f i rst eval uate E[NBP 9 ] . Fi rst con-N, Z

si der the case for i 0 > 1.


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Pr {i ) = - j' " A-+ (z) df j ( z) (f romThm. 2b)1 _oo

2

- - f V 1 (z) dV2 (z) (fromEqns. 4. 1- 4. 4).CO

i . -l i 9- 1- - J* [ l - V z) ] 1 dV2

z ( z) (f romEqn. 3. 2)-00

We may now perf orm the subst i tut i on u V2( z ) , el i mi nat i ng the speci f i cdi s

t r i but i on of bot tomposi t i ons.

Pr i i - j =- I 0- u ) d uz ' 1

i . - l 1

= ( I - x) d xJ o

i - 1 i -1 - g 1

J o

i -1 i 2 - l

4 - 3 ( 1 , , - f - ) ( i 2 > D

where 0( x, y) =F ^ ^ ^ , the beta f unct i on.

Si mi l ar l y we can f i nd t he val ue of Pr {i 2) f or i 2 = 1 and i > 1 f rom

Theorem2a.

Pr {i2

) = f & (z) d (z)-o o 2 2

f V 1 ( z) d V, (z)2

.00

For i 0 85 1 we have

Pr{i 2} =f d V*1 ' ( z )

L 2 i . -l

V1 1 00.00


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by the f undamental theorem of cal cul us. Theref ore

Pr{i 2} =1 ( i 2> 1, i 2 = 1)

By Theorem2c, Pr{( l , 1) } = 1.

Thus

N N N i -1 iE[NBP 2 ] = 1 + 2 1 + S L - | ^

N , /

1 2 i2=2

N 1 N

. N- l NN + N 1 . S5>

1=1 j =1

, N-l N i - iN +J E i E F u3" ' ( l - u ) M du

i =l j =l 0

1 N- l . -1 / NN + 2 i | ' ' ( l - u) N ( SuJ - ' ] duN I=l 0 VJ= 1

1 N _ 1 l n " 2 n

N + i 2 i f ( l - u) N (1-u ) duN I=l JO

E[ NBPN) 2] = N + V j i j - [ *


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Thi s f orm i s qui te adequate f or comput i ng the expected val ue over the range

of branchi ng f actors useful i n game pl ayi ng programs. For smal l val ues of

N, E[NBP 9 ] was computed exactl y usi ng MACSYMA, a symbol i c mani pul at i onN, Z

programdevel oped at MI T [ Bogen, et al . 1972] . These val ues are presented

i n Tabl e 5. 1.

Next we wi l l eval uate Pr f i } f or arbi t rary dept h. A f ew prel i mi nary

def i ni t i ons and l emmas wi l l suppl y the necessary f oundat i ons.

Fi rst we def i ne the operator T( f , k) f or a f uncti on f and non- negat i ve

i nteger k as f ol l ows:

r i f k = 0T(f , k)

l - [ T( f , k-1) ] N i f k > 0

For exampl e, T( V3( x ) , 2) = 1-[T(V3(x ) , 1)]

= l - [ 1- [ T( V3(x) , 0) ]N] N

N N

Lemma 5. 1a

Lemma 5. l b

Lemma 5. l c

Lemma 5. I d

Vk

Vk ( x )

T( VD( x ) , k) , D even, k

T( VD( x ) , k ) , D odd, k

T( VD( x ) , k) , D even, k

T( Vp( x ) , k ) , D odd, k

1, 3, 5, . . . , D-1

0, 2, 4 D-l

0, 2, 4, . . . , D

1, 3, 5, . . ,D

Proof : We wi l l prove 5. 1a by i nduct i on on k. The other proof s are the same.

For k 1, D even, we have f rom Eqn. 3. 2

Vi ( x )

Vi ( x )

V D- 1 ( X )

vZ(x)

1- Vj (x)

T( VD( x ) , 1)


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1 1

Tabl e 5.1. E[ NBPJN, 2

E[NBP . J > N B P 2 C 2 )

3

5 2 1

N B P 2 C 3 )

7 0

5 4 7 0 3 3

N B P 2 C 4 )

4 5 0 4 5

1 7 2 0 2 9 1 1 6 9

N B P 2 < 5 )

9 7 3 4 9 6 1 6

1 4 7 8 6 0 0 1 0 1 7 7 7 1

N B P 2 < 6 >

6 1 7 1 0 9 2 0 0 4 0 0

1 4 8 3 6 3 4 2 7 3 4 3 1 5 2 7 6 1 7

N B P 2 C 7 ) -

4 7 9 1 3 4 8 9 5 5 2 3 4 9 9 8 0

1 2 5 2 6 1 5 1 4 5 6 3 8 0 4 3 8 0 9 7 0 6 7 2 6 9

N B P 2 C 8 )

3 2 4 0 8 6 3 1 6 9 1 4 1 5 0 8 4 0 8 7 4 8 2 5

6 0 6 3 0 1 9 4 2 4 9 2 9 1 7 2 5 1 2 6 4 9 1 1 0 1 9 2 4 9 7 7

N B P 2 ( 9 > . . . .

1 2 9 0 2 8 3 6 1 5 9 7 6 2 0 9 6 8 7 3 7 8 0 7 9 0 1 9 2 0 0

3 9 0 1 5 2 2 6 2 5 9 2 7 7 9 8 4 1 9 6 8 1 7 8 0 9 9 7 1 6 0 6 2 2 1 7 5 1 8 0 9

N B P 2 C 1 0 ) *

6 9 7 2 0 3 7 5 2 2 9 7 1 2 4 7 7 1 6 4 5 3 3 8 0 8 9 3 5 3 1 2 3 0 3 5 5 6 8

2 0 1 9 6 5 4 2 6 4 2 3 8 0 8 7 6 5 6 2 3 8 3 8 5 6 5 6 6 4 5 9 6 1 1 3 2 3 4 8 9 3 0 1 3 8 3 1 1 9

N B P 2 C 1 1 ) - - * '

3 0 8 1 5 2 1 7 6 7 6 5 3 8 6 3 5 0 5 8 4 6 2 6 3 8 2 4 1 6 5 9 5 8 5 8 7 3 5 2 1 1 6 4 8 8 0 0

NBP2 ( 12) =?29^I5234423499941_235877

26468917348837676265384815256420322119583790673790618350""


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Assume , . (x) = T(\L(x), k*) f or a part i cul ar odd k*. Then

VD- k * - 2 ( x ) - 1" >- k*- 1( x ) ( f r m E q n * 3 , 2 )

1 - ( 1 - V D- k * - l ( x ) ) N

- 1- ( 1-VJJ_k*(x))N (f romEqn. 3. 2)

l - [ 1- TN( VD( x ) , k* ) ]N

=l - TN( VD( x ) , k*+l )

= T( VD( x) , k*+2) , provi ng the i nduct i on step. I

We now observe that i f i 1 f or m 2,4, . . . , | .DJ , then Theorem2a maym e

be appl i ed di rectl y.

Pr{i _} = f By ( z ) dAy ( z )

F N B . . ( z ) d( N A ( Z ) )- j&=2, 4, . . . , LDJ e

X" ' k=1, 3, . . , , LDJ o K" '

i.-l i , -1j 1 " n V

&

( z ) d ( n VK

(ZJ 6=2,4 |DI k- l , 3, . . . , LDj

o

V1

f d( n V ( z) , si nce i 1 f or- k=l , 3, . . . , LDJ o

K I even *

V ,00 '

II V ( z ) I b y the Fundamentalk=1, 3, . . . , LD] 0 -

00Theoremof Cal cul us.

Pr{i D} = 1 f or i ] = i . = . . . = U = 1. (Thm. 2c)

.'. Pr{i n} = 1 for i 9 = i ^ = . . . = i | D ] =1 and i 1 for some odd m.J e

For the rest of the devel opment we wi l l assume i m > 1 for some even m.

We are now ready to consi der Pr {i D} f or arbi t rary depth D. FromTheorem

2b we have


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- 29-

-oo D D

- - T n \ , (z ) d( n B , . ( z ) )k=1, 3, . . . fLDJ o

K_ l j 2, 4, . . . , LDJ e*-1

i . - l i . -l- - T n v k ( Z) d( n v * ( z

k=l , 3 LJ)J J 2, 4,...,LDJ *o e

As i n t hecase f or D 2 ,wewi sh toperf orma subst i tut i on whi ch wi l l

el i mi nate theunder l yi ng di str i but i on. For Dodd we canappl y l emmas5. 1b

and5. I d:

Pr{i }- - p n [T(V( z ) , D- k ) ] ^ \( n [T(V( z ) , D-D] 1*- k=l , 3 , . . . , | _Dj * = 2 , 4 , . . . , [ D J

o e

Substi tut i ngu V^(z)9 weobtai n

P r f L } - - ! 1 1 n [T(u, D- k) ] k d ( n [ T ( u , D- ja)] l A ) (5. 3)-1 i . -l

d ( n [T0 k=1, 3 [DJ * * 2 , 4 , . . . , LDJ

' e

f or D odd.

Si mi l ar l y, f or Devenweappl y l emmas5. 1a and 5. 1c:

i , -1 i . -Pr(i }- - J* H [ T(V ( z ) , D- k)] d( n [ T(V ( z ) , D- A) ] 1

- k=1, 3, . . . j&2, 4, . . . , [Djo e

(5. 4)1 i . - 1 i . - l

P r C l } - +J n [T(u, D- k) ] K d( [ T ( u , D - l ) ] ' ) , Deven.' 0k1, 3, . . . , [Dj 2, 4, . . . J D J

o e

Combi ni ng equati ons(5.3) and (5. 4) ,

pr {i D}=(-DY n [ K u . D- k ) ] ^1 d( n [ T(u, D- j t ) ] Xi \

' 0k=1, 3, . . . , [Dj tf2yUt...J DJo ^J e


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Di f f erent i at i ng the second product and observi ng that the termf or

whi ch i ^ 1 di sappear , we have

. i -1Pr{i n} - (- l )

Df II T (u, D- k) )0 k = 1 ' 3 LDJ o , .

S [( n T * (u, D- i )) dTm ( u, D-m)]m=2, 4, . . . , LDj J, 4. . . . , | DJ

i {1 6 Jfcf m .m

(5. 5)

We eval uate the der i vat i ve ofEqn. 5. 5 wi t h the fol l owi ng l emma.

1 i f k = 0k-1

nn=0

Lemma 5. 2; -r- T( u, k) = < k-1 .d U n T ' ( u, n) i f k > 0

Proof (by i nducti on onk) ;

3JT < U '0 )

" '

du

* k -2d * k- 1 N-1 *

Now assume -rT(u,k - 1) = (-N) II T (u, n) f or some f i xed k Thend U n=0

d * d N *

N-1 * d *= -N T W ' (u, k - 1) ^ T( u, k -1)

* k -1= ( - N) k n ^( u . n )

n=0

Appl yi ng thi s l emma to Eqn. 5. 5 we obtai n

i -1 i - - 1

Pr{J n}- ( - UYC n Tk (u, D-k) ) S [( II T * (u, D-A))

D ' 0 k=l , 3, . . . , | _Dj m=2, 4, . . . fLDJ eA=2, 4, . . . , LDJ e i f 1 1 f m

mi -2 D D-m-1 -

( i - 1) Tm (u, D-m)( -N) " n ir' Oi . n) ) ] dum n0


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Movi ng thesummat i on to thef ront , wehave

m=z,tt, . . . , [DJ 0k=l naO

**^1 k / m

Col l ecti ng termsandnoti ci ng that ( - l ) 20" 01 = 1, we f i nal l y have

2, 4, . . . , ] 0 k=0 du (5. 6)

where Pk( m) = 7

0 i f i =1m

i D. k+N- 2 i f k < D-m

i D_k- 2 i f k = D-m

i D_k- 1 i f k > D-m

LD 4

JD i f D i seven

J D-1 i f D i s odd

and i > 1 f or some evenm.m

Eval uat i on ofEqn. 5. 6 requi res i ntegrat i onoff uncti onsof thef orm

1 k j

I k ( j rj 2 J k> - J H Tn( u, k- n+l ) .

0n=l

For exampl e, i f i / 1

1 i +N-2 i -2 i -1Pr{i }= ( i 9- l ) NJ* T ( u,0)T (u, l ) T

1 ( u,2)du* 0

, i +N-2 i -2 KT i . -l= ( i - 1) Nf1 u3 (1- uN) 2 ( l - ( l - uN) N) 1 du

( i 2- l ) N 1 3 ( ^ - 1 , 1 2 - 2 , 1 ^ - 2 ) ,


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These i ntegral s may be eval uated exactl y usi ng the recurrence rel at i ons of

the fol l owi ng l emma.

Lemma 5. 3:

V1

j 2+i

j

i f k = 1

i f k - 2

'1S (-0 ( J 2+AN, J 3, . . . , J k) i f k > 1.

Proof : = [ u du* 0

I 2( j , , j 2) - fV-uV1 u2du

1 V1

k j 4

k 1 K 0 n=1du

= I*1 T ^ (u, k) n T*"(u, k-n+1) duJ

' 0 n=2

1 N j l k j k- (" (1- TN( u, k- 1) ) ' n T (u, k- n+l ) du

J 0 n=2

= f Z ( - 1 ) A( ) [ TN( u, k- 1) ] A n TJ k( u, k- n+1) du

k j ,

0!F0n=2

(f romthe Bi nomi al Theorem)


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S 1("D f!1

) ^ ( u. k - l ) N TJ k( u, k- n+1) dui -0 *' 0 n=2

i ^(2i) A - i v - ^ Vj&=0

I

As another exampl e of the method, we present the di f f erent cases f or

D = 4 and arbi t rary N.

Pr{i j = 1 f or i j - i i j i l f romThm. 2c.

Pr{i , }1 for i 9=i ,=L f romEqn. 5. 2.

i i v 1 v 2 i - ^ + N - 2

Pr{i , } = a , - D i r r , T ' (u, 3) TZ (u, 2) T (u,L)du0

- (IJ-DN2 ( 1 1, 12- 2, 1 - 2, 0) i f i 4=L and 1 2/ 1

Pr{i 4} = ( i 4-L) I 4( i r 1, 0, i 3-L, i 4- 2 ) i f i 2=L and ^ J 1

Pr{i 4} =(IJ-DN2 I 4( i RL, i 2- 2 , i 3+N- 2, i 4+N- 2) +

+ ( i 4-1 ) I 4( i RL, i 2- 1, i 3- 1, i 4- 2) i f 1 and i ^L.

N N N NThen E[NBP , ] - 2 2 2 2 Pr{i . },

i =1 i =1 i =1 i =11 2 3 4

and I 4( J 1, J 2J 3J 4) i s eval uated wi t h Lemma 5. 3.

I t i s possi bl e t o el i mi nate one summat i on f rom thi s method of eval uat

i ng E[ NBPN D ] . We observe that

P r [ 7 l > } = 9 L 2 m i V V V " - ' W >m=2, 4, . . . , | _DJ e

where > 1 f or some even m, t f c= PD_k( m) - i k (Eqn. 5. 6) , and I D i s computed

wi t h a D-2- f ol d summati on. However , the summati on on the i ndex i-j may be

combi ned wi t h the f i rst reduct i on of Lemma 5. 3.


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i ,-". pr - 1, L2,4>L, LDJ e( v, ) i D


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- 3 6 -

7 1 9

E [ N B P J = N B P 3 C 2 ) -

2 , 3

' 1 0 5

5 7 8 7 9 3 8 9 7 9 1

N B P 3 C 3 )

2 9 7 4 5 7 1 6 0 0

1 0 5 8 6 1 2 1 3 4 8 2 7 2 0 8 0 2 3 1 6 8 8 7 9 6 7

N B P 3 C 4 )

2 6 3 8 9 8 8 5 8 0 5 8 6 6 5 6 8 4 7 1 2 3 5 7 5

NBP3 ( 5 ) = 12 1-^I?Z51Z2457756H97 8 7 6 8 5 1 _ 26 9 3 7 8 7 8 7 6 2 3 8 7 5 9 0 4 1 7

1 7 4 1 3 6 0 2 5 7 6 2 2 4 0 9 3 1 6 3 8 2 8 2 9 6 9 7 4 8 1 7 3 4 5 9 0 6 1 1 9 9 7 1 6 4 8

r ~\ 7 7 5 0 3E [N BP J = N B P 4 C 2 ) - - - - - -

' \ 6 4 3 5

3 7 9 7 1 7 6 7 4 9 8 0 5 9 8 5 7 0 5 8 3 3 0 8 7 2 6 5 7 0 0 2 3

N B P 4 ( 3 > .

8 4 0 0 3 7 7 9 4 7 3 6 9 5 8 8 9 4 3 6 9 9 2 4 1 7 2 6 4 0 0

Tabl e 5 . 2 . Exact val ues for E [ N B P n f or D = 3 , 4 .


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the val ues are drawn, we see that the expected number of move generati on f or

a depth D search i s si mpl y t he sumof the expected number of non- termi nal

posi t i ons at each l evel , i . e. ,

E [ N M G N > ] ) ] = 1 + V E [ N B P N > I ]

6. APPLI CATI ON OF THE GAME TREE MODEL TO CHESS

Several assumpt i ons have been made to si mpl i f y t he anal ysi s of the

model whi ch do not conf orm to the propert i es of game t rees i n general .

Fi rst , the model assumes a f i xed branchi ng f actor and f i xed depth. Many

game pl ayi ng programs ( though not al l ) use searches of var i abl e depth and

breadth. Second, the val ues of the bot tomposi t i ons have been assumed

to be i ndependent ; i n pract i ce there are strong cl uster i ng ef f ects. For

exampl e, i n a chess programwi t h an eval uat i on f unct i on whi ch depends

st rongl y on materi al , a subt ree whose parent move i s a queen capture wi l l

have more bot tom posi t i ons i n the range correspondi ng t o the l oss (or wi n)

of a queen than wi l l subt rees whose parent move i s a non- capt ure. The

f i nal assumpt i on i s t hat the probabi l i ty that two bot tom posi t i ons i n the

t ree wi l l have the same val ue i s zero ( cont i nui ty assumpt i on) . I n pract i ce,

game programs sel ect the val ue of the termi nal posi t i on f roma f i ni te (and

somet i mes smal l ) set of val ues.

No attempt has been made to model modi f i cati ons to t he basi c al pha-

beta search, such as f i xed order i ng, dynami c order i ng, or the use of

aspi rat i on l evel s. Thi s model shoul d be vi ewed as an upper bound i n the

sense t hat any program can perf orm thi s wel l i f the moves are no worse t han

randoml y ordered.


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I n thi s secti on we wi l l i nvest i gate the ef f ects of these assumpt i ons

by compari ng the predi cted average search ef f ort wi t h the observed searches

of a chess program. Si nce several exi st i ng chess programs use a f i xed

branchi ng f actor (sel ecti ng the N best moves accordi ng to a stat i c eval ua

t i on funct i on) and f i xed depth (at tempt i ng to resol ve i ssues of qui escence

wi th a stat i c anal ysi s), we wi l l concent rate on the i ndependence assumpt i on

and the cont i nui ty assumpt i on.

I n order to assess the expected ef f ect of the cont i nui ty const rai nt ,

we rel ax i t i n the model so that the val ues f or the eval uat i on functi on

are chosen f romR equal l y l i kel y di st i nct val ues. I f R88 1, we have, i n

ef f ect , per f ect order i ng, si nce equal val ues wi l l produce a cutof f . As

R approaches i nf i ni ty the expected number of bot tomposi t i ons i s predi cted

anal yt i cal l y by thi s paper . The var i at i on of the number of bot tom posi t i ons

wi t h R i s shown i n Fi gure 6.1 f or D 3; these curves were generated usi ng

a Monte Car l o si mul at i on.

The chess programused f or compari sons was a modi f i cat i on of the

Technol ogy Chess Program [Gi l l ogl y, 1972] . The Technol ogy Program (Tech)

i s a "brute f orce" programwhi ch i nvest i gates al l l egal moves to a f i xed

depth; al l chai ns of captures f rom these bot tom posi t i ons are expl ored. The

termi nal posi t i ons are eval uated onl y wi t h respect to materi al , where a pawn

i s consi dered to be wort h 100 poi nts, kni ght and bi shop 330, rook 500, and

queen 900. A number of posi t i onal heur i st i cs are appl i ed st at i cal l y at the

top l evel , and var i ous modi f i cat i ons are made to the basi c t ree search.

Tech was modi f i ed f or thi s anal ysi s to search t rees of f i xed depth

and branchi ng f actor (the branches to be exami ned sel ected randoml y) , and


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the t ree search modi f i cat i ons were del eted. I n addi t i on to Tech' s st andard

eval uat i on f unct i on, an opt i onal eval uat i on term f or mobi l i t y was programmed.

Thi s term i s useful f or the anal ysi s, si nce i t i ncreases the ef f ect i ve range

of eval uat i on and decreases the degree of correl at i on i n subt rees. Most

chess programs have eval uati on f unct i ons whi ch are consi derabl y more compl ex

than Tech' s.

I n order to ensure a r easonabl e mi x of openi ng, mi ddl e and endgame

posi t i ons a compl ete game was anal yzed, consi st i ng of 80 posi t i ons (Spassky-

Fi scher , Reykj avi k 1972, game 21) , Each of these posi t i ons was anal yzed by

the modi f i ed Tech programs f or D = 2, 3, and 4 over the ef f ect i ve range of

branchi ng f actors. Fi gure 6. 2 shows the anal yt i c r esul ts f or these para

meters wi t h the empi r i cal val ues obtai ned usi ng Tech' s standard eval uat i on

f unct i on. At a typi cal poi nt ( = ) , the observed r ange ( R) of

di st i nct bot tomposi t i on val ues i n the t rees var i ed bet ween 1 and 9, wi t h

medi an 5. Thi s agrees wel l wi t h Fi gure 6. 1. To demonst rate the ef f ect of

the i ndependence assumpt i on, these poi nt s were r e- run wi t h the programmodi

f i ed so t hat a val ue whi ch woul d r esul t i n a prune by equal i ty was randoml y

perturbed up or down. Thi s si mul ated an eval uat i on f unct i on that assi gns

uni que val ues f or al l the bot tomposi t i ons. The pertubat i ons changed the

val ue of the posi t i on by at most two poi nt s, si nce there are at most twoal phas or two bet as bei ng kept at any gi ven ti me i n a depth 4 t ree. Si nce

two poi nt s i s smal l compared to the val ue of a pawn (100poi nt s) , t he t i e-

breaki ng procedure does not si gni f i cant l y af f ect the correl at i on among posi

t i ons i n a subt ree. The systemat i c di screpancy between these poi nt s and

the anal yt i c curve must then be due to the assumpt i on of i ndependence.


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- 4 1-

BRANCHI NG FACTOR (N)


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I n Fi gure 6.3 we pl ot the anal yt i c curves wi t h t he data f romthe

eval uat i on functi on whi ch i ncl udes mobi l i t y as wel l as mat er i al . The

range i s much hi gher f or thi s eval uat i on f unct i on, so that the number of

bot tomposi t i ons i s consi derabl y hi gher than the unmodi f i ed versi on. The

range (R) at = f or thi s versi on var i ed between 12 and 82, wi t h

medi an 43. Si nce the ti ebroken poi nt s are onl y sl i ght l y hi gher , i t i s cl ear

that the range i s cl ose to the ef f ect i ve maxi mum range f or that eval uat i on

f uncti on. I t i s of i nterest to not e that the t i ebroken poi nts l i e al most

on the anal yti c l i ne, i ndi cat i ng t hat the i ndependence assumpt i on i s more

near l y correct f or thi s eval uat i on f uncti on.

Compar i son of these graphs i ndi cates two ways i n whi ch t he eval uati on

f uncti on af f ects t he number of bot tomposi t i ons eval uated: (1) as the

range of the eval uat i on f unct i on i ncreases, the number of bot tomposi t i ons

i ncreases; (2) as t he correl at i on among val ues i n the same subt ree i ncreases,

the number of bot tomposi t i ons decreases.

7. EMPRI CAL OBSERVATI ONS

For l arge val ues of N and D, the anal yt i c r esul t presented i n Secti on

5 i s i n i ts present f orm computat i onal l y i nf easi bl e. The growth of computa

t i on ti me i s governed by two f actors. Fi r st , the probabi l i ty t hat a bot tom

posi t i on i s eval uated must be cal cul ated f or each bot tom posi t i on i n the

compl ete t ree, aD- f ol d nested summati on f or D > 2. Second, the expressi on

f or thi s probabi l i ty i s i n the f orm of a recurrence rel at i on ( Lemma 5. 3)

whi ch reduces the i ntegral to a D-2- f ol d nested summat i on of eval uat i ons of

the beta f unct i on and bi nomi al coef f i ci ent s. The number of nest ed summat i ons


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BRANCHI NG FACTOR (N)


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f or the compl ete t ree i s thus 2D- 2. Thesi mpl i f i cati onof theanal yt i cal

resul t to a form f or whi ch thecomputati on eff or t grows at a si gni f i cantl y

sl ower rate remai ns anopen probl em.

I n thehope ofprovi di ng some resul t s ofpracti cal val ue, wepresent

the empi r i cal observat i ons whi ch f ol l ow. The set ofdata poi nt s upon whi ch

the empi r i cal observat i ons arebased were obtai ned i npart f rom theanal yt i cal

f ormul at i on and i npart f romaMont e Car l o si mul at i onof theal pha- betaal

gor i thm. I n the si mul at i on the stat i c val ues f or the l eaf nodes of thetrees

were drawn f romauni f orm randomprobabi l i t y di str i but i on of i ntegers i n the

35

i nterval (0, 2 - 1) . Thesampl e meanand standard devi ati on were obtai ned

from theresul ts of 1000i t erat i ons of theal pha- beta al gor i thm f or each

si zeof the tree. The sampl e mean t hus l i es wi t hi n the i nterval x + Rx^

N, D N, D

at the 95$conf i dence l evel , where x D i s thet rue meanand 0.008 < R < . 013

f or thepoi nt s col l ected i n the si mul at i on. Therat i o of sampl e st andard

devi at i on tosampl e meanwascomputed f or 173poi nt s (thepoi nt s der i ved f rom

the anal yti c f ormul at i on were al so si mul ated to obtai n thei r sampl e st andardde

vi at i ons) , and al l of t hese rat i os l i e i n the i nterval ( . 120, . 215) .

The val uesof l ogE[ NBP ] as a f uncti onof l og N f or f i xed D are

observed to beapproxi matel y a st rai ght l i ne (Fi gure 5. 1) . A l east squares

f i t of a str ai ght l i ne to these data gi ves anapproxi mati on whose root mean

square rel at i ve er ror i sabout . 004 f or theval ues exami ned. Thecor res

pondi ng approxi mati on to theexpect ed number ofbot tomposi t i ons takesthe

L

f orm E[ NBPN D l ^# N >wi t h an RMS rel at i ve error ofapproxi matel y. 015.

Furthermore, theval ues areobserved to beapproxi matel y l i near wi t h

respect to D. A l east squares f i t of a str ai ght l i ne to the


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val ues yi el ds t he approxi mat i on 1^ . 720D + . 227, wi t h an RMS rel at i veerror of approxi matel y . 007.

Tabl e 7. 1:

Depth1 1. 00

2 1. 12

3 1. 30

4 1. 39

5 1.51

6 1. 59

7 1. 67

8 1.71

9 1. 73

I t i s of i nterest to note that the growth rate f or the al pha- beta al gor i thm

( . 720D) i s about mi dway between t hat of perf ect order i ng (0. 5D) and mi ni max

search ( D) . An accurate si mpl e approxi mat i on to the val ues i s not read

i l y apparent . The val ues of f or D=l , 2, . . . , 9 are f ound i n Tabl e 7. 1. The

val ues l i e i n the range [1, 2] .

Thi s empi r i cal anal ysi s of the data avai l abl e to us suggest s an approxi

mat i on of the form E[ NBP^D l N# 7 2 0 D + # 2 7 7

. The l i mi ted number and

arbi t rary sel ecti on of the data poi nt s used i n thi s anal ysi s prevent us f rom

attachi ng hi gh conf i dence t o thi s approxi mat i on f or the general case, al

though f or the poi nt s f romwhi ch i t i s der i ved t he approxi mat i on i s excel l ent .

We do note t hat the approxi mati on agrees wi t h the boundary case D = 1, f or

72 0D + 27 7 QQ7whi ch E[ NBPN - N N * - W , and agrees f ai r l y wel l wi t h


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the boundary case N 1 f or whi ch E[NBP^ ] K .

Sl agl e and Di xon [1969] def i ne a measure of the rel at i ve ef f i ci ency

of a t ree- searchi ng al gor i thm:

l og NBPDR s ,

X l Q gm*m

where NBP *- s the expected number of bot t omposi t i ons exami ned i n a mi ni max

search and NBPx

i s the expected number of bottomposi t i ons exami ned by al

gori thmx.

The val ue of DR l i es between 0 and 1. As an exampl e of the i nterpre

tat i on of DR, DR .6 i ndi cates that the al gor i thmunder consi derat i on can

be used t o search a tree of depth 5 wi t h about the same ef f ort as that r e

qui red by t he mi ni max al gor i thm to search a t ree of depth 3. Usi ng the

empi r i cal l y der i ved approxi mat i on,

277 l o g *DDR Q w . 720 +

1

zL

+ -Ma-j 3w D D l og N

277For l arge N, DR Q w . 720 + - -r - .

Qf-P D

For the search wi t h per f ect order i ng [Sl agl e and Di xon, 1969] ,

D R

1 , l og 2D R P0 2+ D l og N *

For l arge N, DRp Q M j .

The depth rat i o DR i s useful as a measure of the ef f i ci ency of a t ree

searchi ng al gor i thmrel at i ve to the mi ni max al gor i thm. However , the ex

pected number of bot tomposi t i ons exami ned by t he al pha- beta al gor i thmpro

vi des a much t i ghter upper bound f or the expected perf ormance of a good tree-

searchi ng al gor i thm than does the number of bot tomposi t i ons exami ned by the


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mi ni max al gor i thm. Hence, i t woul d be more desi rabl e to redef i neDRsuch

that the perf ormance of the al pha- beta al gor i thm rather than the mi ni max

al gor i thm i s used as the standard of compar i son, i . e. ,

* l og NBPxDR s

xl og NBP

Qf-P

8. CONCLUSI ONS

We have proven that the probabi l i ty of exami nganode P(i^) inagame

tree wi t h cH3 pruni ngi s

Pr{i k}= - f A^( z ) dB ( z ) =fj^^ ^(z)>

>

where ( z) and ( z) are the di st r i but i on f unct i onsofcKi^) and PCi^)

respect i vel y. The i ntegral may be eval uated exact l y usi ng Eqn. 5. 6 and

Lemma 5. 3. Thi s f ormul a i s used to compute the expected val ueof the

number of bot tom posi t i ons to be eval uated i nat ree of arbi t rary (but f i xed)

depth and branchi ng f actor .

For l arge val ues ofDandNLemma 5. 3 i s not computat i onal l y f easi bl e

and so we empi r i cal l y f i t the f ol l owi ng curve toaset ofsi mul at i on poi nt s:

E t N B P ^ y r 7 2 0 1 * - 2 " ,

where i sacoef f i ci ent i nthe i nterval [ 1, 2] dependi ng on dept h. Thi s

shows that the depth rat i o as def i nedbySl agl e and Di xon [1969] f or a-P

i s about

277

DR

o,-P - 7 2 0+ ^'irfor large N

-


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We i nvest i gate t he def i ci enci es of the model wi t h respect to correl a

t i on and possi bl e equal i t y of bot tomposi t i on val ues and demonst rate that

hi gher correl at i on reduces the number of bot tomposi t i ons, as does equal i ty

of bot tomposi t i on val ues (Fi gures 6. 2 and 6, 3) .

A number of i nterest i ng probl ems remai n to be sol ved i n the anal ysi s

of the al pha- beta al gor i thm: The si mpl i f i cat i on of the anal yt i cal expressi on

to a f ormwi t h a smal l er computat i onal growth rate than i ndi cated by Lemma

5. 3 i s an open quest i on. I t woul d al so be usef ul to f i nd an anal yt i cal ex

pressi on f or the var i ance of the number of bot tomposi t i ons.

Extensi on of the model to more general game t rees woul d be an i nter

esti ng goal . The range const rai nt may be rel axed, as shown i n Sect i on 5.

Anal ysi s of thi s model i s consi derabl y more compl i cated than the model we

chose.

The present model i s i nadequate f or model l i ng var i at i ons of the search

strategy such as f i xed order i ng, dynami c order i ng, and aspi rat i on l evel s,

because the model assumes i ndependence of the underl yi ng r andomvari abl es.

One possi bl e extensi on of the model to encompass these requi rements mi ght

be to assi gn a randomnumber to each branch t hroughout the t r ee, and l et

the f i nal eval uat i on of the bot tomposi t i ons be the sumof the val ues of

i t s predecessor . Thi s woul d l ead to an i ntui t i vel y appeal i ng stati c eval ua

t i on functi on that woul d i nt roduce a measure of correl at i on i nto the t ree.


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APPENDI X: NOTATI ON

a k( i d) kth- l evel al pha val ue, where k=0, 2, . . . , LdJ

A, . ( x) di str i but i on f unct i on ofa, _ ( i j ) ; d > k

>

A-? (x) di str i but i on f unct i on of a( i , )a

a( i d) i ax{a0( i d) , a2( i d) , . . . , a( i ) }

e

e

obk( i d) kth- l evel bet a val ue, where k=1, 3, . . . , LdJ

B, . (x) di str i but i on f unct i on of b, ( i , ) ; d > kk , 1k+l k d

B-? (x) di st r i but i on f uncti on of P( i , )a

aP( i d) mi n{b1( i d) , b3( i d) , . . . , b( i }

L J e

P( a, b) (i+b)t h e B 6 t a f u n c t i o n

D depth of game tree

F( x) the survi vor f unct i on, i . e. , F( x) = 1-F(x)

i d i , i 2, , i . Let i g be the empty vector .

1 k j

I k( j rj 2, . . . , j k) J n Tn( u, k- n+1)

0 n=l

L k j o 2

111

N branchi ng f actor of game t ree

p( i d) a node at l evel d i n the game tree wi t h i ndex i d # Number

of el ements i n vector i ndi cates depth of node i n the t ree

For exampl e, p( i ) *-s a l eaf node.


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P k ()

D-k+ N- 2

V k - 2

v. - k

i f i = 1m

i f k < D-m

i f k = D-m

i f k > D-m

T(f , k)

1 - [T(f , k- 1)]

N

i f k - 0

i f k > 0

v( i d) val ue of p( i , ) . Thi s i s the backed- up val ue unl essa

d D, i n whi ch case p( i d> i s a l eaf node and we

use the l eaf node' s stat i c val ue.

Vd( x ) cumul at i ve di st r i but i on f uncti on of v( i )


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References

Bogen, R. A. et al . MACSYMA Users Manual , Proj ect MAC, MI T (September 1972)

Gi l l ogl y, J . J o , "The Technol ogy Chess Program, 11 Art i f i ci al I ntel l i gence 3( 1972) , 145- 163,

Ni l sson, N. J . , Probl em Sol vi ng Methods i n Ar t i f i ci al I ntel l i gence, McGraw-Hi l l ( 1971) .

Parzen, E. , Modern Probabi l i t y Theory and I ts Appl i cat i ons, Wi l ey and Sons,I nc. , N. Y. ( 1960) .

Samuel , A. L. , "Some Studi es i n Machi ne Learni ng Usi ng t he Game of CheckersI BM J ournal 3, ( 1959) , 211- 229.

Shannon, C. E c , "Programmi ng a Computer f or Pl ayi ng Chess,11 Phi l . Mag. 7th

ser i es, V. 41, N. 314 ( 1950) , 256- 275.

Sl agl e, J . R. , "Game Trees, m and n Mi ni maxi ng, and the m and n Al pha- BetaProcedure, 11Al Group Rep. No. 3, UCRL- 4671, Lawrence Radi at i on Laboratory,Uni versi ty of Cal i f orni a ( November , 1963) .

Sl agl e, J . R. and J . K. Di xon, "Exper i ments wi t h Some Programs that SearchGame Tr ees, " J . ACM 16, 2 (Apr i l , 1969) , 189- 207.

Documents

Analysis of the Alpha-beta Pruning Algorithm