COMPUTING A PERFECT STRATEGY FOR
n x n CHESS REQUIRES TIME EXPONENTIAL IN N
A v i e z r i S. Fraenkel
Department of Applied Mathematics
The Weizmann I n s t i t u t e of Science
Rehovot, Israel
David Lichtenstein
Department of Computer Science
Yale U n i v e r s i t y
New Haven, Connecticut 06520, U.S.A.
:Tis a l l a chequer-board of nights and days
where destiny with men f o r pieces plays;
h i t h e r and t h i t h e r moves and mates and slays
and one by one back in the closet lays.
The Rubaiyat of Omar Khayyam
ABSTRACT.
I t is proved that a natural generalization of chess to an
complete in exponential time.
nxn
nxn
board is
This implies that there e x i s t chess-positions on an
chess-board for which the problem of determining who can win from that position
requires an amount of time which i s at least exponential in
I.
n .
INTRODUCTION
From among a11 the games people play, chess towers as the most absorbing and
widely played.
Indeed, i f a t t e n t i o n is r e s t r i c t e d to 2-person games of perfect i n f o r -
mation without chance moves played outside the Orient, the ever rejuvenating i n t e r e s t
in the 1500 year old game has a q u a l i t y of depth and breadth well beyond t h a t of any
potential r i v a l °
I t is noteworthy, then: that in the long s t r i n g of complexity
results f o r games~ chess had yet to appear.
on an
nxn
board is Pspace-hard [ I 0 ] .
a natural generalization of chess to
the f i r s t
Recently J. Storer announced that chess
See also J.M. Robson [7].
nxn
boards is complete in exponential time,
such r e s u l t f o r a " r e a l " game. This implies that f o r any
are i n f i n i t e l y
many positions
~
c > 1
is a constant, and
k ~ l ,
there
such t h a t any algorithm f o r deciding whether White
(Black) can win from that position requires at least
where
We w i l l show that
l~I
cI~l k
is the size of
~
time-steps to compute,
Generalized chess is
thus provably i n t r a c t a b l e , which is a stronger r e s u l t than the complexity results f o r
board games such as Checkers: Go, Gobang and Hex which were shown to be Pspace-hard
[1,3,5,6].
We l e t generalized chess be any game of a class of chess-type-games with one
king per side played on an
nxn
chessboard.
are subject to the same movement rules as in
The pieces of every game in the class
8x8
chess, and the number of White
and Black pawns~ rooks, bishops and queens each increases as some f r a c t i o n a l power
279
of n .
Beyond t h i s growth c o n d i t i o n , the i n i t i a l
position is immaterial, since we
analyze the problem of winning f o r an a r b i t r a r y board p o s i t i o n .
Unfortunately, our constructions seem to v i o l a t e the s p i r i t of
8x8
chess, in
much the same way as the complexity proofs f o r Checkers, Go, Gobang and Hex mentioned
above.
cal
Typical positions in our reduction do not look l i k e larger versions of t y p i -
8x8
bility,
chess endgames. Although we have not t r i e d to answer questions of reacha-
i t seems offhand as though players would have a hard time t r y i n g to reach our
board positions from any reasonable s t a r t i n g p o s i t i o n .
(Reachability may not seem
quite as unfeasible, perhaps, i f we recall the chess rule s t a t i n g that a pawn reaching
the opposite side of the board can become any piece of the same color other than pawn
or king [ 4 ] . )
What we can say, however, is that certain approaches for deciding
whether a position in
8x8
chess is a winning position for White may not be very
promising, namely those approaches which work for a r b i t r a r y positions and generalize
to
nxn
boards.
Such approaches use time exponential in n ,
ful only i f the exponential e f f e c t had not yet been f e l t f o r
Thus, while we may have said very l i t t l e
and hence can be usen=8 .
i f anything about
8x8
chess, we
have, in f a c t , said as much about the complexity of deciding winning positions in
chess as the tools of reduction and completeness in computational complexity allow
us to say.
Our r e s u l t is in l i n e with the suggestion to demonstrate the complexity of
i n t e r e s t i n g board games by imbedding them in families of games [8].
An i n t e r e s t i n g
c o r o l l a r y of our r e s u l t is that i f Pspace ~ Exptime, as the conjecture goes, then
there is no polynomial bound on the number of moves necessary to execute a perfect
strategy.
This is so because Pspace ~ Exptime, and the "game-tree" of chess can be
traversed in endorder to determine the w i n - l o s e - t i e membership of each node (game
position).
Though t h i s takes an exponential amount of time, the memory requirement
at each step is only the depth
p(n)
of the tree - w h i c h is kept on a stack - and
the description of a terminal position.
is in Pspace.
Thus, i f
p(n)
is polynomial, then the game
Since chess is complete in Exptime, i t belongs to the hardest problems
there, hence i t l i e s in E x p t i m e - Pspace i f Pspace ~ Exptime.
For the sake of the u n i n i t i a t e d , we now give a short informal i n t r o d u c t i o n to
the basic notions of computational complexity.
Let S be a subclass of decision
problems ( i . e . problems whose answer is "Yes" or "No").
z2 ,
we say that
~l ~ ~2 ) '
(i)
is polynomiaily transformable (or reducible) to
i f there exists a function
set of instances of
f(1)
~l
I
~2
f(1)
f
from the set of instances of
~2
~l
~2 (notation:
~l
to the
for which the answer is "Yes" i f and only i f
for which the answer is
"Yes".
is computable by a polynomial time algorithm in the size of
"polynomial time a l g o r i t h m " ) .
A decision problem
~l '
such that:
is an instance of
is an instance of
(ii)
For decision problems
~
is S-complete i f :
I
(a
280
(i)
~ E S ,
(ii)
f o r every
~
E S ,
A decision problem
hold.
~
~
~ ~ .
is S-hard i f ( i i )
holds but ( i ) does not necessarily
A decision problem is i n t r a c t a b l e i f i t cannot be decided by a polynomial time
algorithm.
A nondeterministic algorithm is an "algorithm" which can "guess" an e x i s t e n t i a l
s o l u t i o n , such as a path in a tree and then v e r i f y i t s v a l i d i t y by means of a deterministicalgorithm.
Important classes of decision problems are the class P of a l l decision problems
with ( d e t e r m i n i s t i c ) algorithms whose running time is bounded above by a polynomial
in the size
sion problems
I~I
of
~
;
the class NP (nondeterministic polynomial) of a l l deci-
~ with nondeterministic algorithms whose running time is bounded above
by a polynomial in
I~I ;
the class Pspace of a l l decision problems
~
whose
algorithms require an amount of memory space bounded above by a polynomial in
and the class Exptime of a l l decision problems
~
l~I ;
with ( d e t e r m i n i s t i c ) algorithms
whose running time is bounded above by an exponential function in
]~I
The
f o l l o w i n g basic relations hold:
P ~ NP ~ Pspace ~ Exptime
I t is not known whether any of these inclusions is proper, except t h a t
P ~ Exptime.
Furthermore, NP and Pspace are not known to contain any i n t r a c t a b l e decision problems, but Exptime i s .
From the d e f i n i t i o n of
~I E P .
of S.
~
i t follows that i f
~I ~ ~2 '
Therefore the S-complete problems for any
In p a r t i c u l a r f o r
then
72 E P implies
S are the "hardest" problems
S =Exptime, the S-complete problems are a l l i n t r a c t a b l e .
For f u r t h e r d e t a i l s and a formal treatment of t h i s topic the reader is referred to
Garey and Johnson [ 2 ] .
2.
Let
THE REDUCTION
Q be the f o l l o w i n g question:
chess-game on an
n×n
win from t h a t position?
Given an a r b i t r a r y position of a generalized
chessboard from our class of chess games, can White (Black)
Following [2], we define Exptime to be the set of decision
problems with time-complexity bounded above by
input size n .
f o r some polynomial p of the
Since in chess there are s i x d i s t i n c t pieces o f each c o l o r , the num-
ber of possible configurations in
Q E Exptime.
2p(n)
We shall show that
nxn
chess is bounded by
G3 ~ Q ,
where
G3
13n2 ,
is the f o l l o w i n g Boolean game
proved complete in exponential time by Stockmeyer and Chandra [9].
(B) stands for White (Black).
plement.
Every position in
As usual, a l i t e r a l
G3 i s a 4-tuple
hence
Throughout
(%, W-LOSE(X,Y), B-LOSE(X,Y), ~) ,
where
E {W,B} denotes the player whose turn i t is to play from the p o s i t i o n ,
W-LOSE = CII v C12 v . . . v Clp
and
W
is a Boolean variable or i t s com-
B-LOSE = C21 v C22 v . . . v C2q are Boolean
281
formulas in 12DNF, t h a t is each
literals
(I ~ i ~ p ,
of variables
XUY .
Cli
1 ~ j ~ q) ;
and each
and
~
C2j
is a conjunction of at most 12
is an assignment of values to the set
The players play a l t e r n a t e l y .
the value of precisely one variable in
X (Y) .
Player W (B) moves by changing
In p a r t i c u l a r , passing is not per-
mitted.
W (B) loses i f the formula W-LOSE (B-LOSE) is true a f t e r some move of player
W (B) .
Thus W can move from
iff
(W, W-LOSE, B-LOSE, m)
B-LOSE i s false under the assignment
p r e v i o u s l y ) , and m and
m'
to
(B, W-LOSE, B-LOSE, m')
m (otherwise the game already terminated
d i f f e r in the assignment of exactly one variable in X.
I f W-LOS#[ is true under the assignment
m' ,
then
W just lost.
A player who vio-
lates any of the game's rules loses immediately.
In order to show G3 = Q ,
we have to simulate
G3
on an
n×n
chess-board.
S p e c i f i c a l l y , the goal is to construct a p o s i t i o n on the board where only one rook
and two queens per variable can move. A l l other pieces are deadlocked.
Each rook is
permitted to be in only one of two p o s i t i o n s , which have the meaning of assigning the
values of 1 {T)
or 0 (F)
to the corresponding variable.
The p o s i t i o n i n g of the
deadlocked pieces force the queens to move through predefined "channels" in order to
reach the opponent's king, and the p o s i t i o n i n g of the rook determines one of two
possible avenues through which a queen may pass.
The overall construction is such
that those and only those truth-assignments to the variables which win the game G3
for
W (B)
lead the queens of
W (B)
to win the generalized chess game from the
constructed p o s i t i o n .
Our basic structure is the Boolean c o n t r o l l e r .
W (B)
Boolean c o n t r o l l e r f o r a variable
x E X
Figure 1 (2) i l l u s t r a t e s a
(y E Y) .
White c i r c l e s are WP's
(W pawns), black c i r c l e s BP's (B pawns), white squares WB's (W bishops), black
squares BB's (B bishops), and WR, BR, WQ, BQ stand f o r W rook, B rook, W queen,
B queen, respectively.
I T WR is at i t s south position in the WR-channel, as in
Figure I , also called x - p o s i t i o n , then the value of
north position of the WR-channel, denoted by
then the value of
x
is
0 .
x
is
1 .
I f WR is at the
W~ in Figure I , also c a l l e d ~ - p o s i t i o n ,
A s i m i l a r convention is adopted f o r Figure 2 which is
indicated only schematically because a B Boolean C o n t r o l l e r (BBC) is obtained from a
W Boolean C o n t r o l l e r (WBC) by an interchange
W~
Cli ~
B throughout, followed by a 180 ° r o t a t i o n .
C2j ,
x < >y ,
(Here and below,
x<--~y
and
Cli (C2j) denotes
a t y p i c a l clause of W-LOSE (B-LOSE).)
There is one W (B) Boolean C o n t r o l l e r f o r each
x c X
(y £ Y) .
In normal play,
W (B) moves his WR (BR) between the x - p o s i t i o n and the x - p o s i t i o n (y- and j - p o s i t i o n )
in any W (B) Boolean C o n t r o l l e r u n t i l the game G3 w i l l have been decided. I f
W (B) does not abide by these r u l e s , then his opponent can win via the B (W) Normal
Clock (NC) or the B (W) Rapid Clock (RC) mechanisms detailed below.
A global view o f the construction is shown in Figure 3.
number o f l i t e r a l s
in any "And-Clause" in W-LOSE and B-LOSE. Let
an And-Clause consisting of L l i t e r a l s
Cli=l
Let k be the largest
a f t e r a move of W.
Now C l i = l
for some 1 ~ ~ ~ k ~ 12 .
i f and only i f there are
Cli
in W-LOSE be
Suppose that
~
B queens which
282
can reach Cli-channel intersections not under attack in
t=8
moves each:
two
moves in the WBC (Figure l ) or BBC (Figure 2), one move for reaching the W Switch
(Figure 4), four moves in the W Switch and one l a s t move for reaching the C l i channel.
These
~ B queens now proceed down t h i s channel, where
~-l
of them are
captured at the W A l t a r (Figure 5), and the lone survivor passes through a W delayl i n e from where i t emerges into the B Coup De Grace (CDG)-channel to checkmate the
W king (WK) (Figure 6).
The W (B) Switch (Figure 4) is designed to l e t a single B (W) queen pass from a
W (B) Boolean C o n t r o l l e r to the Cli (C2j)-channels.
When a BQ comes down a WBC or a
BBC to an as yet untraversed W Switch, i t captures the WP on the longer diagonal path
and then proceeds down unperturbed to the Cli-channels.
I f , however, a BQ attempts
to pass the W Switch in the opposite d i r e c t i o n , whether previously traversed or untraversed, then, on reaching the northeast corner of the longer diagonal path, the
WP j u s t underneath the captured WP goes north by one square and thus opens up a l i n e
of more than
k
WB's e f f e c t i v e l y covering the shorter diagonal path of the switch,
making i t impassable.
The crossing of Clause-channels with a Clock-channel and two Literal-channels
can be observed from the western part of Figure 5.
If
y £ Cll ,
y ~ Cl2
say, then
a BQ coming down the y-channel can stop unperturbed at the i n t e r s e c t i o n - called
island-with
the Cll-channel.
But i f i t t r i e s to come to rest at the i n t e r s e c t i o n
with the Cl2-channel, called t h r o u g h - i n t e r s e c t i o n , then i t
WP.
is promptly captured by a
The s i t u a t i o n is reversed f o r a BQ coming down the y-channel
(y ~ Cll , y E Cl2).
On the other hand, a BQ coming down a Clock-channel cannot stop unattacked at any
crossing with a Cli-channel; a l l i t s intersections with Clause-channels are throughintersections.
We remark that i f a l i t e r a l
is not used in W-LOSE (B-LOSE), i t s channel is
truncated p r i o r to reaching the W (B) Switch (Figure 3).
Every channel-segment has length at least
U ~ 2(k(t+l)+2)
,
and the shields
around each channel, including truncated ones, also have thickness at least U.
reason for t h i s w i l l become clear l a t e r .
The
(In the f i g u r e s , some segments seem short
and some shields t h i n , which is the r e s u l t of emphasizing the main features at the
expea~se of the standard ones.
But i t should be kept in mind that the true length of
segments and thickness of shields is at least U throughout.)
3.
As was mentioned above, i f
move of W,
then there are ~
THE WINNING SCENARIO
Cli
contains
~
literals
and
Cli = l
following a
BQ's each of which can reach the Cli-channel in
t = 8 moves. The strategy of B is to f i r s t
move a l l
~
BQ's into the Cli-channel
and then to move each of them as far down the Cli-channel towards the B CDG-channel
as W permits. The f i r s t
BQ to pass has to capture the WP located at the W A l t a r
which is backed up by a l i n e containing precisely
~-l
w i l l capture j
Then the ( j + l ) - t h BQ captures a
of the BQ's for some 0 ~ j < L .
WB's (Figure 5).
Thus W
283
W piece at the W A l t a r a f t e r
~t÷j÷i
moves to reach the Cli-channel and
moves: each of the ~
j+l
BQ's requires t
of them make one capture move each.
the ( j + l ) - t h BQ captures a W piece at the A l t a r , i t spends (k-~)t
d e l a y - l i n e consisting of
and r i d i n g the
(k-~)t
B CDG-channel.
~t+j+l+(k-~)t+2 = kt+j÷3
After
moves in a
W
WP's. Two a d d i t i o n a l moves are spent for reaching
Using this strategy,
moves for checkmating the
B thus requires
WK.
Following the departure of the f i r s t BQ from i t s vantage point on some Boolean
C o n t r o l l e r towards a Cli-channel, the WQ on the same Boolean C o n t r o l l e r can enter
the W Clock-channel.
(Figure 6).
Each Clock-channel contains a d e l a y - l i n e of
Since W also captures j
kt-3
moves
BQ's in the Cli-channel and there are six
a d d i t i o n a l moves for entering and leaving the W Clock-channel and r i d i n g the
CDG-channel,
W can checkmate the BK (B king) a f t e r
a margin of one move. Since
k(t+l)+2
j < ~ s k ,
kt+j+3
B can in fact checkmate the WK in at most
(= U/2) moves. Every other move of W , from among the l i m i t e d moves a v a i l -
able to him, is also doomed to f a i l u r e .
This is shown in the next section.
I f , a f t e r W's move which made Cli = l
,
W switches his WR between the x
- p o s i t i o n and the ~ - p o s i t i o n on some WBC, thus possibly unsatisfying W-LOSE,
still
W
moves. Thus B wins with
B can
select the values s a t i s f y i n g W-LOSE by using the B Detour Route (Figure l ) .
This requires an a d d i t i o n a l move of B, but since also W l o s t one move in his extra
WR switching maneuver, the move balance between B and W is preserved, and B can s t i l l
win.
Now suppose that B starts to move BQ's towards some Cli-Channels before the game
G3 has been decided.
We show that W w i l l win i f he activates a W Clock immediately
f o l l o w i n g the departure o f the f i r s t
BQ, and then captures BQ's in the Cli-channels
whene~er possible, otherwise proceeding down the W Clock-channel.
Given t h i s strategy of W, B's only chance to win is to transfer in some C l i channel at least
~
BQ's i f clause Cli comprises
only way a BQ can enter the B CDG-channel.
the Cli-channel, where
(i)
t~ = t
for all
t 'r= t
r
or
t+l .
(I ~ r ~ L) .
BQ must stop at a through-intersection.
~ literals,
since this is the
The r - t h BQ requires
f
tr
moves to reach
There are two cases:
Since W-LOSE is s t i l l
Then a WP captures i t ,
false~ at least one
f o i l i n g B's design.
Now W wins via i t s clock-mechanism a f t e r a possible engagement at the W A l t a r .
(ii)
WBC).
t~ = t + l
for some r
(which means that B uses the B Detour Route in some
I f B again stops at a through-intersection, the s i t u a t i o n is as before.
If
B stops at islands only, then B spends ~ moves in the BQ-WB battles at the W A l t a r ,
(k-~)t
moves in the channel d e l a y - l i n e and two moves f o r reaching and r i d i n g the B
CDG-channel°
Thus B requires at least
moves to checkmate the WK.
~ t r + (k-L)t+~+2 ~ Lt+l+(k-&)t+L+2=kt+~+3
r=l
Now W spends L-l moves in capturing BQ's and kt+3
moves in the W Clock and W CDG-channels.
moves, less moves than B, and so W wins.
Thus W can checkmate the BK in
kt+~+2
284
4.
"ILLEGAL" MOVES
The above analysis - except the l a s t p a r t - w a s
the players do in fact simulate G3.
based on the assumption that
We c a l l a move " i l l e g a l "
i f i t is a legal move
in generalized chess, but is e i t h e r not part of the simulation of G3 altogether, or
i s part but is taken at the wrong time f o r a proper simulation of G3.
Below we con-
sider the nonobvious " i l l e g a l " moves.
I.
The WBC. There are only six pieces that can move: WR, WQ, BQ, two BP's and
one WP (Figure I ) .
A.
Moves of WR.
(i)
Suppose that while the game G3 is s t i l l
WR-channel from i t s normal
undecided, WR leaves the
x or ~ - p o s i t i o n , going east or west.
bizarre e f f e c t o f making both
x =1
and
x= 1
(This has the
as f a r as B-LOSE is concerned, but
leaving x unchanged in W-LOSE.)
I f WR stops in the l i n e of s i g h t of BQ, then BQ captures WR. The timing is
such, as is easy to v e r i f y , that even i f WR's move made B-LOSE t r u e , B can now win
via the B RC-channel except that i f WQ moved to the x - p o s i t i o n a f t e r BQ captured WR,
then BQ has to back up to the B NC/RC-channel i n t e r s e c t i o n and win via the B NCchannel.
I f WR stops elsewhere, then BQ goes d i r e c t l y to the WR/B RC-channel i n t e r -
section and wins via the B RC-channel.
(ii)
Suppose t h a t while G3 is s t i l l
undecided, WR stops w i t h i n the WR-
channel at some location other than the x or ~ - p o s i t i o n .
making
x =I
and
x =I
in both B-LOSE and W-LOSE.)
(This has the e f f e c t of
I f t h i s location is the i n t e r -
section with the B RC-channel, then BQ captures WR and wins again via the B RCchannel,
Otherwise a BP captures WR.
I f now W moves his queen to the x - p o s i t i o n ,
then BQ goes to the B NC/RC-channel i n t e r s e c t i o n and then wins via the B NC-channel
(even i f B-LOSE is now t r u e ) .
B,
Otherwise BQ can again win via the B RC-channel.
Moves of W~(i)
suppose t h a t while G3 is s t i l l
the i n t e r s e c t i o n with the W Clock-channel.
W can win via i t s Clock mechanism.
undecided, WQ moves northwest to
Then BQ w i l l capture WQ, since otherwise
Even i f W now makes B-LOSE true, B can win by
moving southeast to the i n t e r s e c t i o n with the B NC-channel and then proceeding down
t h i s channel.
(ii)
Suppose that WQ moves as in ( i ) in some WBC R, but the move is
made a f t e r W-LOSE has been made true previously by W.
winning, B moves i t out towards the Cli-channels.
I f BQ in R is required for
Otherwise B continues with his
normal winning strategy, ignoring W's move altogether.
(iii)
Suppose that while G3 is s t i l l
undecided, WQ moves down v e r t i c a l l y .
I f i t comes to rest at the B NC/RC-channel i n t e r s e c t i o n , B w i l l capture i t with his
BQ which w i l l subsequently proceed down the B NC-channel and win.
captured by a BP.
Otherwise WQ is
Even i f W now makes B-LOSE true, B can win with his BQ via the
285
B NC-channel.
(iv)
Suppose that WQ moves as in ( i i i ) ,
W-LOSE has previously been made true by W.
same as in ( i i ) ,
(v)
but the move is made a f t e r
Then B's strategy is e s s e n t i a l l y the
so we omit i t .
Once BQ has l e f t a WBC, WQ can neither pass through the L i t e r a l -
channels in W-LOSE nor through the B Clock-channel, because of the BP's defending
the channel corners.
An attempt by WQ to advance in p a r a l l e l to some of these
channel segments from the outside by gnawing i t s way along the shielding WP's and
then s l i p p i n g in at a s u i t a b l e corner, is simply ignored by B, since the length of
each channel-segment is at least
win.
U, which is about twice as long as i t takes B to
Also WQ cannot skip from channel to channel by penetrating through channel-
shields, since these have thickness at least U.
(vi)
Suppose t h a t a f t e r W-LOSE has previously been made true by W, and
WR is in the x - p o s i t i o n , WQ moves to the x - p o s i t i o n in some WBC R.
I f BQ in R is
required f o r winning, B w i l l now move i t towards the Cli-channels via the B Detour
Route.
Otherwise B continues with his normal winning strategy.
I f under the same assumption WR is in the x - p o s i t i o n and WQ advances towards
the x - p o s i t i o n by capturing the BP j u s t southwest of the x - p o s i t i o n , then provided
BQ of R is required for winning, BQ moves out towards the Cli-channels via the xchannel.
I f BQ is not required f o r winning, W's move is ignored as before.
C.
Moves of BQ.
The moves (Bi)-(Bv) have obvious counterparts for BQ in a
WBC and move (Bvi) has a counterpart in a BBC, so we omit the d e t a i l s . Only in the
counterpart of ( B i i ) a s l i g h t l y new s i t u a t i o n may arise:
Suppose that BQ moved to
the B NC/RC-channel i n t e r s e c t i o n and WQ then advanced towards the x - p o s i t i o n - since
WQ is required for w i n n i n g - f i r s t
capturing the BP j u s t southwest of the x - p o s i t i o n .
I f BQ now moves to the o r i g i n a l position of WQ, then WQ captures BQ and then continues
down the x-channel towards the C2j-channels.
the x-channel.
Otherwise WQ continues d i r e c t l y down
A s i m i l a r s i t u a t i o n can arise in the counterpart of ( B i v ) , which W
handles also in the way j u s t described.
If,
before G3 has been decided, BQ advances to
its first
s t a t i o n towards an
x (x)-channel w h i l e WR is in the ~ ( x ) - p o s i t i o n , then BQ is captured by WR. On i t s
next move, WQ w i l l enter the W Clock-channel in the WBC in which the BQ was captured,
and win via i t s Clock-mechanism.
I f BQ makes a move of t h i s type a f t e r B made B-LOSE
true, i t is ignored by W, who continues with his normal winning strategy.
D.
Moves of the Pawns.
(i)
Suppose that while G3 has not yet been decided, the BP j u s t west
of the B NC/RC-channel i n t e r s e c t i o n or the BP two squares north of i t , moves south.
Then WQ goes northwest to a point one square southeast of the W Clock i n t e r s e c t i o n
( c a l l t h i s square K).
W can now win via his Clock since B loses one move on account
of blocking the entrance to the B Clock-channel with his own BP.
286
(ii)
Suppose that while G3 has not yet been decided, the WP j u s t south
of K moves north onto K.
Then BQ moves southeast to the middle of the f i r s t
leg of
the B RC-channel, from where i t can win by going west to the B NC-channel.
II.
Preventi_n~Backlash.
Suppose that B, e i t h e r before G3 has been decided or
a f t e r i t has been decided in W's favor, assembles a squadron of BQ's in the C l i channels in an attempt to break back into some B Clock-channels or i n t o some L i t e r a l channels, with the aim of reaching the C2j-channels via some Boolean Controllers.
I f B succeeds in capturing even one of the WQ's needed for a normal winning strategy
of W, the game's outcome is not clear anymore.
Now W commences executing his normal winning strategy at the l a t e s t one move
a f t e r the f i r s t
BQ is moved towards the Cli-channels.
to break back via some B Clock-channels.
B needs
t+l
Assume f i r s t
that B attempts
moves to place a BQ at a
Cli/B Clock-channel i n t e r s e c t i o n , which is a t h r o u g h - i n t e r s e c t i o n .
ture BQ there.
A f t e r B moved k+l
Then W w i l l cap-
BQ's to such through-intersections and W cap-
tured them (the f i r s t with a WP~ subsequent ones with WB's, see Figure 5), B spent
( k + l ) ( t + l ) moves~ and W spent
( k + l ) t moves pursuing his normal winning strategy and
k+l moves capturing BQ's at t h e i r prospective backlash points.
thickness at least
U > k+l ,
Since shields have
W has a s u f f i c i e n t supply of bishops to do the l a t t e r .
(Note that in Figure 5 the true distance between the three v e r t i c a l channels is much
larger than shown.)
W wins.
I t is thus seen that in at most
k-t+2 $ 6
additional moves,
I f B attempts to break back via some L i t e r a l - c h a n n e l s , then i t again takes
t+l moves to place a BQ at a C l i / L i t e r a l - c h a n n e l i n t e r s e c t i o n , which may be an
island.
At least three additional moves are made by BQ before i t is captured by a
WB in a W Switch.
Thus a f o r t i o r i
and capturing (at most
5.
POLYNOMIALITYOF TRANSFORMATION
Recall our e a r l i e r notation:
(B-LOSE) and
m = IX[ + [Y[ .
encoded in binary,
12 p log p
W wins by pursuing his normal winning strategy
k+l ) BQ's which t r y to break back.
p (q) is the number of And-Clauses in W-LOSE
The subscripts
i
of the l i t e r a l s
xi
and Yi
are
Therefore the length of W-LOSE (B-LOSE) has magnitude about
(12 q log q) ,
and the input size is thus
O((p+q)log(pq)) .
Clearly
m ~ 12(p+q) .
For each variable our construction requires a constant amount of chess-pieces:
The Boolean C o n t r o l l e r , four L i t e r a l - c h a n n e l s , two Clock-channels and four Switches
associated with a variable require a constant amount of chess-pieces since each
channel-segment has length
O(k(t+l))
each channel also have thickness
which is a constant, and the shields around
O(k(t+l))
.
Thus the sequence of m Boolean Con-
t r o l l e r s oriented in a general northwest to southeast d i r e c t i o n (Figure 3), has
length O(m) = O(p+q)
Therefore also the Clause-channels and CDG-channe!s have
length
is also
O(p+q)
O(p+q)
each.
The t o t a l thickness of the Clause-channels with t h e i r shields
I t follows that the construction can be realized on a square
287
board of side
Note.
n = O(p+q) ,
and so the transformation is polynomial.
I f we provide Switches in the Clock-channels in addition to those in the
Literal-channels, we can replace the bishop shields around the Clause-channels by
pawn shields.
The Switches themselves can be redesigned so that they' can operate
without bishops.
I f , in addition, we back up the Altars by queens instead of
bishops, i t seems possible to avoid using bishops altogether.
p o s s i b i l i t y that
nxn
This leads to the
German checkers ("Dame") can be proved Exptime-complete by a
method s i m i l a r to the above proof.
(In German checkers a piece reaching the opposite
side of the board e s s e n t i a l l y becomes a queen rather than a king.
We are told that
t h i s is the rule also for the version of the game as played in the USSR.) Of course
also other board-games (such as
nxn
Go )
may be Exptime-complete.
ACKNOWLEDGEMENTS. We are much indebted to J.M. Robson for putting his finger
on a number of weak spots in e a r l i e r drafts of the paper.
We also l i k e to thank the
referee for his constructive c r i t i c i s m and comments.
REFERENCES
I.
A.S. Fraenkel, M.R.
of checkers on an
on Foundations of
Computer Society,
2.
M.R. Garey and D.S. Johnson, Computers and I n t r a c t a b i l i t y : A Guide to the Theory
of NP-Completeness, W.H. Freeman, San Francisco, 1979.
3.
D. Lichtenstein and M. Sipser, Go is Polynomial-Space hard, J. ACM 27 (1980),
393-401. Also appeared in Proc. 19th Annual Symp. on Foundations of Computer
Science, 48-54, Ann Arbor, MI, October 1978; IEEE Computer So--ciety, Long Beach,
CA, 1978.
4.
M.E. Morrison, e d i t o r , O f f i c i a l Rules of Chess, 2rid ed., David McKay, New York,
1978.
5.
S. Reisch,
Gobang i s t PSPACE-vollstandig, Acta Inform. 13 (1980), 59-66.
6.
S. Reisch,
Hex i s t PSPACE-vollstandig,
7.
J.M. Robson, N by N chess is Pspace-hard,
Australian National University, 1980.
TR-CS-80-09,
8.
H. Samelson, e d i t o r , Queries, No. 4 ( i i i ) ,
190-191.
Notices Amer. Math. Soc. 24 (1977),
9.
L.J. Stockmeyer and A.K. Chandra,
a. Com~ut. 8 (1979), 151-174.
lOo
Garey, DoS. Johnson, T. Schaefer and Y. Yesha, The complexity
n × n board - - preliminary report, Proc. 19th Annual Symp,.
Computer Science, 55-64, Ann Arbor, MI, Octob'er 1978~ IEEE
Long Beach, CA, 1978.
Acta Inform. 15 (1981), 167-191.
Provably d i f f i c u l t
Computer Science Dept.,
combinatorial games, SIAM
J. Storer, A note on the complexity of chess, Proc. 1979 Conference on Information Sciences and Systems, Dept. of Electr. Eng., Johns Hopkins University,
Baltimore, 160-166.
288
W CLOCK-CHANNEL
TO Czj-CLAUSES
IN B-LOSE
x~
A~
WB
o
•
a
•
WHITE
BLACK
WHITE
BLACK
PAWN
PAWN
BISHOP
BISHOP
B RAPIDCLOCK
(RC)- CHANNEL
xV
v~
TO CIi-CLAUSES
B NORMALCLOCK
IN W- LOSE
(NC)-CHANNEL
FIGUREI WHITE BOOLEAN
CONTROLLER
B CLOCK-CHANNEL
289
I
~
W CLOCK-CHANNEL
TO Cll' - CLAUSES
TO C2i- CLAUSES
IN B- LOSE
B CLOCK-CHANNEL
IN W- LOSE
FIGURE 2. SCHEMA OF BLACK BOOLEAN CONTROLLER.
290
.J
hi
z_.
Wl
CON]R(
W SWITCHE'
I SWITCHES
(BBC)
I
~/~r~
9"~)(0~
J ~
So
FIGURE 3. GLOBAL VIEW OF THE CONSTRUCTION
FOR THE CASE:
W - L O S E = Ctl v CI2 v CI3, Ctl = XIA x 2 A Yl , CI2 = x2 ^ Yl , CI3 = xI A x 2
B - LOSE= C21v C ~ v C23, C21: xl ^ Y2, C2"z=;,I ^ ~2^Yl ^ Y2 , C23: xl ^ ~,:z^ Y~
N
I
D
Z
I
i
m
]HANNEL
3 LITERAL
292
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z
Z
"y
--I
LU
Z
Z
Z
0I
-li
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W B
lID
_I
I,
Ll
Z
Z
"I0
I
o,i
~USE DELAY-LINE
LENGTH ( k - [ ) t
! 1011101Ol I01
i ~110~tOl I
I t o l o t ~ I,
J.~ IOlOlOI,
i IolOl,
:l
i
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t
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110
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i
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ilC
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c
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trx
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lli
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i
i
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c ~io
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c
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c
: : Ev
{t'
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-~-
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r" 31o
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FIGURE 5. W CHANNEL CROSSINGS, W ALTAR AND CLAUSE-CHANNEL
DELAY-LINES.
293
B CLOCK-CHANNEL
W B
CLOCK DELAY- LINE
OF LENGTH kt-5
FIGURE 6
LOWER PART OF B CLOCK-CHANNEL WITH
CLOCK DELAY-LINE IMPINGING ON B CDGCHANNEL.