UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN CENTRAL CIRCULATION AND BOOKSTACKS The person borrowing this material is re- sponsible for its renewal or return before the Latest Date stamped below. You may be charged a minimum fee of $75.00 for each non-returned or lost item. Theft, mulilalion, or detacement of library maferlals can be causes for student disciplinary action. All materials owned by the University of Illinois Library are the property of the Stale of Illinois and are protected by Article 16B of f//ino(s Criminal Law and Procedure. TO RENEW, CALL (217) 333-8400. University of Illinois Library at Urbana-Champaign OEC 1 1993 When renewing by phone, write new due date below previous due date. L162 Digitized by the Internet Archive in 2013 http://archive.org/details/computingperfect968frae t^^n % ;iUCDCS-R-79-968 UILU-ENG 79 1716 COMPUTING A PERFECT STRATEGY FOR n X n CHESS REQUIRES TIME EXPONENTIAL IN n by Aviezri S. Fraenkel and David Lichtenstein June 1979 IBeueB6SX9JlHE UIUCDCS-R-79-968 COMPUTING A PERFECT STRATEGY FOR n X n CHESS REQUIRES TIME EXPONENTIAL IN n by Aviezri S. Fraenkel and David Lichtenstein June 1979 DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URB ANA- CHAMPAIGN URBANA, ILLINOIS 61801 COMPUTING A PERFECT STRATEGY FOR n X n CHESS REQUIRES TIME EXPONENTIAL IN n 1 2 Aviezri S. Fraenkel and David Lichtenstein 'Tis all a chequer-board of nights and days where destiny with men for pieces plays; hither and thither moves and mates and slays and one by one back in the closet lays. The Rubaiyat of Omar Khayyam 1. Introduction. From among all the games people play, chess towers as the most absorbing and widely played. Indeed, if attention is restricted to 2-person games of perfect information without chance moves played outside of the Far East, the ever rejuvenating interest in the 1500 year old game has a quality of depth and breadth well beyond that of any potential rival. It is noteworthy, then, that in the long string of complexity results for games, chess had yet to appear. We have shown that a natural generalization of chess to n X n boards is complete in exponential time, the first such result for a "real" game. This implies that for any k >_ 1 , there are infinitely many positions tt such that any algorithm for deciding whether White (Black) can win from that position requires at least C time-steps to compute, where c > 1 is a constant, and [irl is the size of it. Generalized chess is thus provably intractable, which is a stronger result than the complexity results for board games such as Hex, checkers and Go which were shown to be Pspace-hard [2, 3, 5, 6]. In proving the result for chess, it seems that we have highlighted the strengths and the weaknesses of current techniques, and that the further analysis of strategies for board games will need finer tools if it is to obtain interesting results. Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801. On leave from Department of Applied Mathematics, The Weizmann Institute of Science, Rehovot, Israel. 2 Computer Science Division, University of California, Berkeley, California 94720. -2- We have generalized chess to be any game of a class C of chess- type-games with one king per side played on an n x n chessboard. The pieces of every game in C are subject to the same movement rules as in 8 X 8 chess, and the number of pawns, rooks, bishops and queens each increase as some fractional power of n. Beyond this growth condition, the initial position is immaterial, since we analyze the problem of winning for an arbitrary board position. Unfortunately, our constructions seem to violate the spirit of 8x8 chess, in much the same way as the complexity proofs for Hex, checkers and Go mentioned above. Typical positions in our reduction do not look like larger versions of typical 8x8 chess endgames. Although we have not tried to answer questions of reachability, it seems offhand as though players would have a hard time trying to reach our board positions from any reasonable starting position. What we can say, however, is that certain approaches for deciding whether a position in 8 x 8 chess is a winning position for White may not be very promising, namely those approaches which work for arbitrary positions and generalize to n x n boards. Such approaches use time exponential in n, and hence can be useful only if the exponential effect had not yet been felt for n = 8. Thus, while we may have said very little if anything about 8x8 chess, we have, in fact, 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 result is in line with the suggestion to demonstrate the complexity of interesting board games by imbedding them in families of games [7]. An interesting corollary of our result is that if Pspace 4" Exptime, -3- then there is no polynomial bound on the number of moves necessary to execute a perfect strategy. This is so because the existence of a polynomial bound on the number of moves implies that the game is in APtime = Pspace. For the sake of the uninitiated, we now give a short informal introduction to the basic notions of computational complexity. Let S be a class of decision problems (i.e. problems whose answer is "Yes" or "No"). For decision problems IT , tt , we say that 7T is polynomial ly transformable (or reducible ) to tt (notation: tt ^ fT ) , if there exists a function f from the set of instances of tt to the set of instances of TT such that: (i) I is an instance of tt for which the answer is "Yes" if and only if f(I) is an instance of tt for which the answer is "Yes." (ii) f is computable by a polynomial time algorithm in the size of I (a "polynomial time algorithm"). A decision problem tt is S-complete if: (i) TT e s, (ii) for every tt' ^ S, tt' « tt. A decision problem tt is S-hard if (ii) holds but (i) does not necessarily hold. A decision problem is intractable if it cannot be decided by a polynomial time algorithm. A nondeterministic algorithm is an "algorithm" which can "guess" an existential solution, such as a path in a tree and then verify its validity by means of a deterministic algorithm. An alternating algorithm is an "algorithm" which can "guess" an entire subtree soltuion, such as a solution -4- of a game, and verify its validity by means of a deterministic algorithm. See Chandra and Stockmeyer [1]. Important classes of decision problems are the class P of all decision problems ir with (deterministic) algorithms whose running time is bounded above by a polynomial in the input of tt; the class NP (nondeterministic polynomial) of all decision problems tt with nondeterministic algorithms whose verification running time is bounded above by a polynomial in [7t|; the class Pspace of all decision problems TT whose algorithms require an amount of memory space bounded above by a polynomial in | tt j ; the class APtime of all decision problems tt with alternation algorithms whose verification running time is bounded above by a polynomial in | TT | ; and the class Exptime of all decision problems tt with (deterministic) algorithms whose running time is bounded above by an exponential function in |tt|. The following basic relations hold: P C NP C Pspace = APtime C Exptime . It is not known whether any of these inclusions is proper, except that P / Exptime. Furthermore, NP and Pspace are not known to contain any intractable decision problems, but Exptime is. From the definition of °= it follows that if tt °° tt , then tt e P implies tt e P. Therefore the S-complete problems for any S are the "hardest" problems of S. In particular for S = Exptime, the S-complete problems are all intractable. For further details and a formal treatment of this topic the reader is referred to Garey and Johnson [4]. 2. The Reducti on Let Q be the following question: Given an arbitrary position of a generalized chess-game on an n x n chessboard from our class C, can -5- White (Black) win from that position? We shall show that G„ °^ Q, where G_ is the following Boolean game proved complete in exponential time by Stockmeyer and Chandra [8]. Throughout W (B) stands for White (Black). Every position in G^ is a 4-tuple (T, W-LOSE(X,Y), B-LOSE(X,Y), a), where i G {W,B} denotes the player whose turn it is to play from the position, W-LOSE, B-LOSE are Boolean formulas in 12DNF and a is an assignment of values to the set of variables X U y. Player W (B) moves by changing tha value of precisely one variable in X (Y). In particular, passing is not permitted. W (B) loses if the formula W-LOSE (B-LOSE) is true after some move of player W (B). Thus W can move from (W, W-LOSE, B-LOSE, a) to (B, W-LOSE, B-LOSE, a') iff B-LOSE is false under the assignment a (otherwise B already lost), and a and a' differ in the assignment of exactly one variable in X. If W-LOSE is true under the assignment a', then W just lost. Our basic structure is the Boolean controller . Figure 1 (2) illustrates a W (B) Boolean controller for a variable x G x (y €E Y) . The white circles are WP's (W pawns), the black circles BP's (B pawns), and WR, BR, WB, BB, WQ, BO stand for W rook, B rook, W bishop, B bishop, W queen, B queen, respectively. If WR is at the south position of the WR-channel, as in Figure 1, also called the x-position , then the value of x is 1. If WR is at the north position of the WR-channel, indicated by '.•./•> in Figure 1 also called x-position , then the value of x is 0. A similar convention is adopted for Figure 2 which is indicated only schematically because a B Boolean controller is just a 180° rotation of a W Boolean controller and an interchange C^^ "^ ^2 ' ' ^ "^ ^ throughout. -6- There is one W (B) Boolean controller for each x ^ X (y G Y). In normal play, W (B) moves his WR (BR) between the x-position and the x-position (y-and y-position) in any W (B) Boolean controller. If W (B) does not abide by these rules, then his opponent can win via the B (W) clock or the B (W) rapid victory mechanisms detailed below. A global view of the construction is shown in Figure 3. Let k be the largest number of literals in any "And-clause" in W-LOSE and B-LOSE. Let C^^ in W-LOSE be an And-clause consisting of £ literals for some 1 < £ £ k < 12. If C^^ = 1 after a move of V, then there are £ B queens each of which can reach the C -channel in t moves (t = 11 as can be seen in Figures 1, 2, 3: four moves in the literal channel up to the W switch, six moves in the W switch, and one final move to a clause- channel). The C^ -channel contains £-1 W pawns and a delay-line (for details see Figures A, 5). The strategy of B is to bring one BQ at a time into the C -channel, moving it as far down the channel as W permits. Whenever a BQ captures one of the £-1 original WP's on the C^-channel, it can be recaptured by another WP. Now W will capture j of the F.Q's for some < j < £. Then 1 / • 1 ^ Lh ' '"' (j + 1) 1^'? cnpt ores the Inst Wl' in tlie C -channel after (j+l)(t+l) + 9.-2 moves. This Is obviously the case for j = 0. AsRume true for any h satisfying _< h < £-1 and consider the case where W captures lH-1 BQ's. Suppose that W captures tiie first BQ after the latter captured the first q WP's on the C^.-channel (1 £ q £ £-l-h). By the induction hypothesis, the (h+2) BQ captures the last WP in the C^ .-channel after (t+q) + (l-H-i)(t+l) + (£-l-q-]) + 1 = (hf2)(t+l) + £-2 moves, where the last +1 is the move capturing the WP that captured the first BQ. -7- After the (j+1) BQ captured the last WP in the C -cliannel , one move is required for reaching the delay line, where (k-£.)(t+l) moves are spent. Another move is required to ride the B coup de grace (cdg)-channel for administering the coup de grace to the W king (WK) . See Figure 6. Using the above strategy, B thus requires (j+l)(t+l) + I + (k-^Ct+l) = (k-H+j+l) (t+1) + I moves for checkmating tlie WK. Following the departure of the first BQ from its vantage point on some Boolean controller towards a C.. -channel, the WQ on the same li Boolean controller can enter tlie W clock-channel . F.acli clock-channel contains a delay line of k(t+l)-4 moves. Since W also captures j BQ's and there are 4 additional moves for entering and leaving the W clock-channel and one additional move in tlie W cdg-channel, W can checkmate the BK after k(t+l) + j + 1 moves. Since k(t+l) + j + 1 - ( (k-^+j+1 ) (t+l)+£) = (£.-j-l)t >^ 0, it is seen tliat W cannot catch up with B. In a certain sense, the best strategy for W is to capture j = l-l BQ's, but even then he falls short of victory by one move. Every other move of W, from among the limited moves available to him, is also doomed to failure. This is shown in Section 3, wliere the detailed structure of the Boolean controller and backlash prevention are discussed. If, after w's move which made C = 1, W switches his WR between the X. -position and the >; .-position on some W BooJean controller, thus possibly 1 1 unsatisfying U-LOSE, B can still select the values satisfying W-LOSE by using the detour route (Figure 1). This requires an additional move of B, but since also W lost one move in his extra WR switching maneuver, the move balance between B and W is preserved, and B can still win. -8- Now suppose that B starts to move BQ ' s towards some C .-channels while W-LOSE is false. We show that W will win if lie activates a W clock immediately following the departure of the first BQ, and then captures BQ's in the C .-channels whenever possible, otherwise proceeding down the W clock-cluinnei . (;jven this stragcgy of W, B' s only chance to win is to transfer in some C.. -channel at least I BQ's if clause C^ . comprises i literals, since 11 li ' this is tlie only way a BQ can enter the B cdg-channel. If B does not do this, he cannot even prevent W from winning: Every (t+1) move he can delay W by one move by sacrificing one of liis queens. After k(t+l) + k moves, B sacrificed k queens (in more than one clause-channel), and W advanced k(t4-l) moves in his clock, spending an additional k moves in capturing tlie k queens. Thus W can win via tlie clock in liis next move. We may therefore assume that B transfers £ BQ's into some C,. -channel, where C-, . consists of £ literals (1 < £ < k). li — — The i BQ requires t '. moves to reach the C,. -channel, where ^ ^ 1 li t! = t or t + 1, and at least for some i, t '. = t + 1 (1 ^ i £ £) . In the channel, 15 spends 2(£-l) moves in the BQ-WP battles, (k-£) (t+1) moves in the channe: delay line and two moves for reaching the delay line and riding tlie cdg-channel. £ Thus R ran cliorkmatc the WK in .T,, t\ + (k-e,)(t+l) + 2(£-l) + 2 "^ 1 = 1 I £t + 1 + (k-£)(t+l) 4- 2(£-l) + 2 = k(t+l) + £ + 1 moves. Now W spends £-1 moves in capturing BQ's and k(t+l) + I moves in the clock and W cdg-channel. Thus W can checkmate the BK in k(t+l) + £ moves, less moves than B and so W wins. -9- 3. The Details I. The W Boolean Controller (Figure 1) . There are only four pieces that can move: WR, WQ, BQ and a BP at position a, just west of the B clock/B rv-channel intersection. What happens if any of these II I) move illegally? (i) Suppose that while B-LOSE is still false, WR leaves the WR-channel (i.e., it goes east or west), or it goes to the WR/B rv-channel intersection. Then BQ goes to the same intersection, winning in four moves . (ii) Suppose then that while B-LOSE is still false, WR stops in the WR-channel of some W Boolean controller T , at some location other than the X- or x-position or the intersection with the B rv-channel. Then some BP will capture it. If now W makes any move other than moving WQ in T to the x- or x-position, then BQ in T goes to the B rv-channel, leading to a win. Note that W can at most delay this plan of B: by making ii If some further illegal move immediately following B's capture of WR, e.g., moving a WQ to the B clock/B rv-channel intersection, W may attempt to block BQ or defend the WK. But B can win nevertheless. So suppose that WQ moves to the x- or x-position in T . Then BQ in T goes to the B clock-channel intersection. Note that if the WR-move of (ii) left B-LOSE false, B can now win via the B clock-channel emanating from T whatever W's next move is. So suppose that the WR-move made B-LOSE true. There are three possibilities. (a) If WQ in T leaves the WR-channel or goes to the WR/B rv-channel intersection, then the BQ in T can again win via the B rv-channel, (b) If W wastes a move, such as moving WR within the WR-channel in some Boolean controller, B can win via the B clock-channel emanating from T . -10- (c) Otherwise, W moves his queen in some Boolean controller T^ i^ T^ towards a clause-channel. Then B will move his queen to rhe B clock-chimnel intersection in T^. As long as the WQ stays in the WR-channel in T^, B moves his queen down the B clock-channel emanating from T_. Note that for W to win he has to move his queen from the WR-channel in T towards a clause-channel, whence B can win via the B rv-channel in T . If, however, W fails to move his queen from the WR-channel in T , B will win via the B clock-channel emanating from T„. (iii) Suppose that while B-LOSE (W-LOSE) is false, WQ (3q) moves in any direction other than south (north). Then B (W) can win by moving his queen into the B (W) clock-channel. If WQ (BQ) stops anywhere in the vertical channel which it commands other than at the intersection with the B (W) clock-channel, it is captured by a BP (WP) and then B (W) can win via the B (W) clock-channel. In particular, if WQ captures the BP at the southern tip of the channel, it gets captured by the BP at a, and B can win via its clock. If WQ (BQ) moves to the intersection with the B (W) clock-channel, it is captured by the BQ (WQ) , which then proceeds to win via its clock-channel as before. (iv) If WQ (BQ) tries to enter the B (W) clock-channel after the BQ (WQ) left its initial position, it can be captured by a BP (WP). Similarly, if WQ tries to enter the B rv-channel after BQ left its initial position, it gets captured at the intersection marked s at the latest. (v) If the BP at a moves south while W-LOSE is still false, WQ will go northwest to position e, from where it can win via the W clock- channel, since B loses one move on account of blocking his clock-channel with his own BP. \ -11- (vi) Note that if WR is in its x or x-position, then BQ cannot break, into the B rv-channel even if B-LOSE is true and WQ already left its initial position. One such attempt of B might be to move his queen to the vacated initial position of WQ (via the B clock/rv-channel intersection). If WR is not already in the x-position, W will now move there. It is then seen that any attempt of BQ to penetrate into the WR-channel and from there into the B rv-channel is suicidal. We finally remark that if a literal is not used in W-LOSE (B-LOSE), its channel is truncated before the W (B) switch. II. Preventing Backlash (Figure 4) . The crossing of the clause-channels with the literal, clock and rv-channcls raise the danger of backlash: queens leaping from channel to channel. Up to m = |x| + |y| queens of one color might concentrate in one clause-channel and launch a concerted attack. If BQ's try to break through from a clause-channel at a location other than one at which they can legitimately park, the first BQ will be captured by a WP, whose movement opens up a row of 2m WB's which can effectively thwart the attack at that location. In particular, B cannot break from a clause-channel position at which it cannot legitimately park into any B rv or clock-channel. Since we may assume that B might attempt such attacks only when W-LOSE is false, W can then, in addition to repelling the attack, win via a W clock-channel. The situation is more complicated at those clause/literal- intersections at which BQ's may park legitimately. A squadron of BQ's assembled at such intersections, lying on one vertical literal-channel, may try to force its way backwards into a Boolean controller, and from there into an rv-channel, say. The W switches through which each B literal-channel passes are designed to prevent this possibility. -12- We first show that if W just made W-LOSE true, B can win even if W attempts to mobilize the backlash prevention mechanism to his advantage. Suppose then that a BQ comes down a vertical B literal-channel towards the diagonal part of the channel, called W switch , on which there are five W s. After the BQ captured the first two topmost WP's, the WP immediately below the topmost WP can move upward. After four such WP's move upward, a WB can reach the switch. But in the meantime the BQ reaches the end of the switch, and in the next move it can reach safely a C -channel. Note that also four WP's would have to be moved upward below the second W on the switch, and two below the third, fourth and fifth WP on the switch, before a WB could reach the switch, and that whichever of the pawns W decides to move, B can always bring his queen safely down to a clause-channel. Note further that no WB can reach a clause-channel because of the phalanx of WP's which has depth > 2k(t+2) , which is amply more than the number of moves that B requires for winning. We now assume that while W-LOSE is false, B attempts to win by backlashing. There are essentially two ways in which B may attempt to break through from the clause-channels to a Boolean controller. First he may, with the switch not yet traversed, attempt to traverse it upwards. Even if B has a queen at each end of the switch, in addition to the BQ trying to traverse the switch, the attempt is thwarted by the line of 2m WB's just on top of the W phalanx, which will command the switch as soon as either the traversing BQ captured the second WP from top, or two traversing BQ's, coming from the clause-channels, captured the first and the third WP's from bottom. In either case, no BQ can cross the switch. -13- One of several variations of this is when B attempts, with the switch not yet traversed, to break through by positioning one BQ at the location of the bottom WP of the switch, one at the bottom end of the switch and then come down with a (single) BQ from the Boolean controller, capturing part or all of the remaining four WP's on the switch. It can be seen that also this attempt is doomed to failure, though one BQ may succeed in going back to the Boolean controller. Secondly, B may attempt to backlash when the switch is already traversed sufficiently to permit the WP's under some of the WP's on the switch to move up. As an example, consider the case where, while W-LOSE is false, a BQ came down from a Boolean controller through liLeral channel L, and now rests at the location of the fifth (lowest) WP on the W switch. If now another BQ stops at the intersection of a clause-channel and L, then W moves up a WP below the location of the fourth WP on the switch. Whether B now captures this WP with the first BQ or moves up the second BQ to the lower end of the switch or moves a third BQ to a clause-channel intersection with L, W moves up the second WP. thus unleashing a horde of 2m WB's onto the switch. The first BQ ma> safely back up towards the Boolean controller it came from, but no other BQ can get through. Moreover, W can eventually win via a VJ clock-channel. It is easily verified that also none of the other variations of B's attempts to backlash when the W switch is partly or completely traversed can prevent W from winning via its clock mechanism. Needless to say, an attempt of the BQ's to penetrate the W phalanx is futile, since it lias width and depth > 2m. In particular, B cannot break from a clause/literal-channel intersection at which it can park legitimately into a B rv- or clock-channel. -14- III. The Clause-Channel Delay-Lines . A large number of BQ's might attempt to shortcut the delay-lines on a C -channel, in an attempt to reach the vicinity of the WK, either inside or outside- the cdg-channel. To foil this plan, the delay lines are padded with layers of WB's of depth >^ 2m. See Figure 5. IV. The Transformation is Polynomial . It is easy to verify that the number of chessboard squares used in our construction is bounded by a polynomial in the size of W-LOSE and B-LOSE. Since we have shown that W (B) will win from the constructed chess position if and only if B (W) makes B-LOSE (W-LOSE) true, the proof is complete. References 1. A.K. Chandra and L.J. Stockmeyer, Alternation, Proc. 17th Annual Symp. on Foundations of Computer Science , 98-108, IEEE Computer Society, Long Beach, CA, 1976. 2. S. Even and R.E. Tar j an, A combinatorial problem which is complete in polynomial space, Proc. 7th Annual ACM Symposium on Theory of Computing , 66-71, Albuquerque, NM, 1975; Association for Computing Machinery, New York, 1975. 3. A.S. Fraenkel, M.R. Carey, D.S. Johnson, T. Schaefer and Y. Yesha, The complexity of checkers on an n x n board — preliminary report, Proc. 19th Annual Symp. on Foundations of Computer Science , 55-64, Ann Arbor, MI, October 1978; IEEE Computer Society, Long Beach, CA, 1978. 4. M.R. Carey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness , W. H. Freeman, San Francisco ^ 1979. 5. D. Lichtenstein and M. Sipser, Go is Pspace hard, Proc. 19th Annual Symp. on Foundations of Computer Science , 48-54, Ann Arbor, MI, October 1978; IEEE Computer Society, Long Beach, CA, 1978. 6. S. Reisch, Hex ist PSPACE-vollstaridig, manuscript, private communication. 7. H. Samelson, editor. Queries, No. 4 (ill). Notices Amer. Math. Soc. 24 (1977), 190-191. 8. L.J. Stockmeyer and A.K. Chandra, Provably difficult combinatorial games, IBM Report RC 6957 (#29823), IBM T. J. Watson Research Center, Yorktown Heights, NY, 1978. TO Ci, -CLAUSES IN B-LOSE O WHITE PAWN • BLACK PAWN O WHITE BISHOP ■ BLACK BISHOP B CLOCK - CHANNEL B RAPID VICTORY (RV)-CHANNEL Figure 1 . White Boolean controller W RAPID VICTORY (RV)-CHANNEL W CLOCK - CHANNEL TO C2i -CLAUSES IN B-LOSE . I 1 1 1 \ / / / 1 / \ \ \ \ / 1 — \ / \ \ / «l \ ^ -.- \ \ / / \ . / \ X / — 1 1 1 V \ \ / 1 1. \ \ / 1 1 I \ \ 1 1 \ \ \ / r 1 1 \ y- CONTROL BQ w Q-. • - -- DE luu ^OUT R F ■ • -- 1 -1 1 > / \ \ 1 1 / \ / \ \ 1 1 / / / \ \ 1 4- / / \ \ 1 / / / > \ 1 — u, / / / \ / B R V \ / > / / y 1 1 1 / r ii ▼ \ 1 \ \ ' ' W B TO Cl^-CLAUSES IN W-LOSE B CLOCK - CHANNEL Figure 2 . Schema of Black Boolean controller W BOOLEAN CONTROLLER IWBC) W SWITCHES DELAY =(k-l)(t+l) Globe OELAY-Utt* U-4 W B view of the construction case W-LOSE =Ci, + C,2 + C,3, Cn--XiXjyi. Cu'Xzyi. C^'XiXj B -LOSE = C2,+C22 + Cj3 . 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OjO D oi«\ O D C D D Q D o o • \ D ( D •- '"^ r ' qJd o D io»^ s O O t D D D D o • o •^ \oo < D o^s|0^a D o Dl O \o( 3 D D D O o • \o ( D |\]o |"0 D D D O D O • \< 3 O D D o o o 0»\ ( J ^ X' o^D a □ D D O o • '^ S^O O D o o O D • \ ( 3 :\ f c *• E \ ' •Xj o i o 1 d d 'd D O • o « »\o O o o D O a • \ ( D 1 ' '^ 3 \lo'o n|DiDiO o < D •'\0 o o D D n • \ < \ ' 1^ ?! ©K o'OD D o O D o«\ o o D D D D • \ ( D \ ^ 1 |c y o •NJooiDio O O 1 D O • N o D D D D D • \< 3 V^\i 1^ ■> V Di»l\ O OIO o a ( D D O \o ]0_ Q D D D D • ' \,o i \!< 3 |ol o,»Volo O D D O O O • \ o o D D a D D •\ \[ I 3 ioId :o •\jo O D D D D o • N° ODD D D D • \ _ Xj( 3^v.o1o,d: O •\ O D 3 O O □ o o • \ D a D D D • \ , — "^ > yx^a o'D Oj«^ I OIO 3 D D □ o • o •^ \^o D D D D a • \ 1 ,( D^^ODDOQ 0« »No 3 D D n o o o • \o D D D D a • \o — t- 3 ^d/o d'd O D < ^•N O D D o o D •X D Q D a D • \ d' Io^d'dId'dIoId o.^ \[ O O D o o O D • \ D D D c D • NO D lo'oDDDDOt 31 iO • \^o o o o D O D • \ ,0 D a D D D \ s \ ^>i o o c iDiD D < D a ©X o o o D D D • \ ojol D D D D • D |0| o c >'dIo D I D O D o»\ o o D D D D • \ D D D D D ^JL°I c ) O D D I D D O D O • \ o O D D D D • K D D D D d"o OJO D D D D D O .<\o O D D olo D !• >° D D D To ._ 1 1 1 r 1 1 b ' \ ,_ 1 1/ CX3- CHANNE L^ Cl2" ■C HA NN EL ^ Ci 1 " CHA^ JNEL J Clause-channel delay-lines and White pawns in clause-channels _J- . UJ . z . z . < . I - o 1 - _) UJ 2 z < I o 1 > >• cr o »- o > 9 < cr o o o o o o o o m o o o o o o o o o o o o • "^z o • o • o /' \ o o o • o z. • ( :> o • o • • o / • o o o o • o • o / • o o o • o / • o < o . o o • o / • o • N • o o • o / • o • o • N o • o o • o / • o o o • N O K » o o • o / • o o •^ \[ :> • o o • o / • o o > o • o o • o / • o • o /* » o o o • o / • o • o / • c D o • o • o / • o • o / • o o o o / • o • o / • o o / • o • o <; o • o) / o • o • \ o • • o /« \ • o • • \ o • • o / • c > • o o • \ O 1 » • o / • o o - o • \,^ ) • • o / • o ^ o • \ o • o / • o A^ • • • o / • , r ^^^ o o • o / • o r v. / o o • o / • o o • o / • o o o / • o ^-^ o / • o f • o } / o • ^ • o / € \ • o > • o <^ 6 • / • o • o / • o • / • o • • ^ o o OWKO 1 o o 1 T W B Figure 6. Black clock-channel delay-lines. Black coup de grace (cdg)-channel and White king disposition .IBLIOGRAPHIC DATA HEET ]. Report No. UIUCDCS-R-79-968 2. 3. Recipient's Accession No. Tiilc Jnj Subtitle COMPUTING A PERFECT STRATEGY FOR n x n CHESS REQUIRES TIME EXPONENTIAL IN n 5. Report Date June 1979 6. Authot(s ) Aviezri S. Fraenkel and David Lichtenstein 8. Performing Organization Rcpt. No. UIUCDCS-R-79-968 . Performing Organization Name and Address Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois 61801 10. Project/Task/Work Unit No. 11. Contract /Grant No. 2. Sponsoring Or^tanizat ion Name and Address 13. Type of Report & F^eriod Covered research 14. S. Supplementary Notes 6. Abstracts It is proved that a natural generalization of chess to an n x n board is complete in exponential time. 7. Key Words and Document Analysis. 17a. Descriptors games combinatorial games chess complexity computational complexity Exp time completeness Exp time-completeness 7k. Identifiers 'Open-Ended Terms 7e. roSATI Field/Group 8. Availability Statement unlimited 19. Security Class (This Report) UNCI.A.SSIFIED 21. No. of Pages 20 20. Security Class (This Page UNCLASSIFIED 22. Price 3RM NTIS-3S (10-701 USCOMM-DC 40329-P7) hiii; 1 u \9 ¥