Kuo-Yuan Kao

Job-Level Proof Number Search

This paper introduces an approach, called generic job-level search, to leverage the game-playing programs which are already written and encapsulated as jobs. Such an approach is well suited to a distributed computing environment, since these jobs are allowed to be run by remote processors independently. In this paper, we present and focus on a job-level proof number search (JL-PNS), a kind of generic job-level search for solving computer game search problems, and apply JL-PNS to solving automatically several Connect6 positions, including some difficult openings. This paper also proposes a method of postponed sibling generation to generate nodes smoothly, and some policies, such as virtual win, virtual loss, virtual equivalence, flagging, or hybrids of the above, to expand the nodes. Our experiment compared these policies, and the results showed that the virtual-equivalence policy, together with flagging, performed the best against other policies. In addition, the results also showed that the speedups for solving these positions are 8.58 on average on 16 cores.

Design and Implementation of Chinese Dark Chess Programs

Chinese Dark Chess is an old and very popular game in the Chinese culture sphere. This game is a stochastic game with symmetric hidden information. This paper reviews alpha-beta search with chance nodes and proposes heuristics on Chinese Dark Chess programs. We propose an application of nondeterministic Monte Carlo Tree Search with random nodes for tackling partial observation. The proposed methods were implemented in the program Diablo, which won four Chinese Dark Chess tournaments in TAAI 2011/2012, TCGA 2011/2012 computer game tournaments. Diablo also played hundreds of games with different human players and programs based on alpha-beta search. These results show that the nondeterministic MCTS equipped with our heuristics is promising for Chinese Dark Chess.

The Art of the Chinese Dark Chess Program DIABLE

Diable is a famous Chinese dark chess program, which won the Chinese dark chess tournaments in TAAI 2011, TCGA 2011, and TCGA2012 computer game tournaments. Chinese dark chess is an old and very popular game in Chinese culture sphere. This game is played with imperfect information. Most computer Chinese dark chess programs used alpha-beta search with chance nodes to deal with the imperfect information. Diable used a new nondeterministic Monte Carlo tree search model for Chinese dark chess. These tournament results show that the nondeterministic Monte Carlo tree search is promising for Chinese dark chess.

Bitboard knowledge base system and elegant search architectures for Connect6

Efficiency is critical for game programs. This paper improves the search efficiency of Connect6 program by encoding connection patterns and computing the inherent information in advance. Such information is saved in a bitboard knowledge base system, where special bitwise operations are designed. This paper also proposes efficient methods of generating threat moves and the Multistage Proof Number Search. The methods reduce the time complexity of generating threat moves. The search improves the search performance by developing candidate moves in stages according to their importance. In brief, this paper proposes an efficient knowledge base system and elegant search architectures for Connect6. It is expected that the proposed methods can be applied to all kinds of Connect-k games.

XT Domineering: A new combinatorial game

This paper introduces a new combinatorial game, named XT Domineering, together with its mathematical analysis. XT Domineering is modified from the Domineering game in which 1x2 or 2x1 dominos are allowed to be placed on empty squares in an mxn board. This new game allows a player to place a 1x1 domino on an empty square s while unable to place a 1x2 or 2x1 domino in the connected group of empty squares that includes s. After modifying the rule, each position in the game becomes an infinitesimal. This paper calculates the game values of all sub-graphs of 3x3 squares and shows that each sub-graph of 3x3 squares is a linear combination of 8 elementary infinitesimals. These pre-stored game values can be viewed as a knowledge base for playing XT Domineering. Instead of searching the whole game trees, a simple rule for determining the optimal outcome of any sum of these positions is presented.

Incentive Learning in Monte Carlo Tree Search

Monte Carlo tree search (MCTS) is a search paradigm that has been remarkably successful in computer games like Go. It uses Monte Carlo simulation to evaluate the values of nodes in a search tree. The node values are then used to select the actions during subsequent simulations. The performance of MCTS heavily depends on the quality of its default policy, which guides the simulations beyond the search tree. In this paper, we propose an MCTS improvement, called incentive learning, which learns the default policy online. This new default policy learning scheme is based on ideas from combinatorial game theory, and hence is particularly useful when the underlying game is a sum of games. To illustrate the efficiency of incentive learning, we describe a game named Heap-Go and present experimental results on the game.

Solving Nine Layer Triangular Nim

Triangular Nim, one variant of the game Nim, is a common two-player game in Taiwan and China. In the past, Hsu [9] strongly solved seven layer Triangular Nim while some of the authors recently strongly solved eight layer Triangular Nim. The latter required 8 gigabytes in memory and 8,472 seconds. Using a retrograde method, this paper strongly solves nine layer Triangular Nim. In our first version, the program requires four terabytes in memory and takes about 129.21 days aggregately. In our second version, improved by removing some rotated and mirrored positions, the program reduces the memory by a factor of 5.72 and the computation time by a factor of 4.62. Our experiment result also shows that the loss rate is only 5.0%. This is also used to help improve the performance.

Selection Search for Mean and Temperature of Multi-Branch Combinatorial Games

This paper shows a new algorithm to calculate the mean and temperature of multi-branch combinatorial games. The algorithm expands gradually, one node at a time, the offspring of a game. After each step of expansion, the lower and upper bounds of the mean and temperature of the game are re-calculated. As the expanding process continues, the range between the lower and upper bounds is little by little narrowed. The key feature of the algorithm is its ability to generate a path of which the outcome is most likely to reduce the distance between the lower and upper bounds. 1. COMBINATORIAL GAMES Combinatorial game theory studies two-player games with perfect information. The two players are assumed to take turns alternatively, and a game is considered as a sum of local positions, where each player can choose one local position to move at each turn. This section introduces a heap game, named heap-go, for illustrating some key ideas of combinatorial game theory. Heap-go is played on a number of heaps of counters. Each counter has a weight and is coloured either blue or red. Figure 1 shows an example of heap-go setup. (Heap A and B are considered to be blue; heap E is red. Heap C and D are mixed; the top counter of C is red, the two other counters are blue.) Figure 1: An example of heap-go setup. Two players,  and , move alternatively and their legal moves are different.  When it is s turn to move, he can choose any one of the heaps and repeatedly removes the top counter until either he removes a red counter or the heap has become empty.  When it is s turn to move, he can choose any one of the heaps and repeatedly removes the top counter until either he removes a blue counter or the heap has become empty. 1 Department of Information Management National Penghu University, Penghu, Taiwan. Email: stone@npu.edu.tw 2 Department of Computer Science and Information Engineering, National Chiao Tung University, Hsinchu, Taiwan. Email: icwu@csie.nctu.edu.tw, ycshan@java.csie.nctu.edu.tw. 3 Department of Computer Science & Information Engineering, National Dong Hwa University, Hualien, Taiwan. Email: sjyen@mail.ndhu.edu.tw 4 For brevity, we use ‘he’ and ‘his’ whenever ‘he or she’ and ‘his or her’ are meant. 6 C B A D E 2 5 5 5 8 3 2 1 7 8 ICGA Journal September 2012 158 The game is finished if all the counters in all the heaps are removed. The player who removed more total weights is the winner. Heap-go is a two-player game with perfect information. The heaps are the local positions, where each player can choose one local position to move at each turn. The more heaps in a setup, the more options each player has. Although the complexity of the complete minimax game tree of a heap-go game may grow up exponentially with the increase of the total number of heaps, each local position can be represented as a simpler combinatorial game tree where each node represents the state of the local position, each left branch represents a ’s move and each right branch a ’s move at the local position. Figure 2 shows the combinatorial game tree of heap  in Figure 1. The numbers at the terminal nodes are the net scores of the paths from the root to these nodes. ’s scores are counted positive and ’s negative. For example, consider the path .  gets 8 points for the first move (removed 2 counters);  gets 5 points for the second move (removed 1 counter); the net score is 3. Figure 2: The game tree of heap . To follow the terminology of combinatorial game theory, each local position is called a game. If  is a game, then  () represents the set of ’s (’s) options at the game, where each option in  () is a game after ’s (’s) move at . A game  is defined as an ordered pair of sets of games, and expressed as    (1) When  is a terminal node in a game tree, it is represented by a numerical outcome value. For example, heaps ,  and  can be expressed as                   Note that   ,   ,   ,   ,   , and       . In combinatorial game theory, the sum of two games is a game. Let  and  be two games, the sum    is defined as               . (2) When  is a game and  is a number, the sum can be simplified as          (3) -13 -9 2 5 5

Solving 9 Layer Triangular Nim

Triangular Nim, one variant of the game Nim, is a common two-player game in Taiwan and China. In the past, Hsu strongly solved 7 layer Triangular Nim while some of the authors recently strongly solved 8 layer Triangular Nim. The latter required 8 gigabytes in memory and 8878 seconds. Using a retrograde method, this paper strongly solves 9 layer Triangular Nim. In our first version, the program requires four terabytes in memory and takes about 129.21 days aggregately. In our second version, improved by removing some rotated and mirrored positions, the program reduces the memory by a factor of 5.86 and the computation time by a factor of 4.38. Our experiment result also shows that the loss rate is only 5.0%. This is also used to help improve the performance.

Chilled Domineering

This paper introduces a new game named Chilled Domineering, together with its mathematical analysis. Chilled Domineering is modified from the Domineering game by allowing a player placing a 1x1 domino while unable to place a 2x1 domino at a position. After modifying the rule, each position in the game becomes an infinitesimal. Many interesting infinitesimals are found in this new game. This paper calculates the game values of all sub-graphs of 3x3 squares and presents a rule to determine the outcome of any sum of these positions.