Endgame Databases Research Articles

Construction and investigation of Cram endgame databases

In this paper we investigate the game of Cram, which is the impartial version of Domineering. We have built Cram endgame databases for all board sizes < 30 squares. We developed a program that fills the databases with their Combinatorial Game Theory (CGT) values. Since Cram is an impartial game, all CGT values for Cram positions are so-called nimbers, indicated by ∗ n. The nimber value of a position not only directly determines the game-theoretic value (first- or second-player win), but also provides an optimal playing strategy. When analyzing the resulting databases we observed the following facts. Firstly we confirmed that the CGT values of all investigated empty boards are in agreement with results published in the literature. Since the value of an empty board depends completely on the values of many partially filled positions in the database, this is a strong indication that our process of filling the database with CGT values is correct. Secondly, although the series of values for 2 × n boards is completely regular, namely a value ∗ 0 for n being even and ∗ 1 for n being odd, this was not proven formally so far. We were able to provide such a proof. We also investigated the databases for their contents. So far we encountered nimber values up to ∗ 11 among single-fragment Cram positions. It appears that for a ∗ n value to occur a board size with ≈ 3 n squares is needed, some more for very tall boards ( 1 × n), some less for wider boards ( 4 × n). So far no single-fragment Cram positions were encountered with nimber values ⩾ ∗ 12, although we show a construction of larger multi-fragment positions with nimber values from ∗ 12 up to ∗ 15. In a preliminary experiment we incorporated the CGT endgame databases constructed into a simple alpha-beta solver for the game. Results revealed a large improvement in solving power.

Compressing Chinese Dark Chess Endgame Databases by Deep Learning

Endgame databases can be difficult to use in tree search when database sizes remain large even after compression. Given the same endgame, we discover that the compression ratios vary significantly when using different encoding schemes. The intuition is that when a set of positions mapped into a continuous chunk of segments have similar values, block-based compression libraries such as zlib can yield a better compression ratio than cases where segments contain diversified values. However, finding the optimal encoding scheme by exhaustive enumeration is time-infeasible for endgame databases with a large number of pieces. We propose a novel approach using deep learning to obtain an encoding scheme so that the compression ratio is suitable for practical purposes. Our approach can be applied to chess-like games.

Equivalence Classes in Chinese Dark Chess Endgames

Chinese Dark Chess, a nondeterministic two-player game, has not been studied thoroughly. State-of-the-art programs focus on using search algorithms to explore the probability behavior of flipping unrevealed pieces in the opening and the midgame phases. There has been comparatively little research on opening books and endgame databases, especially endgames with nondeterministic flips. In this paper, we propose an equivalence relation that classifies the complex piece relations between the material combinations of each player, and derive a partition for all such material combinations. The technique can be applied to endgame database compression to reduce the number of endgames that need to be constructed. As a result, the computation time and the size of endgame databases can be reduced substantially. Furthermore, understanding the piece relations facilitates the development of a well-designed evaluation function and enhances the search efficiency. In Chinese Dark Chess, the number of nontrivial material combinations comprised of only revealed pieces is 8 497 176, and the number that contain at least one unrevealed piece is 239 980 775 397. Under the proposed method, the compression rates of the above material combinations reach 28.93% and 42.52%, respectively; if the method is applied to endgames comprised of three to eight pieces, the compression rates reach 5.82% and 5.98%, respectively.

New results for Domineering from combinatorial game theory endgame databases

We have constructed endgame databases for all single-component positions up to 15 squares for Domineering. These databases have been filled with exact Combinatorial Game Theory (CGT) values in canonical form. We give an overview of several interesting types and frequencies of CGT values occurring in the databases. The most important findings are as follows.First, as an extension of Conway's [8] famous Bridge Splitting Theorem for Domineering, we state and prove another theorem, dubbed the Bridge Destroying Theorem for Domineering. Together these two theorems prove very powerful in determining the CGT values of large single-component positions as the sum of the values of smaller fragments, but also to compose larger single-component positions with specified values from smaller fragments. Using the theorems, we then prove that for any dyadic rational number there exist single-component Domineering positions with that value.Second, we investigate Domineering positions with infinitesimal CGT values, in particular ups and downs, tinies and minies, and nimbers. In the databases we find many positions with single or double up and down values, but no ups and downs with higher multitudes. However, we prove that such single-component ups and downs easily can be constructed, in particular that for any multiple up n⋅↑ or down n⋅↓ there are Domineering positions of size 1+5n with these values. Further, we find Domineering positions with 11 different tinies and minies values. For each we give an example. Next, for nimbers we find many Domineering positions with values up to *3. This is surprising, since Drummond-Cole [10] suspected that no *2 and *3 positions in standard Domineering would exist. We show and characterize many *2 and *3 positions. Finally, we give some Domineering positions with values interesting for other reasons.Third, we have investigated the temperature of all positions in our databases. There appears to be exactly one position with temperature 2 (as already found before) and no positions with temperature larger than 2. This supports Berlekamp's conjecture that 2 is the highest possible temperature in Domineering.

An Enhanced Solver for the Game of Amazons

The game of Amazons is a modern board game with simple rules and nice mathematical properties. It has a high computational complexity. In 2001, the starting position on a 5 × 5 board was proven to be a first player win. The enhanced Amazons solver presented here extends previous work in the following five ways: by building more powerful endgame databases, including a new type of databases for so-called blocker territories, by improving the rules for computing bounds on complex game positions, by local search to find tighter local bounds, by using ideas from combinatorial game theory to find wins earlier, and by using a df-pn based solver. Using the improved solver, the starting positions for Amazons on the 4 × 5, 5 × 4, 4 × 6, 5 × 6, and 4 × 7 boards were shown to be first player wins, while 6 × 4 is a second player win. The largest proof, for the 5 × 6 board, is presented in detail.

TAAI 2013 COMPUTER GAME TOURNAMENTS

th , 2013. In this event, fifty computer programs from the Netherlands, France, and Taiwan participated in tournaments for ten different games being Go 9x9, Chinese chess, Connect6, Chinese dark chess, Mahjong, NoGo, MiniShogi, Nonogram, Othello, and Double King Dark Chess. The participants and the final results are listed in Table 1. With much pleasure the organizer invited Prof. Yoshimasa Tsuruka of the University of Tokyo for the keynote speech. Yoshimasa is a developer of the well-known Shogi program, GEKISASHI (激指), which is the four-times World Computer Shogi Champion (WCSC). The title of his talk was Recent Advances in Algorithms for Traditional Board and Table Games. In Table 1, the team from the National Chiao Tung University (NCTU) performed best by winning four gold medals. Each of the other teams was awarded at least one gold medal. This showed that each team has focused on specific games during 2013 and did so obtain good achievements. The champion and the runner-up of Chinese chess tournament were SHIGA and CHIMO, respectively. They had also the same ranks in the 17 th Computer Olympiad and in TAAI 2012. SHIGA won all games except that only one game against CHIMO ended up in a draw in this tournament. SHIGA was designed to support the multiprocessor architecture, and its search depth reached 20 plies while using a 4.5GHz 12-core computer. CHIMO used an opening book compiled from 85,000 game records that was then enhanced by the JL-ABS algorithm. Its evaluation function was improved and was tuned better. Both programs were estimated to be near 8-dan. SHARK as a newcomer won the bronze medal. Chinese dark chess is popular in China and Taiwan. It uses the same board and pieces as Chinese chess, but the pieces can be either revealed or unrevealed (Chen et al., 2010). In 2013, nine programs from France and Taiwan participated. The competitions were played automatically on the game-playing platform designed by Jr-Chang Chen. The platform was also used in the 17 th Computer Olympiad. YAHARI, which landed at the third place in the 17 th Computer Olympiad, became the champion; and SGTDARK was the runner-up. Both programs are based on alpha-beta search (ABS) and used endgame databases constructed by retrograde analysis. The third place was for DARKKNIGHT, which also won the bronze medal in TAAI 2012. DARKKNIGHT was based on Monte-Carlo tree search (MCTS) and utilized the concept of chance nodes to deal with flipping unrevealed pieces. The playing strength of each program improved significantly, and the programs based on ABS performed slightly better than those based on MCTS.

Statistical Inference of Rough Set Dependence and Importance Analysis

Statistical inference about dependence degree (DD) and importance degree (ID) of variables in an information system is crucial for variables appraisement and model reconstruction. However, in rough set data analysis (RSDA), the literature is restricted to validate independence or test whether the degree is significantly big, while the fixed value test and interval estimation for related measurements have been ignored. Because these important issues have not been addressed, we cannot determine whether the data support expert opinions and compare the features in depth. To enhance the integrity of statistical inference for DD and ID in an RSDA, fixed value tests and interval estimations of DD and ID are presented in this paper. With multinomial distribution as the carrier for statistical information in the databases, the fixed value test of DD is successfully transformed into a restricted estimation of multinomial distribution and a goodness-of-fit test for distributions. The fixed value test and interval estimation algorithms for DD and ID are then presented in detail and illustrated with examples. Explicit expressions for the DD and ID interval estimation, DD confidence curves, and the limit theory for DD and ID are shown. Furthermore, the effectiveness and discrimination of the proposed algorithms are validated using the Car evaluation, Tic-Tac-Toe endgame, and Fisher's Iris databases.

Extending the Applicability of Pattern and Endgame Databases

For most high-performance two-player game programs, a significant amount of time is devoted to developing the evaluation function. An important issue in this regard is how to take advantage of a large memory. For some two-player games, endgame databases have been an effective way of reducing search effort and introducing accurate values into the search. For some one-player games (single-agent domains or puzzles), pattern databases have been effective at improving the quality of the heuristic values used in a search. This paper introduces new ways to extend the utility of pattern and endgame databases. Through the use of abstraction: (1) single-agent pattern databases can be applied to two- or more-player games; knowledge of the capabilities of one player (being oblivious to the opponent) can be an effective evaluation function for a class of game domains, and (2) endgame database positions can be viewed as an abstraction of more complicated positions; database lookups can be used as evaluation function features. These ideas are illustrated using the games of Chinese Checkers, Chess, and Thief and Police. For each domain, even small databases can be used to produce strong game play. This research has relevance to the recent interest in building general game-playing (GGP) programs. For two- or more-player applications where pattern and/or endgame databases can be built, abstraction can be used to automatically construct an evaluation function.

BEST PLAY IN FANORONA LEADS TO DRAW

Fanorona is the national board game of Madagascar. The game's complexity is approximately the same as that of checkers. In this article, we present a search-based approach for weakly solving this game. It is a well-chosen combination of Proof-Number search and endgame databases. Retrograde analysis is used to generate the endgame databases in which every position with 7 or fewer pieces on the board has been solved. Then, a Proof-Number search variant, PN2, exploits the databases to prove that the game-theoretical value of the initial position is a draw. Future research should develop techniques for strongly solving the game.

CHECKING INDEFINITELY IN CHINESE-CHESS ENDGAMES1

Retrograde analysis has been widely used to solve many problems. In Western chess, it has been successfully applied to construct 6-piece endgame databases (Thompson, 1996). The classical algorithm first determines all terminal positions, e.g., checkm

THE PERFECT 7-PIECE CHECKERS DATABASE1

Many research teams and individuals have computed endgame databases for the game of chess which use the distance-to-mate metric, enabling their software to forecast the number of moves remaining until the game is over. This is not the case for the ga

VERIFICATION OF ENDGAME DATABASES

This article studies the verification of endgame databases using algorithms with limited memory space. To verify a constructed endgame database, the value of each position is checked for consistency against the values of the successor positions. Henc

FAST, MEMORY-EFFICIENT RETROGRADE ALGORITHMS

In this paper, we discuss and compare algorithms used to generate endgame databases, focussing on resource consumption, particularly space requirements. We present a fast, memory-efficient retrograde algorithm that was used to generate Chinese-chess

Solving Chinese chess endgames by database construction

Abstract This paper describes an improved retrograde algorithm used for work on constructing Chinese chess endgame databases. Significant differences from databases for (western) chess are examined, and their influence on database construction noted. Results from Chinese chess yield surprises for human Chinese chess experts, overturning previous human analysis over many years by top players. The aegp-aaee ending is a theoretical win, and not as previously believed, a draw. We also consider the computational resources required to perform these computations.

Exhaustive and Heuristic Retrograde Analysis of the KPPKP Endgame

Although the techniques for constructing chess endgame databases are well understood, it is a permanent challenge to extend the size of the endgames that can be computed with the resources currently available. Ken Thompson’s pioneering attack on 5-pi

Knowledgeable Encoding and Querying of Endgame Databases

Knowledgeable Encoding and Querying of Endgame Databases

Endgame Databases and Efficient Index Schemes for Chess

Endgame databases have become an integral part of modem chess programs during the past few years. Since the early 1990s two different kinds of endgame databases are publicly available, namely Edwards’ so-called tablebases and Thompson’s collection of

Chess Endgames and Neural Networks

The existence of endgame databases challenges us to extract higher-grade information and knowledge from their basic data content. Chess players, for example, would like simple and usable endgame theories if such holy grail exists: endgame experts wou

SOLVING NINE MEN'S MORRIS

The game of Nine Men's Morris is a draw. We obtained this result using a combination of endgame databases (1010 states) and search. Our improved algorithm for computing endgame databases allowed the game to be solved on a personal computer. Other games have been solved using knowledge‐based methods to dramatically prune the search tree. Nine Men's Morris does not seem to profit from such methods, making it the first nontrivial game solved in which almost the entire state space has to be considered.

Bits and pieces: constructing chess endgame databases on parallel and vector architectures

An endgame database for chess encodes optimal lines of play for a specific endgame involving a small number of pieces. The computation of such a database is feasible for as many as six pieces provided the inherent parallelism in the problem is fully exploited. Two computer architectures which can do this are the SIMD CM-2 and the vector multiprocessor YMP. For each machine the computer programs operate on sets of chess positions, each position represented by a single bit. The high-level algorithm is an iterative backtracking procedure. After describing endgame databases and classifying the complexity of endgames we outline the algorithms and give some details of their implementation in Fortran for parallel and vector architectures. For endgames with five or more pieces it is important to use the symmetries of the chess board to reduce storage requirements, and we indicate briefly how this can be done for the vector architecture. Some timing comparisons are presented.