chapter 1 the game of chess

8/10/2019 Chapter 1 the Game of Chess

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Chapter 1

T H E G A M E O F C H E S S

HERBE RT A SIMON

Carnegie-Mellon University

JONATHAN SCHAEFFER

University of Alberta

o n t e n t s

1 Int roduct ion2 Hu m an chess p lay3 Co mp uter chess : Origins4 Search versus knowledge

4 1 Search

4 2 Knowledge

4 3 A tale of two programs

5 Co mp uter chess p lay6 The future7 Othe r games8 Conclus ionRefe rences

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5

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Handbook of Garne Theory Volume 1 Edited by R.J. Aumann and S. Hart Elsevier Science Publishers B.V. 1992. All rights reserved


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H A Simon and J Schaeffer

1 I n t r o d u c t i o n

The game of chess has sometimes been referred to as the Drosophila ofartificial intelligence and cognitive science re se ar ch - a standard task tha tserves as a test bed for ideas about the nature of intelligence and computation-al schemes for intelligent systems. Both machine intelligence - how to programa computer to play good chess artificial inte lli gen ce) -an d humanintelligence - how to understand the processes that hum an toasters use to playgood chess cognitive sc ience )- are encompassed in the research, and we will

comment on both in this chapter, but with emphasis on computers.From the standpoint of von Neumann-Morgenstern game theory [vonNeumann and Morgenstern 1944)] chess may be described as a trivial game. Itis a two-person, zero-sum game of perfect information. Therefore the rationalstrategy for play is obvious: follow every branch in the garne tree to a win, loss,or draw - the rules of the garne guarantee that only a finite number of moves isrequired. Assign 1 to a win, 0 to a draw, and -1 to a loss, and minimaxbackwards to the present position.

For simple games, such as tic-tac-toe and cubic, the search space is smallenough to be easily exhausted, and the games are readily solved by computer.

Recent ly, the game of connect-four , with a search space of 1013 positions, wassolved; with perfect play, the player moving first white) will always win[Uiterwijk et al. 1989)]. This result was not obtained by a full search of thespace, but by discovering properties of positions that, whenever present in aposition, guarantee a win. In this way, large sub-trees of the game tree couldbe evaluated without search.

The only defect in trying to apply this optimal strategy to chess is thatneithe r human beings nor the largest and fastest computers that exist or thatare in prospect) are capable of executing it. Estimates of the number of legallypossible games of chess have ranged f rom 1043 to 102. Since even the smallernumbers in this range are comparable to the number of molecules in theuniverse, an exploration of this magnitude is not remotely within reach ofachievable computing devices, human or machine, now or in the future.

In the literature of economics, the distinction has sometimes been madebetween substantive rationaUyand procedural a. k.a . computational) rationali-ty. Substantive rationality is concerned with the objectively correct or bestaction, given the goal, in the specified situation. Classical game theory hasbeen preoccupied almost exclusively with substantive rationality.

Procedural rationality is concerned with procedures for finding good actions,

taking into account not only the goal and objective situation, but also theknowledge and the computa tional capabilities and limits of the decision maker.


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C h 1 : T h e G a m e o f C h e s s

T h e o n l y n o n t r i v i a l t h e o r y o f c h e s s i s a t h e o r y o f p r o c e d u r a l r a t i o n a l i t y i nc h o o s i n g m o v e s . T h e s t u d y o f p r o c e d u r a l o r c o m p u t a t i o n a l r a t i o n a l it y isr e l a t i v e l y n e w, h a v i n g b e e n c u l t i v a t e d e x t e n s i v e l y o n l y s in c e t h e a d v e n t o f th ec o m p u t e r ( b u t w i t h p r e c u r s o r s , e . g . , n u m e r i c a l a n a l y si s ). I t i s c e n t r a l t o s u c hd i sc ip l ines a s a r t i f i c i a l i n t e l l i gence and ope ra t ions r e sea rch .

D i f f i c u l t y i n c h e s s, t h e n , i s c o m p u t a t i o n a l d i f fi c u lt y. P l a y i n g a g o o d g a m e o fc h e s s c o n s i s t s i n u s i n g t h e l i m i t e d c o m p u t a t i o n a l p o w e r ( h u m a n o r m a c h i n e )t h a t i s a v a i l a b l e t o d o a s w e i l a s p o s s i b l e . T h i s m i g h t m e a n i n v e s t i n g a g r e a td e a l o f c o m p u t a t i o n i n e x a m i n i n g a f e w v a r i a t io n s , o r i n v e s ti n g a li tt lec o m p u t a t i o n i n e a c h o f a la rg e n u m b e r o f v a r i at io n s . N e i t h e r s t r a te g y ca n c o m ec l os e t o e x h a u s t i n g t h e w h o l e g a r n e t r e e - t o a c h i e v i n g s u b s ta n t iv e r a t i o n a l it y.

2 Hu ma n chess p lay

To g e t a n i n i ti a l i d e a o f h o w t h e t a s k m i g h t b e a p p r o a c h e d , w e c a n l o o k a tw h a t h a s b e e n l e a r n e d o v e r t h e p a st h a l f c e n t u r y a b o u t h u m a n c h e s s p l a y,w h i c h h a s b e e n i n v e s t i g a t e d in s o m e d e p t h b y a n u m b e r o f p s y c h ol o g is t s. T h e r ei s a c o n s i d e r a b l e u n d e r s t a n d i n g t o d a y a b o u t h o w a g r a n d m a s t e r w i n s g a m e s ,b u t n o t e n o u g h u n d e r s t a n d i n g , a l a s, t o m a k e i t e a s y t o b e c o m e a g r a n d m a s t e r.

F i r s t , s in c e th e p i o n e e r i n g s t u d i e s o f t h e D u t c h p s y c h o l o g i s t , A d r i a a n d e

G r o o t , w e h a v e k n o w n t h a t a g r a n d m a s t e r , e v e n i n a d i ff ic u lt p o s i ti o n , c a r ri e so u t a v e r y m o d e s t a m o u n t o f se a r c h o f th e g a rn e t re e , p r o b a b l y s e l d o m m o r et h a n 1 00 b r a n c h e s [ d e G r o o t ( 1 9 65 ) ]. E v e n i f t h i s i s a n u n d e r e s t i m a t e b y a no r d e r o f m a g n i t u d e ( i t p r o b a b l y i s n o t ) , 1 0 3 i s a m i n i s c u l e n u m b e r c o m p a r e dw i t h 1 04 3. I n f a c t , d e G r o o t f o u n d i t v e r y d i f f ic u l t t o d i s c r i m i n a t e , f r o m t h es ta t is ti c s o f s e a r c h , b e t w e e n g r a n d m a s t e r s a n d o r d i n a r y c lu b p l ay e r s . T h e o n l yr e l ia b l e d i f f e r e n c e w a s t h a t t h e g r a n d m a s t e r s c o n s i s te n t ly s e a rc h e d m o r er e l e v a n t v a ri a ti o n s a n d f o u n d b e t t e t m o v e s t h a n t h e o t h e r s - n o t a v e ryi n f o r m a t i v e r e s u l t .

I t s h o u l d n o t s u r p r i s e u s t h a t t h e s k i l l e d c h e s s p l a y e r m a k e s s u c h a l i m i t e ds e a r c h o f t h e p o s s i b i li ti e s . T h e h u m a n b r a i n i s a v e r y s lo w d e v i c e ( b y m o d e r ne l e c t r o n i c s t a n d a r d s ) . I t t a k e s a b o u t a m i l l is e c o n d f o r a s ig n a l t o c r o s s a si n g les y n a p s e i n t h e b r a i n , h e n c e t e n t o a h u n d r e d m i l li s ec o n d s f o r a n y t h i n g

i n t e r e s t i n g t o b e a c c o m p l i sh e d . I n a s e ri o u s c h e ss g a m e , a p l a y e r m u s t m a k em o v e s a t t h e r a t e o f t w e n t y o r th i r t y p e r h o u r , a n d o n l y t e n m i n u t e s o r aq u a r t e r h o u r c a n b e a l l o t t e d t o e v e n a d i f fi c u lt m o v e . I f 10 0 b r a n c h e s i n th eg a m e t r e e w e r e e x a m i n e d i n f if t e en m i n u t e s , t h a t w o u l d a l lo w o n l y n i n es e c o n d s p e r b r a n c h , n o t a la rg e a m o u n t o f t im e f o r a s y st e m t h a t o p e r a t e s a th u m a n s p e e d s .

A s e c o n d t h i n g w e k n o w i s t h a t w h e n g r a n d m a s t e r s l o o k a t a c h e s s b o a r d i nt h e c o u r se o f a g a m e , t h e y s ee m a n y f a m i l ia r p a t t e r n s o r c h u n k s ( p a t t e r n s


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t h e y h a v e o f t e n s e e n b e f o r e i n t h e c o u r s e o f p l a y ). M o r e o v e r, t h e y n o t o n l yi m m e d i a t e l y n o t i c e a n d r e c o g n i z e t h e s e c h u n k s w h e n e v e r t h e y e n c o u n t e r t h e m ,bu t they have ava i l ab le in memory in fo rmat ion tha t g ives the chunks s ign i f i -c a n c e - t ei ls w h a t o p p o r t u n i t ie s a n d d a n g e r s t h e y s ig n al , w h a t m o v e s t h e ysugges t . The capab i l i ty fo r r ecogn iz ing chunks i s l ike a ve ry power fu l index tot h e g r a n d m a s t e r ' s e n c y c l o p e d i a o f c h e ss . W h e n t h e c h u n k i s r e c o g n i z e d , t h er e l e v a n t i n f o r m a t i o n s t o r e d w i t h i t i s e v o k e d a n d r e c a l l e d f r o m m e m o r y.

T h e g r a n d m a s t e r 's v a s t s t o r e o f c h u n k s s e e m s t o p r o v i d e th e m a i n e x p l a n a -t io n f o r t h e a b il it y to p l a y m a n y s i m u l t a n e o u s g a m e s ra p id l y. I n s t e a d o fs e a r c h in g t h e g a m e t r e e , t h e g r a n d m a s t e r s i m p ly m a k e s p o s i t io n a l l y s o u n dm ove s un ti l the op po ne n t m ake s an in fe r io r m ove , c rea t ing a (usua l ly s l igh t )

w e a k n e s s . T h e g r a n d m a s t e r a t o n c e n o t i c e s t h i s c l u e a n d d r a w s o n t h ea s s o c i a t e d m e m o r y t o e x p l o i t t h e o p p o n e n t ' s m i s t a k e .R o u g h e s t i m a t e s p l a c e t h e g r a n d m a s t e r 's s t o r e o f c h u n k s o f c h es s k n o w l e d g e

a t a m i n i m u m o f 5 0 ,0 0 0 , a n d t h e r e a r e p e r h a p s e v e n t w i c e t h a t m a n y o r m o r e[S imon and Gi lm ar t in (1973) ]. Th i s num ber i s com para b le in m agni tude to thetyp ica l na t ive l anguage voca bu la r i e s o f co l l ege -e duca ted pe r sons .

T h e r e is g o o d e v i d e n c e t h a t i t t a k e s a m i n i m u m o f t e n y e a r s o f i n te n s eapp l i ca t ion to r each a s t rong g randmas te r l eve l in chess . Even p rod ig ies l ikeB o b b y F i sc h e r h a v e r e q u i r e d t h a t . ( T h e s a m e p e r i o d o f p r e p a r a t i o n is r e q u i r e df o r w o r l d c l a s s p e r f o r m a n c e in a ll o f t h e d o z e n o t h e r f i el ds t h a t h a v e b e e n

exam ined . ) P re sum ably , a l a rge pa r t o f th is decad e o f t r a in ing i s r equ i red toaccumula te and index the 50 ,000 chunks .

I t h a s s o m e t i m e s b e e n s u p p o s e d t h a t t h e g r a n d m a s t e r n o t o n l y h a s s p e c ia lkno w ledg e , bu t a l so spec ia l cogn i t ive capab i l it i e s, e spec ia l ly the ab i li ty to g raspthe pa t t e rns on chess boards in menta l images . An in te res t ing and s impleexp er im ent , which has be en c a r r i ed ou t by severa l inves t iga to r s and i s eas i lyr e p l i c a te d , m a k e s v e r y c l e ar t h e n a t u r e o f c h e s s p e r c e p t i o n [ d e G r o o t ( 1 96 5 ) ].( T h e a n a l o g o u s e x p e r i m e n t h a s b e e n p e r f o r m e d w i t h o t h e r g a m e s , l i k e b r i d g e ,wi th the same resu l t . )

Al low a sub jec t in the l abora to ry to v iew a chess pos i t ion f rom a we l l -p layedgarne (wi th pe rh aps 25 p ieces on the bo ard ) fo r f ive to t en seconds . T henr e m o v e t h e p i e c e s a n d a s k th e s u b j e c t to r e c o n s t r u c t t h e p o s it io n . I f t h e s u b j e c ti s a toas te r o r g randm as te r, 90% or mo re o f the p ieces (23 ou t o f 25 , say) wi llbe rep lac ed cor rec t ly. I f the s ub jec t i s an o rd ina ry p layer, on ly abo u t s ix wil l ber e p l a c e d c o r r e c t l y. T h i s i s a n e n o r m o u s a n d s t a r t i n g d i f f e r e n c e b e t w e e nexce l l en t and ind i ffe ren t chess p layers . Som eth ing a bou t the i r eyes?

N e x t r e p e a t t h e e x p e r i m e n t , b u t p la c e th e s a m e p ie c e s o n t h e b o a r dtr n d o m [Chase and S imon (1973) ] . The o rd ina ry p layer wi l l aga in rep laceabou t s ix on the r igh t squares . Now the toas te r o r g randmas te r wi l l a l so on ly

rep lace abou t s ix cor rec t ly. C lea r ly the exper t ' s p re -eminence in the f i r s tv e r s i o n o f th e e x p e r i m e n t h a s n o t h in g t o d o w i t h vi su a l m e m o r y. I t ha s t o d o


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Ch 1: The Game of Chess

with familiarity. A chessboard from a well-played garne contains 25 nearlyunrelated pieces of information for the ordinary player, but only a half dozenfamiliar patterns for the master. The pieces group themselves for the expertinto a small number of interrelated chunks.

Few familiar chunks will appear on the randomly arranged board; hence, inreme mber ing tha t position, the expert will face the same 25 unrelated pieces ofknowledge as the novice. Evidence from other domains shows that normalhuman adults can hold about six or seven unrelated items in short-termmemory at the same time (equivalent to one unfamiliar telephone number).There is nothing special about the chess master's eyes, but a great deal that isspecial about his or her knowledge: the indexed encyclopedia stored in

memory.In summary, psychological research on chess thinking shows that it involves

a modest amount of search in the garne tree (a maximum of 100 branches, say)combined with a great deal of pattern recognition, drawing upon patternsstored in memory as a result of previous chess training and experience. Thestored knowledge is used to guide search in the chess tree along the mostprofitable lines, and to evaluate the leaf positions that are reached in thesearch. These estimated values at the leaves of the miniature game trees arewhat are minimaxed (il anything is) in order to select the best move. For thegrandmaster, vast amounts of chess knowledge compensate for the very limitedability of the slow human neurological har dwa re to conduct extensivesearches in the time available for a move.

3 Co m pute r chess: Or ig ins

The idea of programming computers to play chess appeared almost simulta-neously with the birth of the modern computer [see Newell and Simon (1972)for a detailed history to 1970 and Welsh and Baczynskyj (1985) for a moremodern perspective]. Claude Shannon (1950), the creator of modern informa-tion theory, published a proposal for a computer chess program, and A.M.Turing (1953) published the score of a garne played by a hand-simulatedprogram. A substantial number of other designs were described and actuallyprogrammed and run in the succeeding decade.

There was a close family resemblance among most of the early programs.The task was viewed in a game-theoretic framework. Alternative moves wereto be examined by search through the tree of legal moves, values were to beassigned to leaf nodes on the tree, and the values were to be minimaxedbackwards through the tree to determine values for the initial moves. The

move with the largest value was then selected and played. The same search andevaluation program represented both the player and the opponent.


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Of course the programs represented only the crudest approximations to theoptimal strategy of von Neumann-Morgenstern game theory. Because of thesevere computational limits (and the early programs could examine at most afew thousand branches of the tree), a function for evaluating the (artificial)leaves had to be devised that could approximate the true garne values of thepositions at these nodes.

This evaluation function was, and remains, the most vulnerable Achilles heelin computer chess. We shall see that no one, up to the present time, hasdevised an evaluation function that does not make serious mistakes from timeto time in choosing among alternative positions, and minor mistakes quitefrequently. Although the best current chess programs seldom blunder outright,

they orten make moves that masters and grandmasters rightly regard asdistinctly inferior to the best moves.In addition to brute force search, two other ideas soon made their appear-

ance in computer chess programs. The first [already proposed by Shannon(1950)] was to search selectively, rathe r than exhaustively, sacrificing complete-ness in the examination of alternatives in order to attain greater depth alongthe lines regarded as most important. Of course it could not be known withcertainty which lines these were; the rules of selection were heuristic, rules ofthumb that had no built-in guarantees of correctness. These rules could beexpressed in terms of position evaluation functions, not necessarily identicalwith the function for evaluating leaf nodes.

Turing made the important distinction between de ad positions, whichcould reasonably be evaluated in terms of their static features, and livepositions having unresolved dynamic features (pieces en prise, and the like)that required further search before they could be evaluated. This distinctionwas incorporated in most subsequent chess programs.

The second departure from the basic game-theoretic analogue was to seekways of reducing the magnitude of the computation by examining branches inthe best order. Here there arose the idea of alpha-beta search, whichdominated computational algorithms for chess during the next three decades.

The idea underlying alpha-beta search is roughly this. Suppose that, after apartial search of the tree from node A, a move, M1, has already been foundthat guarantee s the player at A the value V. Suppose that afte r a partial searchof one of the other moves at A, M2, a reply has been found for the opponentthat guarantees him or her a value (for the player at A) of less than 17. Thenthere is no need to carry on further search at the subnode, as M2 cannot bebetter than M1. Roughly speaking, use of this procedure reduces the amountof computation required to the square root of the amount without thealgorithm, a substantial reduction. The alpha-beta algorithm, which has many

variant forms, is a form of the branch-and-bound algorithm widely used insolving various combinatorial problems of management science.While the alpha-beta algorithm provides a quite powerful tree-pruning


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able to find very deep mating combinations (up to eight moves, 15 ply, deep)with search trees usually well under 100 branches.

An interesting successor to this program, which combines search and knowl-edge, with emphasis on the latter, is Wilkins' program PARADISE [Wilkins(1982)]. PAR ADI SE built small hum ano id trees, relying on extensive chessknowledge to guide its search. Within its domain it was quite powerful.However, like MATER, its capabilities were limited to tactical chess problems(checkmates and wins of material), and it could not perform in real time.

During this same period, stronger programs that adhered more closely to thestanda rd design described earlier were designed by Kotok, Greenblatt, and

others [described in Newell and Simon (1972, pp. 673-703)]. Gradual improve-

ments in strength were being obtained, but this was due at least as much toadvances in the sizes and speeds of computer hardware as to improvements inthe chess knowledge provided to the programs or more efficient searchalgorithms.

4 S e a r c h v e r s u s k n o w l e d g e

Perhaps the most fundamental and interesting issue in the design of chessprograms is the trade-off between search and knowledge. As we have seen,

human players stand at one extreme, using extensive knowledge to guide asearch of perhaps a few hundred positions and then to evaluate the leaves ofthe search tree. Brute-force chess programs lie at the opposite extreme, usingmuch less knowledge but considering millions of positions in their search. Whyis there this enormous difference?

We can view the human brain as a machine with remarkable capabilities, andthe computer, as usually programmed, as a machine with different, but alsoremarkable capabilities. In the current state of computer hardware and soft-ware technology, computers and people have quite different strengths andweaknesses. Most computer programs lack any capability for learning, whilehuman brains are unable to sum a million numbers in a second. When we aresolving a problem like chess on a computer, we cater to the strengths ratherthan the weaknesses of the machine. Consequently, chess programs haveevolved in a vastly different direction from their human counterparts. They arenot necessarily better or worse; just different.

4 1 Search

Progress in computer chess can be described subjectively as falling into threeepochs, distinguished by the search methods the programs used. The pre-1975


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Ch 1: The Garne of Chess

p e r i o d , d e s c r i b e d a b o v e , c o u l d b e c a l l e d t h epioneer ing era .C h e s s p r o g r a m swere s t i l l a nove l ty and the peop le work ing on them were s t rugg l ing to f ind af r a m e w o r k w i t h i n w h i c h p r o g r e s s c o u l d b e m a d e ( m u c h l i k e G o p r o g r a m m e r st o d a y ) . B y t o d a y s s t a n d a r d s , m a n y o f t h e s e a r c h a n d k n o w l e d g e t e c h n i q u e su s e d i n t h e e a r l y p r o g r a m s w e r e a d h o c w i t h s o m e e m p h a s i s o n s e l e c t i v e ,k n o w l e d g e - b a s e d s e a r c h . S o m e ( n o t a l l ) p r o g r a m s s o u g h t t o e m u l a t e t h ehu m an app roac h to chess , bu t th i s s t r a t egy ach ieved on ly l imi ted success .

T h e 1 9 7 5 - 1 9 8 5 p e r i o d c o u l d b e c a l l e d t h etechnology era.I n d e p e n d e n t l y,s e v e r a l r e s e a r c h e r s d i s c o v e r e d t h e b e n e f it s o f d e p th - fi rs t , a l p h a - b e t a s e a r c hwi th i t e ra t ive deep en ing . Th i s was a r e l i ab le , p red ic tab le , and ea s i ly im-p l e m e n t e d p r o c e d u r e . A t t h e s m e ti m e , a s t ro n g c o r re l a ti o n w a s o b s e r v e d

b e t w e e n p r o g r a m s p e e d ( as m e a s u r e d b y p o si ti o ns c o n s i d er e d p e r s e c o n d ) a n dpro gram per fo rm anc e . In i ti a lly, it was es t ima ted tha t an add i t iona l p ly o fs e a r c h ( in c r e a s in g t h e d e p t h o f se a r c h b y a m o v e b y W h i t e o r B l a c k ) c o u l di m p r o v e p e r f o r m a n c e b y a s m u c h a s 25 0 ra t in g p o i n ts . 1 A t t hi s r a te , o v e r c o m -i ng t h e g a p b e t w e e n t h e b e s t p r o g r a m s ( 1 80 0 ) o f t h e m i d d l e 1 9 70 s an d t h e b e s thu m an p layers (2700) req u i red on ly 4 p ly o f add i tiona l sea rch . S ince an ex t rap l y c o st s ro u g h l y a f a c t o r o f 4 - 8 in c o m p u t i n g p o w e r, a ll o n e h a d t o d o w a sw a i t f o r t e c h n o l o g y t o s o lv e th e p r o b l e m b y p r o d u c i n g c o m p u t e r s 4 , 0 0 0 ti m e sf a s t e t t h a n t h o s e t h e n a v a i la b l e , s o m e t h i n g t h a t w o u l d s u r e ly b e a c h i e v e d i n af e w y e a r s .

U nfo r tun a te ly , the ra t ing sca le is no t l inea r in am oun t o f sea rch , and l a t e re x p e r i e n c e i n d i c a te d t h a t b e y o n d t h e m a s t e r l e v e l ( 22 0 0 p o in t s ), e a c h p l y w a sw or th on ly an add i t iona l 100 po in t s . A nd , o f course , i t i s wide ly be l i e ved tha tt h e r a t e o f g ai n w il l d e c r e a s e e v e n f u r t h e r a s c h e s s p r o g r a m s r e a c h b e y o n d t h ebase o f the g ran dm as te r (2500) l eve l.

A l l t h e t o p p r o g r a m s t o d a y r e f l e c t t h i s f a s c i n a t i o n w i t h b r u t e - f o r c e a l p h a -b e t a s e a r c h , h a v i n g a n i n s a ti a b le a p p e t i t e f o r s p e e d . T h e m a j o r c o m p e t i ti v ec h e s s p r o g r a m s r u n o n s u p e r- c o m p u t e r sCray B l i t z ) ,s p e c i a l p u r p o s e h a r d w a r eBel l e , H i t ech , Deep Though t ) ,and mul t i -p rocessors Phoen ix ) .T h e b e s t c h e ss

prog ram s reac hed the l eve l o f s t rong mas te r s l a rge ly on the coa t ta i ls o ft e c h n o l o g y r a t h e r t h a n b y m e a n s o f m a j o r i n n o v a t io n s i n s o f t w a r e o r k n o w l -edge eng ineer ing .

S ince 1985, com pu te r chess has h it upo n a whole hos t o f new ideas tha t a rep r o d u c i n g a n e m e rg i n galgori thm era.Th e l imit o f e ff i c iency in a lp ha -b e tas e a r c h h a d b e e n r e a c h e d ; n e w a p p r o a c h e s w e r e c a l l e d f o r [ S c h a e f f e r ( 1 9 8 9 ) ] .I n q u i c k s u c c e s s i o n , a n u m b e r o f i n n o v a t iv e a p p r o a c h e s t o s e a r c h a p p e a r e d .W h e r e a s t r a d i t i o n a l a l p h a - b e t a s e a r c h m e t h o d s e x a m i n e t h e t r e e t o a f i x e d

1 C h e ss p e r f o r m a n c e i s u s u a l ly m e a s u r e d o n t h e s o - c a l le d E l o s c a l e w h e r e a s c o r e o f 2 0 0 0r e p r e s e n t s E x p e r t p e r f o r m a n c e 2 2 0 0 M a s t e r l ev e l p e r f o r m a n c e a n d 2 5 0 0 G r a n d m a s t e r p e r-f o r m a n c e . T h e r e a r e b o t h A m e r i c a n a n d I n t e r n a t i o n a l c h e s s r a t in g s w h i c h d if f e r b y p e r h a p s 1 0 0p o i n t s b u t t h e a b o v e a p p r o x i m a t i o n w i ll b e s u f fi c ie n t f o r o u r p u r p o s e s .


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1 H A S i m o n an d J S c h a e ff e r

depth (with some possible extensions), the new approaches attempt to expandthe tree selectively, at places where additional effort is likely to produce amore accurate value at the root. Alpha-beta relies on depth as the stoppingcriterion; removal of this restriction is the most significant aspect of the newideas in search.

Alpha-beta returns the maximum score achievable (relative to the fallibleevaluation function) and a move that achieves this score. Little information isprovided on the quality of other sibling moves. Singular extensionsis anenhancement to alpha-beta to perform additional searches to determine whenthe best move in a position is significantlybetter than all the alternatives[An ant har aman et al. (1988)]. These s ingular or forced moves are then

re-searched deeper than they normally would be. Consequently, a forcingsequence of moves will be searched to a greater depth than with conventionalalpha-beta.

The conspiracy numbersalgorithm maintains a count of the number of leafnodes in the tree that taust change value (or conspire to cause a change in theroot value [McAllester (1988)]. Once the number of conspirators required tocause a certain change in the root exceeds a prescribed threshold,that value isconsidered unlikely to occur and the corresponding move is removed fromconsideration. The search stops when one score for the root is thresholdconspirators better than all other possible scores.

Min/max approximationuses mean value computations to replace thestandard minimum and maximum operations of alpha-beta [Rivest (1988)].The advantage of using mean values is that they have continuous derivatives.For each leaf node, a derivative is computed that measures the sensitivity ofthe root value to a change in value of that node. The leaf node that has themost influence on the root is selected for deeper searching. The searchterminates when the influence on the root of all leaf nodes falls below a setminimum. When min/max approximation and conspiracy numbers are used,alpha-beta cut-offs are not possible.

Equi-potentialsearch expands all leaf nodes with a marginal utility greatertha n a prescribed threshold [Ana ntha raman (1990)]. The utility of a node is afunction of the search results known at all interior nodes along the path fromthe tree root to that node, together with any useful heuristic informationprovided by the knowledge of the program. The search terminates when allleaf nodes have a marginal utility below the prescribed threshold.

In addition to these procedures, a combination of selective search withbrute-force search has re-emerged as a viable strategy.

Although many of these ideas have yet to make the transition from theory topractice, they are already strongly influencing the directions today s programs

are taking. In some sense, it is unfortunate that computer chess received thegift of alph a-bet a search so early in its infancy. Although it was a valuable and


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Ch. 1: The Garne of Chess 11

p o w e r f u l t o o l , i t s v e r y p o w e r m a y h a v e i n a d v e r t e n t l y s l o w e d p r o g r e s s i n c h e s sp r o g r a m s f o r a d e c a d e .

4.2. Knowledge

I n v i e w o f t h e o b v i o u s i m p o r t a n c e o f e v a l u a t i o n f u n c ti o n s , it m a y s e e ms u r p ri s in g t h a t m o r e e f f o r t ha s n o t b e e n d e v o t e d t o in t e g ra t in g k n o w l e d g e i n ar e l ia b l e m a n n e r i n to c h e s s p r o g r a m s . T h e r e a s o n i s s im p l e : p r o g r a m p e r-f o r m a n c e is m o r e e a s il y e n h a n c e d t h r o u g h i n c r e a s e s i n s p e e d ( w h e t h e r h a r d -w a r e o r s o f t w a r e ) t h a n t h r o u g h k n o w l e d g e a c q u i s i t i o n . Wi t h p r e s e n t t e c h -

n iques , cap tu r ing , encod ing and tun ing chess knowledge i s a d i ff i cu l t , t ime-c o n s u m i n g a n d a d h o c p r o c e s s. B e c a u s e p e r f o r m a n c e h a s b e e n t h e s o l e m e t r i cb y w h i c h c h e s s p r o g r a m s w e r e j u d g e d , t h e r e h a s b e e n l it tl e in c e n t iv e f o rs o l v i n g t h e k n o w l e d g e p r o b l e m .

A c q u i r in g k n o w l e d g e b y h a v in g h u m a n g r a n d m a s t e r s a d v i se c h e s s p r o -g r a m m e r s o n t h e w e a k n e s s e s o f t h e ir p r o g r a m s h a s n o t p r o v e d e f f e c ti v e : t h et w o s i d e s ta l k d i f f e r e n t l a n g u a ge s . C o n s e q u e n t l y, f e w c h e s s p r o g r a m s h a v eb e e n d e v e l o p e d i n c o n s u l t a t i o n w i t h a s t r o n g h u m a n p l a y e r.

D e e p e r s e a r c h h a s h a d t h e u n e x p e c t e d e f f e c t o f m a k i n g c h e s s p r o g r a m sa p p e a r m o r e k n o w l e d g e a b l e th a n t h e y re a l ly a re . A s a t ri vi al e x a m p l e ,c o n s i d e r a c h e s s p r o g r a m t h a t h a s n o k n o w l e d g e o f t h e c o n c e p t o ffork as i tu a t i o n i n w h i ch a s i ng l e p i e c e t h r e a t e n s t w o o f th e o p p o n e n t s p i e c e ss i m u l t a n e o u s l y. S e a r c h in g a m o v e d e e p e r r e v e a l s t h e c a p t u r e s w i t h o u t th e n e e dfor r epresen t ing the fo rk in the eva lua t ion o f the base pos i t ion . In th is man nerd e e p s e a r c he s c a n c o m p e n s a t e f o r t h e a b s e n c e o f so m e i m p o r t a n t k i n d s o fkno w ledg e , b y de tec t ing and inves t iga t ing the e ffec t s o f the unn o t i ced fea tu re s .

A r t h u r S a m u e l c o n s t r u c t e d , i n t h e 1 9 5 0 s , a p o w e r f u l p r o g r a m f o r p l a y i n gc h e c k e r s [ S a m u e l ( 1 9 6 7 )] . T h e p r o g r a m s k n o w l e d g e w a s e m b o d i e d i n ac o m p l e x e v a l u a t i o n f u n c t i o n ( w h i c h w a s c a p a b l e o f s e l f - im p r o v e m e n t t h r o u g hl e a r n in g p r o c e s s e s ) , a n d t h e p r o g r a m s p r o w e s s r e s te d s q u a r e l y o n t h e a c c u r a c yo f it s e v a l u a t i o n s . T h e e v a l u a t i o n f u n c t io n w a s a w e i g h t e d s u m o f te r m s , a n dthe w e igh t o f each t e rm cou ld be a l t e red on the bas i s o f i ts in f luence on mo vest h a t p r o v e d ( r e t r o s p e c t i v e l y ) t o h a v e b e e n g o o d o r b a d .

T h e p u b l i s h e d d e s c r i p ti o n s o f S a m u e l s p r o g r a m p r o v i d e a c o m p l e t e d e s c ri p -t i o n o f t h e c h e c k e r s k n o w l e d g e i n c o r p o r a t e d i n i t. ( C o m p a r a b l e d e t a i l e di n f o r m a t i o n is n o t a v a i l a b le f o r m o s t c h e s s p r o g r a m s . ) H o w e v e r, S a m u e l sp r o g r a m is n o w t h r e e d e c a d e s o ld , a n d t e c h n o l o g y h a s im p r o v e d m a c h i n es p e e d b y s e v e r a l o r d e r s o f m a g n i t u d e . C h e c k e r s p r o g r a m s t o d a y u s e le s s t h a nh a l f o f th e k n o w l e d g e o f S a m u e l , t h e p r o g r a m s d e p e n d i n g o n s e a r c h t o r e v e a l

t h e r e s t. T h e s a m e h o l d s t ru e f o r c h e s s p r o g r a m s .T h a t s k il le d h u m a n p l a y e r s p r o c e s s c h es s p o si ti o n s i n t e r m s o f w h o l e g r o u p s


12/17

2 H A S i m o n a nd J S c h a ef fe r

o f p i e c e s , o r c h u n k s , is re c o g n i z e d a s im p o r t a n t b y c h e s s p r o g r a m m e r s ; b u t ,w i t h f e w e x c e p t i o n s , n o o n e h a s s h o w n h o w t o a c q u i r e o r u s e c h u n k s i np r o g r a m s , p a r ti c u la r ly u n d e r t h e r e a l -t im e c o n s t ra i n ts o f to u r n a m e n t c h e s sg a m e s . O n e o f t h e e xc e p ti o n s is C a m p b e l l' s C H U N K E R p r o g r a m , w h i ch isa b l e t o g r o u p t h e p i e c e s i n k i n g a n d p a w n e n d g a m e s a n d r e a s o n a b o u t t h echun ks in a me an ingfu l way [Cam pbe l l and Ber l ine r (1984) ]. Al tho ugh thes e a r c h t r e e s b u i l t b y C H U N K E R a r e l a rg e r th a n w o u l d b e b u i lt b y a sk i ll e dh u m a n p l a y e r, C H U N K E R , i n t h e l im i t e d cla ss o f p o s i ti o n s i t c o u l d h a n d l e ,a c h i e v e d a le v e l o f p e r f o r m a n c e c o m p a r a b l e t o th a t o f a g r a n d m a s t e r.

On ly recen t ly has s ign if ican t e ffo r t in the des ign o f chess p rogram s bee nr e - d i r e c te d t o w a r d s t h e a c q u i s i ti o n a n d u s e o f c he s s k n o w l e d g e . K n o w l e d g e i s

use fu l no t o n ly fo r pos i t ion eva lua t ion , b u t a l so to gu ide the d i rec t ion o f sea rche f f o r t . T h e Hitech c h e s s p r o g r a m , o n e o f t h e t w o o r t h re e s t ro n g e s t p r o g r a m sin e x i s t e n c e t o d a y, e m p l o y s h ig h - s p e e d , s p e c i a l - p u r p o s e p a r a ll e l h a r d w a r e , b u ta l s o i n c o r p o r a t e s m o r e c o m p l e t e a n d s o p h i s t i c a t e d c h e s s k n o w l e d g e t h a n a n yo t h e r p r o g r a m b u i l t t o d a t e [ B e r l i n e r a n d E b e l i n g ( 1 9 8 9 ) ] . O v e r a t h r e e - y e a rp e r i o d , a n d w i t h o u t a n y c h a n g e i n h a r d w a r e t o i n c r e a s e i t s s p e e d , t h e p r o g r a mi m p r o v e d i n s tr e n g t h f r o m a n e x p e r t t o a s tr o n g m a s t e r s o l el y o n t h e b a s i s o fs o f t w a r e i m p r o v e m e n t s - p r in c ip a ll y i m p r o v e m e n t s o f it s ch e s s k n o w l e d g e .

4.3. A tale of two programs

I t i s in te res t ing to compare the cur ren t two bes t chess p rograms . Bo th o r ig ina tea t C a r n e g i e - M e l l o n U n i v e r s i t y a n d b o t h u s e s p e c i a l - p u r p o s e V L S I p r o c e s s o r s ,bu t tha t i s where the s imi la r i t i e s end .Deep Thoughtc o n c e n t r a t e s o n s p e e d ; aspec ia l ly des igned , s ing le ch ip chess m ach ine tha t can ana lyze 500 ,000 pos i t ionspe r secon d . F ur the r, s ince the bas ic des ign i s eas i ly r epro duc ib le , on e can tunmu l t ip le cop ies in pa ra l l e l, ach iev ing even be t t e r pe r fo rm anc e ( the sys temc u r r e n t l y u s e s u p t o 6 ) . A s a c o n s e q u e n c e , u n d e r t o u r n a m e n t c o n d i t i o n s t h ep r o g r a m c a n l o o k a h e a d a l m o s t 2 m o v e s ( o r p l y ) d e e p e r t h a n a n y o t h e rp r o g r a m . H o w e v e r, c o n s i d e r a b l e w o r k r e m a i n s t o b e d o n e w i t h its s o f t w a r e toa l low i t to overcome the many g la r ing gaps in i t s chess knowledge .

Hitech a l so uses spec ia l -purp ose ch ips ; 64 o f them , one fo r each sq uare ont h e b o a r d . H o w e v e r ,Hitech s speed is less than that of a s ingleDeep Thoughtp r o c e s s o r. Hitech s s t re n g t h l ie s in its k n o w l e d g e b a s e . T h e h a r d w a r e h a s b e e ne n h a n c e d t o i n c l u d e p a t t e r n r e c o g n i z e r s t h a t r e p r e s e n t a n d m a n i p u l a t e k n o w l -edge a t a l eve l no t poss ib le inDeep Thought. C o n s e q u e n t l y, Hitech plays am o r e h u m a n - l ik e g a m e o f c h e ss , w i t h ou t m a n y o f t he m a c h i n e t e n d en c i e st h a t u s u a l l y c h a r a c t e r i z e c o m p u t e r p l a y. H o w e v e r,Deep Thought st r e m e n d o u s

s p e e d a l l o w s i t t o s e a r c h t h e g a m e t r e e t o u n p r e c e d e n t e d d e p t h s , o f f e n f i n d i n gi n p o s i t i o n s u n e x p e c t e d h i d d e n r e s o u r c e s t h a t t a k e b o t h o t h e r c o m p u t e rp r o g r a m s a n d h u m a n p l a y e r s b y su r p ri s e.


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Ch 1: The Game of Chess 13

A game between Hitech and eep Thought is analogous to a boxing matchbetween fighters with quite diffe rent styles. Hitech has the greater finesse, andwill win its share of rounds, whereas eep Thought has the knock-out punch.Unfortunately for finesse, no orte remembers that you out-boxed your oppo-nent for ten rounds. All they remember is that you were knocked out in theeleventh.

5 C o m p u t e r c h e ss p l ay

The current search-intensive approaches, using minimal chess knowledge, have

not be en able to eliminate glaring weaknesses from the machines play. Nowthat machines can wage a strong battle against the best human players, theirgarnes are being studied and their soff spots uncovered. Chess masters, giventhe opportunity to examine the games of future opponents, are very good atidentifying and exploiting weaknesses. Since existing computer programs donot improve their play through learning, 2 they are inflexible, unable tounderstand or prevent the repeated exploitation of a weakness perceived by anopponent.

Learning, in the current practice of computer chess, consists of the pro-grammer observing the kinds of trouble the program gets into, identifying thecauses, and then modifying the program to remove the problem. The pro-grammer, not the program, does the learningt

The biggest shortcomings of current chess programs undoubtably lie in theirknowledge bases. Programs are too quick to strive for short-term gains visiblein their relatively shallow search trees, without taking into account thelong-term considerations that lie beyond the horizons of the search. Theabsence of funda ment al knowledge-intensive aspects of human chess play, suchas strategic planning, constitute a major deficiency in chess programs. Inaddition, they are unable to learn from mistakes and to reason from analogies,so that they cannot improve dynamically from game to garne.

On the other hand, chess programs are not susceptible to important humanweaknesses. The ability to calculate deeply and without computational (asopposed to evaluational) errors are enormous advantages. They are also freefrom psychological problems that humans have to overcome. Humans, tunedto playing other humans, are frequently flustered by computer moves. Mach-ines do not subscribe to the same preconceptions and aesthetics that humansdo, and offen discover moves that catch their opponents oft guard. Manygames have been lost by human opponents because of over-confidence inducedby the incorrect perception that the machine s moves were weak.

20ther than to remember previous games played, so that they can avoid repeating the samemistake in the same openi ng position but not in similar positions).


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14 H .A . S imo n and J . Schaeffe r

F u r t h e r, a c o m p u t e r h a s n o c o n c e p t o f f e a r a n d i s q u i t e c o n t e n t t o w a l ka l o n g a p r e c i p ic e w i t h o u t w o r r y i n g a b o u t t h e c o n s e q u e n c e s o f e r r o r. A h u m a n ,a lways m indfu l o f the f inal ou tco m e of the ga rne , can be f r igh tened aw ay fromh is b e s t m o v e b e c a u s e o f f e a r o f th e u n k n o w n a n d t h e r is k o f lo s in g .

O n l y r e c e n tl y h a v e p e o p l e q u e s t i o n e d w h e t h e r t h e h u m a n m o d e l o f h o w t op l a y c h e s s w e l l i s t h e o n l y g o o d m o d e l o f , i n d e e d , e v e n t h e b e s t m o d e l . T h ek n o w l e d g e a b l e c h e s s p l a y e r i s in d o c t r i n a t e d w i t h o v e r 1 00 y e a r s o f c h e s s th e o r yand p rac t i ce , l eav ing li tt le oppo r tun i ty fo r b reak ing aw ay f rom c onve n t iona lt h in k i n g o n h o w t o p la y ch e s s w e l l. O n t h e o t h e r h a n d , a c o m p u t e r p r o g r a ms ta r ts wi th no p recon cep t ion s . Inc reas ing ly, the ev ide nce sugges t s tha t com pu-t e r e v a l u a t i o n f u n c t i o n s , c o m b i n e d w i t h d e e p s e a r c h e s , m a y y i e l d a d i f f e r e n t

m o d e l o f p l a y t h a t i sbetter t h a n t h e o n e h u m a n s c u r r e n t l y u s eIn fac t , i t i s qu i t e l ike ly tha t c om pute r pe r fo rm anc e is inh ib i ted by the b iasesi n t h e h u m a n k n o w l e d g e p r o v i d e d to p ro g r a m s b y t h e p r o g r a m m e r. S i nc e am a c h i n e s e le c t s m o v e s a c c o r d in g t o d i f fe r e n t c ri te r ia t h a n t h o s e u s e d b y p e o p l e( these c r i t e r i a may have l i t t l e r e l a t ion to any th ing a chess p layer "knows"ab ou t chess ) , i t i s no t su rp r i s ing tha t the ma ch ines ' s ty les o f p lay a re" d i f f e r e n t " a n d t h a t h u m a n s h a v e t r o u b l e b o t h i n p l a y i n g a g a i n s t t h e m a n dimprov ing the i r p rograms .

6 T h e f u t u r e

In 1988, De ep Thoughtb e c a m e t h e f i rs t c h e s s p r o g r a m t o d e f e a t a g r a n d m a s t e rin a se r ious tournament ga rne . S ince then i t has been shown tha t th i s was no t af luke , fo r tw o m ore g rand m as te r s have fa ll en . 3De ep Thoughti s a c k n o w l e d g e dto be p lay ing a t the In te rna t iona l Mas te r l eve l (2400+ ra t ing) andHitechis notfa r beh ind .

H o w e v e r, in 19 89 , W o r l d C h a m p i o n G a r r y K a s p a r o v c o n v in c i ng l y d e m o n -s t r a te d t h a t c o m p u t e r s a r e n o t y e t a t h r e a t f o r th e v e r y b e s t p l a y er s . H e s t u d i e dall of De ep Thought spubl i shed ga rnes and ga ined ins igh t in to the s t r eng thsa n d w e a k n e s s e s o f t h e p r o g r a m ' s p l a y. I n a tw o - g a m e e x h i b it io n m a t c h ,K a s p a r o v d e c i s i v e l y b e a tDeep Thoughtin both garnes .

I t i s d i ff i cu l t to p red ic t when compute r s wi l l de fea t the bes t human p layer.Th i s im por tan t eve n t , so long sough t s ince the in it ia l op t imism of S imon andN ew el l (1958) was d i sappo in ted , wi ll be a l andm ark in the h i s to ry o f a rt if ic i alin te l l igence . Wi th the cons tan t ly improv ing t echno logy, and the po ten t i a l fo rmass ive ly pa ra l l e l sys tems , i t i s no t a ques t ion " i f " th i s even t wi l l occur bu t

3 T h e G r a n d m a s t e r B e n t L a r s e n w a s d e fe a t e d i n N o v e m b e r 1 98 8. S u b s e q u e n t ly, G r a n d m a s t e r sR o b e r t B y r n e t w i c e ), a n d To n y M i l e s h a v e f a ll e n . A s o f A p r i l 1 9 90 ,Deep Thought sr e c o r d

a g a i n s t G r a n d m a s t e r s u n d e r t o u r n a m e n t c o n d i t i o n s w a s 4 w i n s , 2 d r a w s , 4 l o s s e s ; a g a i n s tIn te rna t ion a l Mas te r s , 11 wins , 2 d raws , 1 loss.


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Ch. 1: The Game of Chess 15

r a t h e r w h e n . A s t h e b r u te - f o r c e p r o g ra m s s e a rc h e v e r m o r e d e e p l y, t h ei n a d e q u a c y o f t h e ir k n o w l e d g e is o v e r c o m e b y th e d i s c o v e r ie s m a d e d u r in gs e a r c h . I t c a n o n l y b e a m a t t e r o f a f e w y e a r s b e f o r e t e c h n o lo g i c a l a d v a n c e se n d t h e h u m a n s u p r e m a c y a t c h e s s .

C u r r e n t l y, c o m p u t e r c h e s s r e s e a r c h a p p e a r s t o b e m o v i n g i n t w o d i v e rg e n td i rec t ions . Th e f i rs t con t inues the b ru te - fo rc e approac h , bu i ld ing fas te r andf a s t e r a l p h a - b e t a s e a r c h e r s .Deep Thought, with i t s amazing abi l i ty to considermi l l ions o f chess pos i t ions pe r second , bes t ep i tomizes th i s approach .

T h e s e c o n d d i r e c t i o n i s a m o r e k n o w l e d g e - i n t e n s i v e a p p r o a c h .Hitech hasm a d e a d v a n c e s i n t h e d i r e c t i o n , c o m b i n i n g e x t e n s i v e c h e s s k n o w l e d g e w i t hf a s t, b r u t e - f o r c e s e a r c h . C o m m e r c i a l m a n u f a c t u r e r s o f c h e s s c o m p u t e r s , s u c h

as Mephisto, l imi ted to mach ines tha t consumers can a ffo rd , r ea l i zed long agot h a t t h e y c o u l d n e v e r e m p l o y h a r d w a r e t h a t w o u l d c o m p e t e w i th t h e p o w e r f u lr e s e a r c h m a c h i n e s . To c o m p e n s a t e f o r t h e ir l a c k o f s e a r c h d e p t h ( r ou g h l y1 0 , 0 0 0 p o s i t i o n s p e r s e c o n d ) , t h e y h a v e d e v o t e d c o n s i d e r a b l e a t t e n t i o n t oa p p l y i n g k n o w l e d g e p r o d u c t i v e l y. T h e r e s u l t s h a v e b e e n i m p r e s s i v e .

W h i c h a p p r o a c h w i l l w i n o u t ? A t t h e 1 9 8 9 N o r t h A m e r i c a n C o m p u t e r C h e s sC h a m p i o n s h i p , Mephisto, runn ing on a commerc ia l ly ava i l ab le mic ro -proces -s o r, d e f e a t e d Deep Thought. Mephisto i s r a t ed 2159 on the Swedish Ra t ingLis t on the bas i s o f 74 ga rnes aga ins t hum an opp one n t s . ) The ve rd ic t is no t in .

7 O t h e r g a m e s

M a n y o f t h e l e s s on s le a r n e d f r o m w r it in g c o m p u t e r c h e s s p r o g r a m s h a v e b e e ns u b s t a n t i a t e d b y e x p e r i e n c e i n c o n st r u ct in g p r o g r a m s f o r o t h e r g a m e s : b r i d g eb i d d i n g a n d p l a y, O t h e l l o , b a c k g a m m o n , c h e c k e r s , a n d G o , t o m e n t i o n a f e w.I n b o t h b a c k g a m m o n a n d O t h e ll o , t h e b e s t p r o g ra m s s e e m t o m a t c h o r e x c e e dt h e t o p h u m a n p e r f o r m a n c e s .

B a c k g a m m o n is e s p e c ia l ly in t e re s t in g s in c e th e i n t e r p o l a t io n o f m o v e sd e t e r m i n e d b y t h e t h r o w o f d ic e m a k e s t r e e s e a r c h a l m o s t fu t il e . A s t h e b a s i sf o r e v a lu a t i n g a n d c h o o s i n g m o v e s , p r o g r a m s r e ly o n p a t t e r n r e c o g n i t io n , a s d ot h e ir h u m a n c o u n t e r p a r t s , a n d e x a c t p r o b a b i l it y c a lc u l a ti o n s, f o r w h i c h th ehu m an usua l ly app rox im ates o r uses in tu it ion . B er l ine r ' s cons t ruc t ion o f aw o r l d - c l a s s b a c k g a m m o n p r o g r a m u s i n g p a t t e r n r e c o g n i t i o n [ B e r l i n e r ( 1 9 8 0 ) ]p a v e d t h e w a y f o r b u i ld i n g a r ic h b o d y o f c h e ss k n o w l e d g e i n t oHitech.

8 C o n c l u s i o n

W e h a v e s e e n t h a t t h e t h e o r y o f g a m e s th a t e m e rg e s f ro m t hi s r e s e a r c h is q u i t er e m o t e i n b o t h i t s c o n c e r n s a n d i t s f i n d i n g s f r o m v o n N e u m a n n - M o rg e n s t e r n


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16 H . A . S i m o n a n d J. S c h a ef fe r

theory, To arrive at actual strategies for the playing of games as complex aschess, the garne must be considered in extensive form, and its characteristicfunction is of no interest. The task is not to characterize optimality orsubstantive rationality, but to define strategies for finding good moves-procedural rationality.

Two major directions have been explored. On the one hand, one can replacethe actual garne by a simplified approximation, and seek the game-theoreticaloptimum for the app rox ima tion - which may or may not bear any closeresemblance to the optimum for the real garne. In a garne like chess, usually itdoes not. On the other hand, one can depart more widely from exhaustiveminimax search in the approximation and use a variety of pattern-recognition

and selective search strategies to seek satisfactory moves.Both of these directions produce at best satisfactory, not optimal, strategiesfor the actual game. There is no a priori basis for predicting which will performbetter in a domain where exact optimization is computationa lly beyond reach.The experience thus far with chess suggests that a combination of the two maybe b e s t - with computers relying more but not wholly) on speed of computa-tion, humans relying much more on knowledge and selectivity.

What is emerging, therefore, from research on garnes like chess, is acomputational theory of garnes: a theory of what it is reasonable to do when itis impossible to de termine what is best - a theory of bounde d rationality. Thelessons taught by this research may be of considerable value for understandingand dealing with situations in real life that are even more complex than thesituations we encounter in c he ss - in dealing, say, with large organizations,with the economy, or with relations among nations.

e f e r e n c e s

An antha ram an, T. (1990) A s tat is t ica l study of se lec tive min-max search , PhD thes is , Carnegie-

Mel lon Univers i ty.An antha ram an, T. , M.S . Cam pbel l and F.H. H su (1988) S ingular ex tens ions : Adding se lec t iv i tyto bru te - force search ing ,Artificial Intelligence4: 135-143.

Be r l i ne r, H . J . ( 1980) Com pu te r back gammon ,Scientif ic AmericanJune : 64-72 .Ber l iner, H .J . and C. Ebe l ing (1989) Pa t te rn know ledge and search : The SU PR EM arch i tec ture ,

Articial lntel l igence38: 161-198.Cam pbel l , M .S . and H .J . Ber l iner (1984) Us ing chunking to p lay chess pawn end gam es ,Artificial

Intelligence 23: 97-120.Chase , W .G. and H .A . S im on (1973) Percept ion in chess ,Cognit ive Psychology4: 55-81 .de Groot , A.D. (1965)Thought and choice in chess .The Hague : Mou ton .M cAl les te r, D .A . (1988) Conspi racy numbers for min-max search ,Artificial Intelligence35:

287-310 .Nau , D.S . (1983) Pa thology in garne t rees rev is i ted and an a l te rna t ive to min imax ing ,Artificial

InteUigence 21: 221-244.NeweU, A. and H.A. S imon (1972)H um an p rob lem so Iv ing .Englew ood Cl i ff s , NJ : Pren t ice-Hal l .


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New el l , A . , J .C . Shaw and H.A . S imon (1958) 'Chess -p lay ing program s and the problem ofcomplex i t y ' ,IBM Journal o f Research and Development 2: 320-335.

Rives t , R .L . (1988) 'Garne t ree search ing by min /max approximat ion ' ,Artificial InteUigence 34:77 -96 .

Sam uel , A .L. (1967) 'Som e s tud ies in machine learn ing us ing the gam e of checkers : I I - Re centprogress ' , IBM Journal of Research and Development 11: 601-617.

Schaeffe r, J . (1989) 'The h is tory heur is t ic and a lph a-b e ta search enhancements in prac t ice ' ,IEEETransactions on Pattern Analysis and Machine lntelligence 11: 1203-1212.

Shann on, C .E . (1950) 'P rogram ming a d ig i ta l com puter for p lay ing chess ' ,PhilosophicalMagazine 41: 356-375.

S imo n, H .A . and K . Gi lmar t in (1973) 'A s imula t ion mem ory for chess pos i t ions ',CognitivePsychology 5: 29-46 .

S imon , H .A . and A . N ewel l (1958) 'Heur i s t ic p roblem so lv ing: Th e hext advance in opera t ionsresearch ' , Operations Research 6: 1-10.

Tur ing , A .M . (1953) 'Dig i ta l comp uters appl ied to games ' , in : B .V. Bow den, ed . ,Faster thanthought. London : Pu tnam.

Ui te rw i jk , J .W.J .M . , H .J . van den Her ik and L .V. Al l i s (1989) 'A k nowledge-based app roach toconnec t - four. The garne i s over : W hi te to m ove wins ' ,Heuristic programming in artificialintelligence. New York : W i ley.

von Neum ann, J . and O. M orgens te rn (1944)Theory o f garnes and economic behavior. Pr ince ton ,NJ: Pr ince ton Univers i ty Press .

Welsh, D. and B. Baczynskyj (1985)Computer chess H. Dubuque , IA : W.C . B rown Co .Wilk ins , D.E . (1982) 'Us ing knowledge to cont ro l t ree search ing ' ,Artificial Intelligence 18: 1-5.

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