GENERALIZED SHOGI AND CHESS ARE CONSTANT-TIME TESTABLE

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1 GENERALIZED SHOGI AND CHESS ARE CONSTANT-TIME TESTABLE Hiro Ito Atsuki Nagao Teagun Park School of Informatics and Engineering, The University of Electro-Communications (UEC), 1-5-1, Chofugaoka, Chofu, Tokyo, , Japan. CREST, JST, Tokyo, Japan. Abstract We present constant-time testing algorithms for the generalized shogi (Japanese chess) and chess. These problems are known to be EXPTIME-complete. A testing algorithm (or a tester) for a property accepts an input if it has the property and rejects it if it s far from having the property in high probability (e.g., at least /3) by reading only a constant part of the input. A property is said to be testable if there is a tester. The generalized shogi (and chess) problem is, given any position on n n board with O(n) pieces, for testing the property the player who moves first has a winning strategy. We present that this property is testable for both shogi and chess. The shogi tester is one-sidederror and the chess tester is surprisingly no-error! In the last decade, many problems have been found to be testable. However, almost all of such problems are in class NP. This is the first result on constant-time testability of EXPTIME-complete problems. 1 Introduction In this paper we present constant-time algorithms for generalized shogi 1 and chess. Generalized games and puzzles are often treated from the view point of computational complexity. In many cases they are found to be in hard classes, e.g., NP-complete, PSPACE-complete, EXPTIME-complete, etc. It is known that the generalized shogi and chess are in EXPTIME-complete [1, 4]. In other hand, in the area of algorithm theory, constant-time algorithms are rapidly developed recently. A constant-time algorithm solves a problem in a constant-time, i.e., the time complexity is constant, which is independent of the input size n. Property testing is the best studied in the area of constanttime algorithms. A testing algorithm (or a tester) for a property accepts an input if it has the property and rejects it if it s far from having the property 1 Shogi is the highly popular chess-type game in Japan. There have been professional players at least 400 years ago. It is said that more than 0 million Japanese can play it [9] and there is a federation of professional shogi players, named Nihon Shogi Renmei, of which there are at present more than 00 professional players. in high probability (e.g., at least /3) by reading a constant part of the input. A property is said to be testable if there is a tester. In the last decade, many properties are found to be testable or non-testable [, 3, 5, 6, 7, 8, 10, 11, 1, 13, 15]. In this paper, we study a testability of the generalized shogi and chess. Note that almost all the properties that have been s- tudied so far in the area of property testing are in class NP. We show that the generalized shogi and chess are testable. This is the first result on testable problems in EXPTIME-complete. Methods.1 Position and oracle We explain mainly the generalized shogi. The argument on chess, which is close to it, will be shown later. The generalized shogi is a shogi played on a n n board with O(n) pieces including two k- ings. Black is the player who moves first, and white is the player who follows black. Any given position S, the problem is for determining whether or not black wins if the both player do their best. A position is defined by fixing which piece is on which cell on the board in which direction 3 and the number of each kind of captured pieces of each players. The basic rule is the same with the original shogi and it is omitted here (see manuals, e.g. [9]). In what follows we add some detailed definitions that are necessary to understand the discussions in this paper. In shogi, there are 8 different kind of pieces: king (, ou), rook (R, hisha), bishop (B, kaku), gold (G, kin), silver (S, gin), knight (N, kei), lance (L, kyo), and pawn (P, fu). They are distinguished by giving piecekind-numbers like as king is 0, rook is 1,..., pawn is 7. There are only two kings: one is of black and the other is of white. For each of the other piece-kinds (i.e., rook, bishop, gold, silver, knight, lance, and pawn), In shogi and go, the first player is represented by black and the second player is represented by white. This is the opposite to the way of chess. 3 In shogi, black and white players use the same kind of pieces. Who is the owner of each piece is appeared by the direction. 015 ISORA IET 1 Luoyang, China, August 1 4, 015

2 there are at most cn pieces, where c is a fixed number. Piece-numbers from 1 to cn are given to each piece of in each piece-kind. That is, each piece has it s own piece ID (k, l) consisting of a piece-kind-number k {0,..., 7} and a piece-number l {1,..., cn }. An algorithm can know the given position thorough the following oracles. Piece oracle: A given piece ID (k, l), piece o- racle answers a tuple (p, i, j, r), which shows the information of the piece having the ID: (i) the owner p {0, 1, } (1 means black, means white, and 0 means no owner, i.e., the piece is not appeared in the position), (ii) the place (i, j), i, j {0, 1,..., n } of it (i and j represent the column number and the row number, respectively, if it s on the board, and i = 0 if it s a captured piece), and (iii) the piece is promoted or not (r = 1 if it is promoted and r = 0 otherwise). This oracle is represented by q 1 (k, l) = (p, i, j, r). Position oracle: A given coordinate (i, j), i, j {1,..., n }, position oracle answers (p, k, l, r), which shows the information of the piece being in the cell: (i) the owner p {0, 1, }, (ii) the piece ID (k, l), and (iii) the piece is promoted or not. This oracle is represented by q (i, j) = (k, l, p). There is another oracle that asks the number of captured pieces q. That is, a given owner p and a piece-kind-number k, it answers the number of captured piece of the kind. E.g., q (1, 3) = 5 represents that black player is capturing 5 golds. Both q and q are called position oracle. When we explicitly identify the position S, we represent the oracles like as q 1 (k, l; S), q (i, j; S), and q (p, k; S). Each camp (opponent s promotion area) is the first n/3 rows, i.e., white s camp consists of rows from 1 to n/3 and black s camp consists of rows from n n/3 + 1 to n. We introduce an assumption that all pieces can be arranged on the board simultaneously, and 7cn+ n follows. For simplicity, we assume that c 1/8. A position S is called a winner if black has a winning strategy, i.e., black will win if they play starting from S and do their best to win.. Draw and fouls In shogi, some rules on fouls and draws exist. A draw occurs in sen-nichi-te (prepetual moves) or so-nyugyoku (entering both kings). Sen-nichi-te occurs when the same position appears four times. In the generalized shogi, the game also ends in a draw if it drops in a sen-nichi-te. 4 We don t care about so-nyuu-gyoku, 4 In the rule of Nihon Shogi Renmei, a sen-niti-te of perpetual check is a foul, and a player who were checking loses. In this paper we omit this rule for simplicity. But even if we introduce this rule, the same result is obtained. since from the rule of sen-nichi-te, the game must ends in finite steps. The following cases are fouls. Nifu (double pawn): Two or more unpromoted pawns of a same player never be in the same column simultaneously. Dead end: Pawns, lances, and nights never be to moved to, or dropped onto, cells from which they have no next moves. So black s (white s, resp.) unpromoted pawns and lances can never be in the first (the n th, resp.) row and black s (white s, resp.) knight can never be in the first nor the second (the n th nor ( n 1)th, resp.) rows. Uchi-fu-zume (pawn drop mate): No pawn (fu) can be dropped to deliver immediate checkmate. A player who plays these fouls loses. In our game, a position S is given as an input, and S may include one of these fouls. Uchi-fu-zume never appeared in S, since it depends on a move, not a position. Thus we must consider nifu and dead end. We define that if only one player does such a foul in S, he/she loses, and if both players do such fouls in S, the game ends in a draw..3 Distance between positions A position is fixed by making a query to the piece oracle for every pieces. From c 1/8, the number of different queries for the position-oracle is at most 8cn n. (1) That is, a position is fixed by fixing at most n data. The distance between two positions S and S is defined as dist(s, S ) := {(k, l) q 1(k, l; S) q 1 (k, l; S )}. () n The numerator is the number of pieces (k, l) such that the answers for q 1 (k, l; S) and q 1 (k, l; S ) are different. From (1), 0 dist(s, S ) 1 follows. Two positions S and S are called isomorphic if we can make S the same with S by changing only their piece-numbers. (Note that changing piece-kindnumbers is not allowed.) For example, positions S and S in Fig. 1 are isomorphic. The set of positions that are isomorphic to a position S is denoted by [S]. Let W be the set of winners. W clearly closes under isomorphism, i.e., if S W, then [S] W. A set of positions that is closed under isomorphism is called a property. Note that W is a property. For a position S and a property P, the distance between them is defined as dist(s, P) := min S P dist(s, S ). (3) For a positive number ϵ > 0, S is called ϵ-far from P if dist(s, P) > ϵ, and ϵ-close to P otherwise. 015 ISORA IET Luoyang, China, August 1 4, 015

3 L3 P3 P1 L1 L P1 P4 L3 G S1 1 G1 S 1 P4 P G1 S P5 P3 G S S3 S S1 S3 P5 L P L1 Figure 1: Two positions S and S are isomorphic..4 Tester The number of queries an algorithm makes to the o- racles is called query complexity. A property testing algorithm, or tester for short, for a property P is an algorithm that accepts every position S from P with probability at least /3, and rejects every position that is ϵ-far from P with probability at least /3 with constant query complexity, which is independent of n (but it may depend on ϵ). Moreover, a tester that always (i.e., with probability 1) accepts S P is called onesided-error. A one-sided-error tester that always rejects S that is ϵ-far from P is called no-error. If there is a tester for a property, the property is called testable. 3 Results 3.1 The main theorem We show the following theorem in this paper. Theorem 1 There is a one-sided-error tester whose query complexity is O(ϵ ) for the generalized shogi problem, i.e., the generalized shogi problem is testable with one-sided-error. 3. Nifu-free We prepare some lemmas before proving Theorem 1. A position is called nifu-free if it includes no nifu (double pawn). A promise problem of the generalized shogi such that every given position includes no nifu (double pawn) is called Nifu-free generalized shogi. The set of nifu-free positions is denoted by N. Note that N is also a property. Lemma 1 For any nifu-free position S N and any 0 < ϵ 1, if n > max{1/c, 36/ϵ } (where n is the size of S), then S is ϵ-close to W. Proof: Since S includes no nifu-foul, if black plays foul in S, then it is a dead end. Black s pieces that play this foul is in the first or second rows and thus the number of such pieces is at most n. Let S be a position made from S by removing these foul pieces from the board and making them black s captured pieces. Clearly dist(s, S ) n. Next, let S be a position made from S by replacing pieces in cells (i, j), 1 i 4, 1 j 3 as shown in Fig.. In this configuration, black king is safe and L G S Figure : White king will be checkmated in the next black s move shown by the arrow. white king will be checkmated in the next black s move (moving gold from (3, ) to (, )). The pieces that were in these cells in S are moved to black s captured pieces, and the pieces appeared in these cells in S are moved from other places. Note that from the assumption of n > 1/c, every kind of pieces exists at least one. Then, dist(s, S ) = 19. After the first (black s) move from S, white is checkmated, black king is not checked, and black plays no foul. Hence the winner of S is black, i.e., S W. Therefore, R N dist(s, W) dist(s, S ) dist(s, S ) + dist(s, S ) n Here, if n 5, then n + 19 < 6 n, and thus from n > 36/ϵ (> 5), follows. dist(s, W) < 6 n = 6n n < ϵn Q.E.D. 015 ISORA IET 3 Luoyang, China, August 1 4, 015

4 Lemma For any real number 0 < ϵ 1, there is a one-sided-error tester with the query complexity O(ϵ ) for any given position S for testing on property N. We show a tester for Lemma as follows. Procesure DetectingNifu begin 1. Choose a column j and two rows i and i such that i i uniformly at random.. Query q 1 (i, j) and q 1 (i, j) and knows the pieces in (i, j) and (i, j). If both of the pieces are black s pawn, then output reject and stops. 3. Iterate steps 1 4/ϵ times. If all of the iterations don t reject, output accept and stops. end. The proof of Lemma will be done by showing that the above algorithm correctly works as follows. Proof of Lemma : If n max{1/c, 4/ϵ }, the algorithm can get the perfect information of S by calling q 1 (k, l) for every piece ID (k, l) and thus it can check all possible branches of the game tree in constant time. Note that from the rule of sen-nichi-te (perpetual moves), the game must end in finite steps. Then we consider the case of n > max{1/c, 4/ϵ } and prove that DetectingNifu works correctly. Since the algorithm rejects only when it finds nifu (double pawns), it always accepts S N, and thus it is onesided-error. Next, assume that S is ϵ-far from N. That is, there must be more than ϵn black s pawn on the board. DetectingNifu selects (i, j) and (i, j) with i i u- niformly at random. Therefore the probability P that the algorithms rejects in one comparison is expressed as P = p/(rf(r)), where p is the number of pairs of black s pawns in a same column, f(x) = ( x ) = x(x 1)/, and r = n. Let x j denote the number of black s pawns being on row j and let x denote the total number of black s pawns being on the board. That is, x x r = x > ϵn and p = f(x 1 ) + + f(x r ). Since f(x) is convex, from Jensen s inequality, it follows that Therefore P = p = f(x 1 ) + + f(x r ) ( x ) ( ) ϵn rf rf r n = rf(ϵ n). p rf(r) f(ϵ n) f( n) = ϵ n(ϵ n 1) n( n 1) = ϵ n 1/ϵ n 1. From n > 4/ϵ, n 1/ϵ > n/ holds and P > ϵ n > ϵ n 1 follows. Thus the algorithm rejects with probability more than ϵ / in one iteration of step. Since line is iterated 4/ϵ times, the probability of that S is not rejected through all iterations is less than ( ) 4/ϵ ( ( ) /ϵ ) 1 ϵ = 1 ϵ ( e 1) 1 < 3. Note that x, 1 + x e x is used in the first inequality. Therefore S, which is ϵ-far from N, is rejected with probability at least /3. Q.E.D. 3.3 Proof of the main theorem From Lemmas 1 and, a proof of Theorem 1 is obtained as follows. Proof of Theorem 1: problem as follows. Procesure TestingShogi begin We give an algorithm for this 1. If n 144/ϵ, then call q 1 (k, l) for every piece ID (k, l), get the perfect information of S, check all possible branches of the game tree, and get the correct answer.. If n > 144/ϵ, then by calling DetectingNifu to test whether S is nifu-free or ϵ/-far from nifufree. If S is accepted, then output accept, and otherwise, output reject, and stop. end. We prove that the above algorithm works correctly as follows. If n 144/ϵ, the correctness is clear and the query complexity is O(ϵ ), and thus in what follows we assume that n > 144/ϵ. First, we consider the case of S W. Thus S is nifu-free. Then it s accepted by DetectingNifu in step, and hence it s accepted with probability 1. Next, we consider the case of that S is ϵ-far from W. It is enough to show that S is rejected with probability at least /3. Assume that S is not rejected in probability more than or equal to /3. It follows that S is not ϵ/-far from nifu-free, i.e., there is a nifu-free position S N such that dist(s, S ) ϵ/. From n > 144/ϵ = 36/(ϵ/) and Lemma 1, it follows that S is ϵ/-close to W, i.e., there is a position S W such that dist(s, S ) ϵ/. Therefore dist(s, S ) dist(s, S ) + dist(s, S ) ϵ/ + ϵ/ = ϵ. Thus S must be ϵ-close to W, contradiction. From Lemma, the query complexity is O(ϵ ). Q.E.D. 015 ISORA IET 4 Luoyang, China, August 1 4, 015

5 3.4 Chess It is known that the generalized chess is also in EXPTIME-complete [4]. The generalized chess is defined as similarly to shogi: a n n board is used, there are two kings, and for each other piecekind (i.e., queen, bishop, knight, rook, pawn) there are at most cn pieces. Oracles are also similarly defined. The property to be tested is that, given any position S, white 5 (the player who moves first) has a winning strategy starting from S. To avoid complicated description we omit to describe the detailed definitions here, since readers can imagine them easily and small difference on the definitions might not have an effect on the result. We prove the following theorem. Theorem There is a no-error tester whose query complexity is O(ϵ 1 ) for the generalized chess problem, i.e., the generalized chess problem is testable with noerror. Proof sketch: Let S be a given position. Chess has no foul like as nifu (double pawn) in shogi, and we don t need to check whether white plays foul in S. Let S be the position made from S by changing pieces in cells (i, j), 1 i 4, 1 j 5 like as shown in Fig. 3. The pieces that were in these cells in S Q R N P Figure 3: Black king will be checkmated in the next white s move shown by the arrow. are moved to captured pieces, and the pieces appeared in these cells in S are moved from other places. In S, white king is safe and by the next white s move (moving queen from (3, ) to (, )) black king will be checkmated, i.e., S is a winner. The distance between S and S is at most = 8. Thus, if n 8/ϵ, then dist(s, S ) 8 ϵn, and hence S is ϵ-close to winner. Therefore it is enough to accept it. If n < 8/ϵ, it is enough to read all information by calling piece oracle for all pieces. It needs O(ϵ 1 ) queries. This algorithm always accepts a winner, and furthermore, if a given position S is ϵ-far from winner, then R B 5 In chess, the player who moves first is white. n < 8/ϵ and the algorithm knows the complete information of S. Therefore, this algorithm is no-error. Q.E.D. 4 Conclusion We showed that the generalized shogi and chess are testable. By using a similar discussion, all chess type problems, i.e., xiangqi (Chinese chess) are expected to be testable, and proving it looks not difficult. Another interesting problem is that testability of the generalized GO, which is also known to be EXPTIMEcomplete [14]. We conjecture that it is also testable. However, a similar way used for shogi and chess doesn t look applicable to GO, and an algorithm we are considering for GO is completely different from them. Proving the correctness of the algorithm may be the next work. Acknowledgement We would like to thank the Algorithms on Big Data project (ABD14) of CREST, JST, the ELC project (MEXT AENHI Grant Number ), and JSPS AENHI Grant Numbers and through which this work was partially supported. References [1] H. Adachi, H. amekawa, and S. Iwata: Shogi on n n board is complete in exponential time, IEICE Journal, Vol. J70-D, No. 10, 1987, pp (In Japanese) [] N. Alon, E. Fischer, I. Newman, and A. Shapira: A combinatorial characterization of the testable graph properties: it s all about regularity, SIAM J. Comput., Vol. 39, No. 1, 009, pp [3] I. Benjamini, O. Schramm, and A. Shapira: Every minor-closed property of sparse graphs is testable, Proc. STOC 008, ACM, 008, pp [4] A. S. Fraenkel and D. Lichtenstein: Computing a perfect strategy fo n n chess requires time exponential in n, Journal of Combinatorial Theory Series A, Vol. 31, 1981, pp [5] O. Goldreich (Ed.): Property Testing Current Research and Surveys, LNSC 6390, 010. [6] O. Goldreich and D. Ron: Property testing in bounded degree graphs: Proc. STOC 1997, ACM, 1997, p- p [7] O. Goldreich, S. Goldwasser, and D. Ron: Property testing and its connection to learning and approximation: Journal of the ACM, Vol. 45, No. 4, July, 1998, pp [8] A. Hassidim, J. A. elner, H. N. Nguyen, and. Onak: Local graph partitions for approximation and testing, Proc. FOCS 009, IEEE, pp. 31. [9] T. Hosking: 4 Great Games Rules and strategy for beginners; Published by The Shogi Foudation, Cornwall, England; First Printed by Redwood Books, 1998; Reprinted by Antony Rowe Limited, CPI Group, ISORA IET 5 Luoyang, China, August 1 4, 015

6 [10] H. Ito, S. iyoshima, and Y. Yoshida: Constant-time approximation algorithms for the knapsack problem, Proceedings of the 9th Annual Conference on TAMC, pp (01). [11] H. Ito, S. Tanigawa, and Y. Yoshida: Constant-Time Algorithms for Sparsity Matroids, Proc. ICALP (1), LNCS 7391, 01, pp [1] I. Newman and C. Sohler: Every property of hyperfinite graphs is testable, Proc. STOC 011, ACM, 011, pp [13] R. Levi and D. Ron: A quasi-polynomial time partition oracle for graphs with an excluded minor, Proc. ICALP 013 (1), LNCS, 7965, Springer, 013, pp [14] J.M. Robson: The complexity of GO, Proc. IFIP1983, 1983, pp [15] Y. Yoshida: A characterization of locally testable affine-invariant properties via decomposition theorems, Proc. STOC 014, ACM, 014, pp ISORA IET 6 Luoyang, China, August 1 4, 015

Lecture 19 November 6, 2014

6.890: Algorithmic Lower Bounds: Fun With Hardness Proofs Fall 2014 Prof. Erik Demaine Lecture 19 November 6, 2014 Scribes: Jeffrey Shen, Kevin Wu 1 Overview Today, we ll cover a few more 2 player games