Fred CHAMP, positional-chess analyst

Int. J. Man-Machine Studies (1976) 8, 517-529

Fred CHAMP, positional-chess analyst R. H.

ATKIN,W.

R. HARTSTONAND I. H. WITI'EN

University o f Essex, U.K. (Received 10 June 1976)

In this paper we give the results of further research into the computer simulation of positional play in chess. A well-defined hierarchical approach is used to produce a vector mapping for the positional evaluation. It is illustrated by an analysis of a grandmaster game, Karpov vs. Spassky.

Introduction In earlier papers (Atkin, 1972; Atkin & Witten, 1975) the idea that the game of chess corresponds to certain well-defined structures in multidimensional space has been developed. Precisely, if S, W, B denote the sets (Squares of the Board}, (White men (pawns and pieces)}, (Black men (pawns and pieces)} respectively, then the rules of the game induce a (typical) relation F = Fw c W × S. Such a relation defines two simplicial complexes KW(S) and its conjugate KS(W), where dim KW(S) = 26, and which can be represented by suitably connected convex polyhedra in E 53. The move-properties of the individual pieces {P, N, B, R, Q, K} are expressed in F by saying that ( W , Sj) ~ Fw if and only if Wt "attacks" Sj. Slight variations in F can be introduced to allow for specific tactical or positional approaches, for example, we can allow a piece Wt to "see through" other pieces or pawns (to varying extents) when deciding on the relation F. When the game is in mode (I, J), White having made I moves and Black having made J moves, we characterize the state of affairs by the following mappings: (i) on KW(S), stval : S ~ Z (Z ---- non-negative integers) this to give the "strength value" of each square St ~ S; (ii) on KW(S), pval : W-+Z this to give the "piece value" of each Wt x W; (iii) on KS(W), cval : W-+Z this to give the "control value" of each Wi ~ W; (iv) on KS(W), sval : S-+Z this to give the "square value" of each St ~ S. Each such mapping is a graded pattern on the appropriate structure in E 5a, viz.,

and

stval is a 0-pattern, n °, on the vertices of KW(S), pval(Wt) is a t-pattern, r~t, where t is the dimension of Wt in KW(S), cval is a 0-pattern, n °, on the vertices of KS(W), sval(S3 is a t-pattern n t, where t is the dimension of St in KS(W). 517

518

R.H.

ATKIN, W.

R. H A R T S T O N A N D I. H . W I T T E N

In addition we allow for the "inverse" relation between pairs of these by requiring and

pval (W). cval(W) = h (a fixed constant) sval(S), stval(S) = k (a fixed constant)

(A) (B)

remembering that values are rounded off to give integers. Since, in KW(S), each W is a t-simplex (for some value of t) and is written W = (S~0S~I... S~,) we take pval(w) =

y~ stval (s,)

(c)

S 6W it

and similarly,

sval(S)

=

~

cval (W~).

(D)

W 6 S f

The mutual relation between these four mappings is represented by the diagram: stval,

, sval

C

D A

pval,

, cval

The simplest way to enter this diagram, and the one we have used to-date in this research, is via the comer at "pval" by the classical values (notice we use the American " N " to represent Knight!): pval {P, B, N, R, Q, K} = {1, 3, 3, 5, 9, 100}. Notation: Fred CHAMP is the name of our computer program because he is a CHessAdapted Multi-dimensional Player (known as Fred when he is wearing his cloth cap).

A hierarchical a p p r o a c h t o t h e s t r u c t u r e We pursue the idea (Atkin, 1976) that the concepts associated with positional judgement in chess can be contained in, but require, a hierarchical view of the game--and therefore of the structures KW(S) etc. and mappings {nN} already introduced. We denote this hierarchy (which uses the notion of cover sets, rather than partitioning) by H and refer to its possible levels via the following scheme. H : N-level (N + 1)-level (N+2)-level

squares St; pieces W , Bj sets of squares; sets of pieces sets of sets of squares; sets of sets of pieces

In the first instance we can get an intuitive idea of the significance of H by associating the N-level with the level of tactical decisions; the (N+l)-level with the level of (first order) positional judgements, and the (N+2)-level with the level of (second order)

FRED CHAMP

519

positional judgements. This is because tactics is concerned with the level of precise moves, of deciding to place a piece on a single square, whilst positionalplay is concerned with the control of a file (a set of squares) or of the centre, or of the whole King-side (an (N+2)-level matter), and so forth. Precisely, we use the following elements (which can always be altered if required) at the various levels.

N-level: the set of squares S; the set of White/Black men W/B. (N+l)-level: a set S' c P(S); a set W' = P(W) (similarly for B) where the elements are as follows:

Elements of S' Name of element

Symbol

Subset of S

centre QR-file QN-file QB-file Q-file K-file KB-file KN-file KR-file diagonal W1 diagonal B1 diagonal W2 diagonal B2 diagonal W3 diagonal B3 diagonal W4 diagonal B4 diagonal W5 diagonal B5 diagonal W6 diagonal B6 diagonal W7 diagonal B7 diagonal W8 diagonal B8 diagonal W9 diagonal B9 diagonal Wl0 diagonal BI0 diagonal WI l diagonal Bll diagonal W12 diagonal B12 diagonal W13 diagonal B13 enemy K-simplex

$1' S~' S/ $4' Ss' S/ $7' $8' $9' $1o' S.' $1~' Sxs' S~4' S~5' Sx~' S~7' $18' S~9' S~0'. $2~' S~' S~3' S~,' S,b' S,/ S,7' S,8' S~9' $30' Ss~" $8,' $3~' S,4' $35' $3/

own K-sitnplex

$8,'

(d4, d5, e4, e5 ) (al, a2, a3, a4, a5, a7, a7, a8} {bl, b2, b3, b4, b5, b6, b7, b8} (cl, c2, c3, c4, c5, c6, c7, c8) (dl, d2, d3, d4, d5, d6, d7, d8) (el, e2, e3, e4, e5, e6, e7, e8) {fl, f2, f3, f4, f5, f6, f7, f8) {gl, g2, g3, g4, g5, g6, g7, gS} (hl, h2, h3, h4, h5, h6, h7, h8} {bl, a2} (cl, b2, a3} {dl, c2, b3, a4} {el, d2, c3, b4, a5} {fl, e2, d3, c4, b5, a6} {gl, f2, e3, d4, c5, b6, a7} (Ill, g2, f3, e4, d5, c6, b7, a8} {h2, g3, f4, e5, d6, c7, b8} {h3, g4, f5, e6, d7, c8} {h4, g5, f6, e7, dS} {h5,g6, f7, e8} {h6, g7, f8} (h7, g8} {gl, h2} {fl, g2, h3} (el, f2, g3, h4} (dl, e2, f3, g4, h5} (cl, d2, e3, f4, g5, h6} {bl, c2, d3, e4,'f5, g6, h7} {al, b2, c3, d4, e5, f6, g7, h8} {a2, b3, c4, d5, e6, f7, g8} {a3, b4, c5, d6, e7, f8} (a4, b5, c6, d7, e8} (a5, b6, c7, d8} (a6, lo7, c8} {a7, b8} block of 5 × 5 squares around enemy K with inner 3 × 3 block given triple weighting block of 5 × 5 squares around own K with inner 3 × 3 block given triple weighting

R. H. ATKIN, W. R. HARTSTON AND I. H. WITTEN

520 Name of element

Subset of S

Symbol

weak enemy P-control

S3s'

weak own P-control

$39'

squares of ranks, 3, 4, 5, 6 which cannot be defended by an enemy pawn (unless that Pawn should change files by a capture) squares of ranks, 3, 4, 5, 6 which cannot be defended by an own pawn (unless that pawn should change files by a capture)

Note: the triple weighting in S31',$8~'is incorporated into the mapping "stval", which otherwise takes a value unity on each square.

Elements of W" (B' defined similarly) Name of element King Queen Rooks Black-square Bishops White-square Bishops Knights Q-side pawns K-side pawns centre pawns central pawns

Symbol

Subset of W

Wl' W/ Ws' W,' Ws' W/ W/ W/ W/ Wxo'

{WK} {WQ} {WQR, WKR, WK} {WQB} {WKB } {WQN, WKN } {pawns on Q-side} {pawnson K-side} {pawns on d4, d5, e4, e5} {pawns on QB-, Q-, K-, KB-files}

(N+2)-level: a set S " c Pz(S); a set W " c P2(W), as follows: Element of S" Name of dement

Symbol

Composition in terms of sets of S'

K-files

51"

B-squares

Sl"

{57', $8 t, 59'} {511t, 513' , 815r, 817' , 819 t, 521/, St$', SZ5', SZT', 829'1 5111',

Q-files w-squares

$3"

{S/, S / , $4'}

$4"

{Sxo',$12', Sx,', $1/, $18', SI0', S~l', Sz,', St6', Sl/, Ss0', Ss~', Ss,'}

centre weak enemy P-control enemy K-simplex own K-simplex weak own P-control

S/' S/' $7" S 8" S/'

{S/, S/, S/} {Sss'} {S31'} {SsT'} {SIg'}

$8/, $8/}

Elements of W" (B" defined similarly) Name of element

Symbol

King and Knights heavy pieces pawn power

Wl" W~" Ws"

Composition in terms of sets of W' {Wx', W/} {Wl', Ws', W / , W / } {W/, W / , W / , Wxo'}

FRED CHAMP

521

Relations F, A, As we move from the N-level, at which F is defined, to the ( N + l ) - l e v e l we naturally induce a relation A c W ' × S', defined as follows. I f a e W ' and b e S' then a A b iff there exists some W~ e a n W and some Sj e b n S such that Wi F S~. In the same way we naturally induce a relation ~ c W " × S", defined as follows. I f A e W " and B ~ S " then A Y~ B iffthere exists some a e A n W ' and some b e B n S" such that a A b. It now follows that we have the following structures, with their chess connotations: Level

Structures

N

KW(S) KS(W) KW'(S') KS'(W') KW"(S") KS"(W")

(N + 1) (N +2)

Significance White's tactical view of Board Board's tactical view of White White's 1st order positional view of Board Board's 1st order positional view of White White's 2nd order positional view of Board Board's 2rid order positional view of White

Relations Fw Fw -x Aw Aw-1 ~w Y-w-1

Hierarchical patterns The set of patterns, denoted {nN}, at the N-level give rise to induced patterns at the higher levels. We denote these by {XN+l} and {ns+~}. This paper is concerned with positional judgement derived from a study of the "square-value" pattern, at different hierarchical levels. N : sval ~ {~N}, on the structure defined by F, KW(S) Nq-1 : sval z {nN+l}, on the structure defined by A, K S ' ( W ' ) N q - 2 : sval z {nS+~}, on the structure defined by Y, K S " ( W " ) . The computer algorithms for the evaluation of these patterns are listed below.

At the N-level (i) pval ( p l a y e 0 : = /fplayer = K then 100 else/fplayer = Q then 9 else/fplayer = R then 55 else/fplayer ----B then 3 else if player = N then 3 else 1; (i) cval (player) : = 200/pval(player); (iii) sval(square, side):= resultof

begin result: = 0; for man ~ side do /fxrayattack(man,square) then result : = result q-cval(man);

end

522

R . H . ATKIN, W. R. HARTSTON AND I. H. WITTEN

where xrayattack (man, square) is true only if the man could legally capture an opposing man on that square, were one there, if all other pieces (but not pawns)were removed from the board. A t the ( N + l)-level

super(i, side) : = resultof begin result : -----0; for Sj' ~ Sl" do f o r square e Sj' do result : = result+sval(square, side); end A t the (N+2)-level

grand(i) : = if i = 1 then lO0*(super(7,ownside)-super(8,enemyside)/(super(7,ownside) +super(8,enemyside)) else if i = 2 then 100*(super(8,ownside)-super(7,enemyside))/(super(8,ownside) +super(7,enemyside)) else i f i = 3 then 100*(super(6,ownside)-super(9,enemyside))/(super(6,ownside)+ super(9,enemyside) + 1) else i f i = 4 then 100*(super(9,ownside)-super(6,enemyside))/(super(9,ownside) + super(6,enemyside) + 1) else i f i = 5 then 100*(super(1,ownside)-super(1,enemyside))/(super(1,ownside)+super(1,enemyside)) else if i = 6 then 100*(super(3,ownside)-super(3,enemyside))/(super(3,ownside)+super(3,enemyside)) else if i = 7 then 100*(super(2,ownside)-super(2,enemyside))/(super(2,ownside)+ super(2,enemyside)) else i f i = 8 then 100*(super(4,ownside)-super(4,enemyside))/(super(4,ownside) + super(4,enemyside)) else i f i = 9 then 100" (super(5,ownside)-super(5,enemyside))/(super(5,ownside) + super(5,enemyside)) [Note: this effectively forms the percentage ratio (W--B)/(W +B) or (W--B)/(W + B + 1), where W and B are the sval(~) values for White and Black respectively.]

A meta-hierarchy In order to play a game of chess, or even to analyse a game already played, it is necessary to be able to change the hierarchy H. This is so because H is defined by F (and A, Y,) and F is itself determined by the mode (I, J)~ by the "state of the game". A move p, by either White or Black, immediately alters (I, J)-~(I' J') and so it alters H, which is therefore determined by the move I1, and should be written H(Ix). In order to cope with this, Fred CHAMP (like all human players) must be able to exist in a condition equivalent to being "aware of all possible hierarchies H(Ix), as la varies". This suggests that we need another level o f hierarchy, which we shall call (H q-1), from which H(tt) can be studied. If we allow possible levels of M, (M-t-l) . . . . in (Hq-1) we get the following schema:

ERE]) CHAMP

523

H+1,.,,~/ M

M+I M+2 /

H--I~/ N ~1~-N÷I~N÷2

/

At the level (M, H + 1) the hierarchy H (the state of the game) can be understood. The level (M + 1, H-k-1) will contain sets of things found in (M, H + 1)--that is to say, it will contain a set of hierarchies H(la), or a set of moves or a strategy. The level (M +2, H + 1) will contain a set of strategies, etc. Only by moving into the ( H + I ) hierarchy can we hope to build Fred C H A M P into a Chess Master. But we can see that the level (M, H + 1) is sufficient to make him into a Chess Analyst, since at this level we have the information needed to understand the hierarchy H(la)~for any particular p. Positional analysis of a game is therefore associated with the (M, H + 1)-level, briefly denoted as M-level. At this level Fred C H A M P examines the (N+l)-level pattern ~s+l, given by super O, side) and the (N+2)-level pattern nN+2, given by grand(i). These patterns give vectors whose components have the following names: ENY-K OWN-K ENY-WK OWN-WK

(Enemy K-field) (Own K-field) (Enemy Weak.squares) (Own Weak squares)

K-SIDE Q-SIDE BL-SQ WH-SQ CENTRE

(Kind-Side) (Queen-Side) (Black Squares) (White Squares) (Centre)

Illustrative example In this study Fred C H A M P gives a positional analysis of the quiescent conditions in the game Karpov vs. Spassky (Xlth Match Game, Moscow, 1974). The score of the game is as follows: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

d2-d4 c2-c4 N-f3 N-c3 B-g5 B-h4 e2-e3 B-e2 B'f6 c4"d5 0-0 R-cl a2-a3

: : : : : : : : : : : : :

N-f6 e7-e6 d7-d5 B-e7; h7-h6 0-0 b7-b6 B-b7 B'f6 e6*d5 Q-d6 d7-a6 N-d7

See note: A B C D E F G H

I J K

524

(14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35)

R . H . ATKIN, W. R. HARTSTON AND I. H. WITTEN

b2-b4 N-el N-d3 a3-a4 N-c5 a4-a5 g2-g3 e3-e4 R-el N(c3)*e4 B-h5 Q-f3 N-c3 Q'c6 N-d5 R-e7 R'c7 Q'h6 R'f7 Q'f4 Q-c7+ N-f4

: : : : : : : : : : : : : : : : : : : : : :

b6-b5 c7-c6 N-b6 B-d8 B-c8 B-c7 N-c4 B-h3 d5*e4 Q-g6 Q-h7 f7-f5 g7-g6 g6*h5 f5-f4 Q-f5 R(aS)-e8 R-f7 K'f7 R-e2 K-f8 Resigns

L M N P

Q R S T

U

NOTES BY F. C H A M P

Each note is our interpretation of the 9-vectors 7rN+I and nN+z, where the vector components are ordered in the sequence, = {ENY-K, OWN-K, ENY-WK, OWN-WK, K-SIDE, Q-SIDE, BL-SQ, WH-SQ, CENTRE} (Each note refers to the position after Black's move.) (A) Values of nN ÷1for White and Black Wh : 0 BI : 0

3918 4428

0 0

0 0

1706 1640

1526 1526

4838 3874

4314 5322

1628 1626

Neither side is exerting pressure on the other's K-field; there are no weak squares. White has a slight advantage on the K-side; the Q-side is even;White gains on the black-squares but this is balanced by Black on the white-squares; the centre is evenly contested.

Values of nN +~(-kvefor White, --re for Black) --99

100

0

0

2

0

11

--9

0

The vector shows what percentage difference now exists between the sides consequent on the previous two moves. In this ease it shows the effect from the initial (zero-move) position. Thus White has gained an 11% advantage on the black-squares, compared with a gain of 9 % for Black on the white-squares. Similarly we notice -t-2 % (for White on the K-side. The --99% is a rounding-off "value" for --100%.

FRED CHAMP

(B)

525

IrN+l Wh : 0 BI : 0

3718 4116

0 0

0 0

1706 1706

1570 1724

4838 3734

4402 6122

1828 1892

Each K-field has been slightly weakened; Black has gained equality on the K-Side, increased his control on the white-squares, lost a little in control of the black-squares. ~N+2

--99

100

0

0

0

--4

13

--15

--1

These are the percentage differences expressing the above remarks. C) '~N+I

Wh : 0 B1 : 0

3916 3804

0 0

0 0

1838 1706

1570 1724

5894 3778

4006 6298

2158 2224

The positional play now expresses the fact that White is increasing his control on the K-Side and the black-squares (so the intersection of these two sets provides him with tactical targets), whilst Black is trying to counter on the Q-Side and on the white-squares. The Centre control is almost evenly balanced. ~N+2

--99

100

0

0

4

--4

22

--21

--1

White's percentage increment is now greater on the black-squares than is Black's on the white-squares. (D) ~N+X Wh : 0 B1 : 0

4048 4002

0 0

0 0

1838 1904

1702 1724

5498 4174

5062 6298

2488 2224

~N+~

--99

100

0

0

--1

0

14

--10

6

Now Black is contesting control on the K-Side/white squares whilst White increases his score on the black-squares and centre. (E) '/~N+1

Wh : 462 B1 : 0

4114 3802

0 0

0 0

1904 1944

1702 1724

5894 4654

5062 5898

2620 2224

12

17

8

~N-t-2

--77

100

0

0

0

0

White is using his positional advantage on the K-Side/black squares (by the move 5. B-g5 to weaken the Enemy-K-field, inducing 5 . . . h 7 - h 6 ) . White gains further advantage in the centre and on the black-squares whilst neither player acquires any percentage increment on their previous zones (K-Side and Q-Side). Neither side has any weak squares at this stage.

526

R . H . ATKIN, W. R. HARTSTON AND I. I-L WITTEN

(F) ~N+* Wh : 132 B1 : 0 nN+~ --90

4048 2686

0 0

0 0

1904 1910

1636 1724

5498 4532

5062 5898

2488 2218

100

0

0

0

--2

10

--7

6

Black is now fighting to recover his loss on his OWN-K field by reducing White's score on it--but at the price of a severe loss of absolute score there (from 3802 to 2686). White continues to show a significant relative advantage (10 ~ ) on the black-squares. He has now traded his K-Side advantage for an E N Y - K advantage. (G) nN+x Wh : 132 B1 : 0

3736 2686

0 0

88 198

1970 1910

1702 1790

6298 5332

4658 5230

2754 2218

~N+2

--90

100

--99

98

2

--2

8

--5

11

The first time in the game we see that each side has introduced weak-squares (which cannot be defended by a pawn), but that the opponent has not so far been able to score on them. White has increased his control of the centre (from 8 ~o to 11 ~o) whilst Black has not been able to reduce White's score on his (Black's) K-field. (H) nN+l Wh : 132 B1 : 0

3934 2620

0 0

88 264'

2168 1910

1702 1922

6298 5332

5054 5296

2754 2218

98

6

--5

8

--1

11

~N+2

--89

100

--99

Black's position is deteriorating on his own K-field (2686 to 2620, without reducing White's score of 132 on it), on the K-Side, on the white-squares (5 ~o to 1 ~o). The only compensation is a slight increase in control on the Q-Side (from 2 ~o to 5 ~o), but the tactical danger is all on the K-Side. On the other hand Black has increased his own score on his own weak-squares (from 198 to 264), showing that he is well aware of the danger. (I) nN+l Wh : 0 B1 : 88

2886 2070

--99

94

66 0

88 286

1910 1402

1634 1768

5506 5310

4476 3418

2216 1730

--62

98

15

--3

2

13

12

'/I~N+ 2

After the exchanges initiated by White (9. B'f6, etc.) we see that Black has gained some attack on White's K-field (I00~o dropped to 94~o), but lost positional ground on the white-squares (where he previously had an advantage) and on the Q-Side (from 5 ~o to 3 ~o advantage). White has lost his score on Black's K-field (132 to 0) but has increased his control on the K-Side plus black-squares plus centre, and now has a 13 ~o advantage on the white-squares. Since White also shows a score of 66 on Black's weak-squares (against Black's score of 286 there) the results of the exchanges are positionaUy favourable for White.

FRED CHAMP

527

(J) ~N+I Wh : 0 BI : 88

2886 2070

106 0

88 326

94

--50

98

1910 1402

1834 5706 1808 4910

4636 3898

2216 1730

15

1

9

12

fftN+~

--99

7

Now White has an advantage on all fronts, except that Black has a score of 88 on White's K-field (action of the Queen)--but White's control there is 94 ~ (instead of 100 ~o). White has increased his score on Black's weak-squares (66 to 106) and Black has countered by defending these (286 to 326). The weak-squares in question are c6 (attacked indirectly by White's Rook and defended by Bishop, Knight and Queen). K) ~N+I

Wh : 0 B1 : 88

2886 2334

106 0

I10 194

1910 1534

1834 6106 4234 1874 5702 3502

2216 1796

94

--29

99

11

0

10

~N+2

--99

3

9

Black has reduced his score on c6 (weak-square) by N-d7, although this increased his score on the black-squares (White's drop from 7~o to 3 ~o) and the centre. White's overall positional advantage is maintained. (L) ~N+I Wh : 0 B1 : 88

2688 2070

746 732

344 216

1778 1402

1992 6026 4104 2018 1 9 1 8 3 2 6 6 5 6 9 6 1930

94

55

--36

12

2

StN+2

--99

30

--15

2

White's control on the black-squares is a large 30 ~ advantage (due to Pawn and Knight moves by both sides). White now has 55 ~ advantage on Black's weak-squares (the set b6, c6, c7, d6) whilst Black enjoys a 36 ~o advantage on White's weak-squares (the set a3, a4, b3, c3, c4). The positional struggle is now about weak-black-squares (White's pressure) and weak-white-squares (Black's counter pressure). (M) ~N+I Wh : 0 B1 : 88

2688 1938

746 754

410 348

1778 1402

1992 5626 2116 3266

4504 5696

2018 1600

94

36

--29

12

--2

--11

12

~N+2

--99

27

The struggle continues around the weak-squares. Notice that if Black captures the Pawn on a4 he loses score on White's weak-square c4. The Bishop move, B-d8, has reduced Black's score on his own K-field. Both Black and White have reduced their advantage on the other's weak-squares. (N) nN+l Wh : 66 B1 : 418

2424 1872

946 1154

674 348

1646 1600

1992 4970 1984 3530

5160 6158

2150 1930

528

R.H. ATKIN, W. R. HARTSTON AND I. H. WITTEN

~N+2

--92

71

46

--26

1

0

17

--8

5

Apart from the threatened tactical mate (which Fred CHAMP ignores) we see that White is winning the positional struggle for control of the weak-squares, there being a continuing trade-off between K-Side, Q-Side, etc. scores and that control.

(0) ~s+l Wh : 66 B1 : 550

2024 1872

946 1088

784 480

1646 1578

1992 4970 1 9 1 8 4410

5 1 6 0 2150 5 4 3 2 2062

57

32

--16

2

2

--2

~N+2

--92

6

2

Black's attack on White's K-field has resulted in an increase in white-weak-square scores (squares f3 and h3). Black's attack has reduced White's control on his own K-field (from 71 ~o to 57 ~o), but Black's advantage is still restricted to ENY-WK and WH-SQ. (Q) ~N+I Wh : 516 B1 : 528

2130 1938

1078 1220

586 436

1710 1578

1728 6106 1652 3992

3440 4720

1806 1552

60

42

--35

4

2

--15

8

~N+2

--57

21

White has now increased his control on ENY-K, OWN-K, ENY-WK, K-Side, BL-SQ, and CENTRE. Black has improved his position on ENY-K, ENY-WK, and WH-SQ. (R) ~N+I

Wh : 582 Bl : 528

2130 1512

1512 1198

586 590

1658 1658

1 6 8 4 6 1 2 8 3616 1542 4072 4228

1916 1554

60

43

--34

3

4

10

~N+2

--43

20

--7

Black's tactical counter-attack has not resulted in a decisive shift in positional control. Now White is steadily increasing or maintaining his advantage again. Black is slipping in control of ENY-K, ENY-WK and WH-SQ.

(S) ~N+X Wh : 296 B1 : 528

1690 1200

2044 798

608 846

1i82 1140

2072 1408

5240 3404

3982 3960

1982 1198

52

41

--13

--10

19

21

0

25

~N+2

--59

White has sacrificed a piece to produce a large increase in absolute score on ENY-WK (now 2044), on Q-Side (now 2072). Black is left with an advantage on the K-Side (for the first time in the game)--due chiefly to the advanced Pawns there. White's attack on ENY-WK is now immediately on the sixth rank and (via the Rook) threatens the K-field with a possible R-e7.

FRED CHAMP

529

T) ~N+I

Wh : 748 BI • 884

1370 1200

2304 886

542 1128

1314 2 0 7 2 6 3 3 6 1520 1518 4174

2966 3952

1718 1284

22

34

--24

--6

--13

14

~N+~

--22

15

21

The struggle is now very sharp, each side trying to take tactical advantage of his positional plus--and ignoring the rest. But White's absolute score on the ENY-WK is 2304 (against 2044) and on BL-SQ it is 6336 (against 5240). The largest improvement for White occurs in ENY-K (748 against 296) and a consequent drop for Black from 59 ~o to 22 ~o. The rest is a tactical exploitation for White. (O) ~N+I

Wh : 682 Bl : 772

1118 750

2362 842

476 908

1172 774

1952 1160

6120 1932

2018 3640

1274 1064

18

44

--27

20

25

52

--28

9

nN+~ --4

White's tactical exploitation has resulted in a material advantage of three Pawns as well as a large positional advantage across most of the vector; Black is left with some control over the Q-Side and WH-SQ (a theme throughout the game) whilst White's massive 44 ~o advantage on ENY-WK, together with the tactical mobility of his Knights is too m u c h for Black--who has inadequate defensive resources for his K-field after 35. N-f4 (due to the positions of Knight and Bishop).

Can Fred C H A M P become a player? We fully anticipate that the answer to this is Yes. As indicated above this will require a Strategy at the level (M + 1, H + 1), and preliminary results so far tested are encouraging. In addition it might be worth exploring a set of such strategies, a selection from which would be made by appealing to a super-strategy located in (M+2, H + I ) . In any event we must also require a tactical moveselector at level (N, H), based on the positional assessment and selected strategy. The chief requirement is probably a search for captures, since the quiescent condition is always submitted to the positional analysis. This research has been supported by a grant from the Science Research Council.

References ATKIN, R. H. (1972). Multidimensional structure in the game of chess. International Journal of Man-Machine Studies, 4, 341-362. ATKIN, R. H. (1976). Positional play in chess by computer. In Advances in Computer Chess I, ED. CLARKE,M. Edinburgh University Press. ATKIN,R. H. ~; WITTEN, I. H. (1975). A multidimensional approach to positional chess. International Journal of Man-Machine Studies, 7, 727-750.