Alpha-beta Search Research Articles

The Analysis of Alpha Beta Pruning and MTD(f) Algorithm to Determine the Best Algorithm to be Implemented at Connect Four Prototype

Connect Four is a two-player game which the players take turns dropping discs into a grid to connect 4 of one’s own discs next to each other vertically, horizontally, or diagonally. At Connect Four, Computer requires artificial intelligence (AI) in order to play properly like human. There are many AI algorithms that can be implemented to Connect Four, but the suitable algorithms are unknown. The suitable algorithm means optimal in choosing move and its execution time is not slow at search depth which is deep enough. In this research, analysis and comparison between standard alpha beta (AB) Pruning and MTD(f) will be carried out at the prototype of Connect Four in terms of optimality (win percentage) and speed (execution time and the number of leaf nodes). Experiments are carried out by running computer versus computer mode with 12 different conditions, i.e. varied search depth (5 through 10) and who moves first. The percentage achieved by MTD(f) based on experiments is win 45,83%, lose 37,5% and draw 16,67%. In the experiments with search depth 8, MTD(f) execution time is 35, 19% faster and evaluate 56,27% fewer leaf nodes than AB Pruning. The results of this research are MTD(f) is as optimal as AB Pruning at Connect Four prototype, but MTD(f) on average is faster and evaluates fewer leaf nodes than AB Pruning. The execution time of MTD(f) is not slow and much faster than AB Pruning at search depth which is deep enough.

The Analysis of Alpha Beta Pruning and MTD(f) Algorithm to Determine the Best Algorithm to be Implemented at Connect Four Prototype

Connect Four is a two-player game which the players take turns dropping discs into a grid to connect 4 of one’s own discs next to each other vertically, horizontally, or diagonally. At Connect Four, Computer requires artificial intelligence (AI) in order to play properly like human. There are many AI algorithms that can be implemented to Connect Four, but the suitable algorithms are unknown. The suitable algorithm means optimal in choosing move and its execution time is not slow at search depth which is deep enough. In this research, analysis and comparison between standard alpha beta (AB) Pruning and MTD(f) will be carried out at the prototype of Connect Four in terms of optimality (win percentage) and speed (execution time and the number of leaf nodes). Experiments are carried out by running computer versus computer mode with 12 different conditions, i.e. varied search depth (5 through 10) and who moves first. The percentage achieved by MTD(f) based on experiments is win 45,83%, lose 37,5% and draw 16,67%. In the experiments with search depth 8, MTD(f) execution time is 35, 19% faster and evaluate 56,27% fewer leaf nodes than AB Pruning. The results of this research are MTD(f) is as optimal as AB Pruning at Connect Four prototype, but MTD(f) on average is faster and evaluates fewer leaf nodes than AB Pruning. The execution time of MTD(f) is not slow and much faster than AB Pruning at search depth which is deep enough.

Implementation of Minimax with Alpha-Beta Pruning as Computer Player in Congklak

Congklak is one of the most popular traditional board games in Indonesia. It is said to train mathematical aspect of the player. Interestingly, nowadays more and more people are becoming less interested. Nowadays, people are paying less attention to traditional things and instead focus more to digital and electronic devices, including Indonesian. Traditional things are beginning to be forgotten from the minds of Indonesian. To re-capture Indonesian’s heart for traditional things, one of the ways is to digitize them. Hopefully by making a video game of Congklak more people, especially Indonesian, become more interested in Congklak as one of Indonesian traditional. Minimax Algorithm with its improvement, Alpha-Beta pruning is an option to make an intelligent computer player for Congklak. By applying Minimax Algorithm to the computer player, it becomes intelligent enough to play Congklak as a good opponent; making the game more fun. Keywords — congklak, minimax algorithm, alpha-beta pruning, artificial intelligence

Conspiracy number search with relative sibling scores

For some two-player games (e.g. Go), no accurate and inexpensive heuristic is known for evaluating leaves of a search tree. For other games (e.g. chess), a heuristic is known (sum of piece values). For other games (e.g. Hex), only a local heuristic — one that compares children reliably, but non-siblings poorly — is known (cell voltage drop in the Shannon/Anshelevich electric circuit model). In this paper we introduce a search algorithm for a two-player perfect information game with a reasonable local heuristic.Sibling Conspiracy Number Search (SCNS) is an anytime best-first version of Conspiracy Number Search based not on evaluation of leaf states of the search tree, but — for each node — on relative evaluation scores of all children of that node. SCNS refines CNS search value intervals, converging to Proof Number Search. SCNS is a good framework for a game player.We tested SCNS in the domain of Hex, with promising results. We implemented an 11-by-11 SCNS Hex bot, DeepHex. We competed DeepHex against current Hex bot champion MoHex, a Monte Carlo Tree Search player, and previous Hex bot champion Wolve, an Alpha–Beta Search player. DeepHex widely outperforms Wolve at all time levels, and narrowly outperforms MoHex once time reaches 4 min/move.We tested the strength of SCNS features: most critical is to initialize leaves via a multi-step process. Also, we show a simple parallel version of SCNS: it scales well for 2 threads but less efficiently for 4 or 8 threads.

A Synthetic Player for Ayὸ Board Game Using Alpha-Beta Search and Learning Vector Quantization

Game playing especially, Ayὸ game has been an important topic of research in artificial intelligence and several machine learning approaches have been used, but the need to optimize computing resources is important to encourage significant interest of users. This study presents a synthetic player (Ayὸ) implemented using Alpha-beta search and Learning Vector Quantization network. The program for the board game was written in Java and MATLAB. Evaluation of the synthetic player was carried out in terms of the win percentage and game length. The synthetic player had a better efficiency compared to the traditional Alpha-beta search algorithm.

IMPLEMENTASI ALGORITMA NEGASCOUT PADA PERMAINAN NINE MEN’S MORRIS

Application development nowadays many are using technology/knowledge of Artificial Intelligence study. This research tries to make an intelligence agent that can play Nine Men’s Morris game using Negascout algorithm. The purpose of this research are to compare the effectiveness of Negascout algorithm in exploring the best move along with time that required, compared with Alpha-Beta Pruning algorithm. The result from this research are Negascout algorithm explores fewer nodes and require less time in the search of the best move compared with Alpha-Beta Pruning algorithm.

MCTS-Minimax Hybrids

Monte Carlo tree search (MCTS) is a sampling-based search algorithm that is state of the art in a variety of games. In many domains, its Monte Carlo rollouts of entire games give it a strategic advantage over traditional depth-limited minimax search with αβ pruning. These rollouts can often detect long-term consequences of moves, freeing the programmer from having to capture these consequences in a heuristic evaluation function. But due to its highly selective tree, MCTS runs a higher risk than full-width minimax search of missing individual moves and falling into traps in tactical situations. This paper proposes MCTS-minimax hybrids that integrate shallow minimax searches into the MCTS framework. Three approaches are outlined, using minimax in the selection/expansion phase, the rollout phase, and the backpropagation phase of MCTS. Without assuming domain knowledge in the form of evaluation functions, these hybrid algorithms are a first step towards combining the strategic strength of MCTS and the tactical strength of minimax. We investigate their effectiveness in the test domains of Connect-4, Breakthrough, Othello, and Catch the Lion, and relate this performance to the tacticality of the domains.

Autonomous Chess Playing Robot

Autonomous Chess Playing Robot is a robot that can challenge a human's chess playing ability in a tangible environment with its varying difficulty level. As any other autonomous robot, ACPR is based upon three pillars, viz. electronics, programming, and mechanical design. The robot consists of a customized chess board below which LDR sensors have been fabricated on a PCB (Printed Circuit Board). Multiplexers also being laid on PCB provide a way for one-way serial communication between sensors and microcontroller. Linear sliders facilitate horizontal and vertical movement of the gripper to move a piece from one position to other. The linear sliders use rack and pinion mechanism while the gripper uses the concept of four bar linkage mechanism. The crucial element of the robot i.e. the intelligence has been incorporated with the help of MinMax algorithm supported by alpha-beta pruning. The microcontroller is connected to a computer terminal by a USB and the chess engine code runs on the computer terminal itself.

Job-Level Alpha-Beta Search

An approach called generic job-level (JL) search was proposed to solve computer game applications by dispatching jobs to remote workers for parallel processing. This paper applies JL search to alpha-beta search, and proposes a JL alpha-beta search (JL-ABS) algorithm based on a best-first search version of MTD(f). The JL-ABS algorithm is demonstrated by using it in an opening book analysis for Chinese chess. The experimental results demonstrated that JL-ABS reached a speed-up of 10.69 when using 16 workers in the JL system.

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.

Ultra-weak solutions and consistency enforcement in minimax weighted constraint satisfaction

The task at hand is that of a soft constraint problem with adversarial conditions. By amalgamating the weighted and quantified constraint satisfaction frameworks, a Minimax Weighted Constraint Satisfaction Problem (formerly Quantified Weighted Constraint Satisfaction Problem) consists of a set of finite domain variables, a set of soft constraints, and a min or max quantifier associated with each of these variables. We formally define the framework, suggest three solution concepts, and propose a complete solver based on alpha-beta pruning techniques. We discuss in depth our novel definitions and implementations of node, arc and full directional arc consistency notions to help reduce search space on top of the basic tree search with alpha-beta pruning for solving ultra-weak solutions. In particular, these consistencies approximate the lower and upper bounds of the cost of a problem by exploiting the semantics of the quantifiers and reusing techniques from both Weighted and Quantified Constraint Satisfaction Problems. Lower bound computation employs standard estimation of costs in the sub-problems used in alpha-beta search. In estimating upper bounds, we propose two approaches based on the Duality Principle: duality of quantifiers and duality of constraints. The first duality amounts to changing quantifiers from min to max, while the second duality re-uses the lower bound approximation functions on dual constraints to generate upper bounds. Experiments on three benchmarks comparing basic alpha-beta pruning and the six consistencies from the two dualities are performed to confirm the feasibility and efficiency of our proposal.

Epaminondas: Exploring Combat Tactics

Epaminondas is a two-person, zero-sum strategy game that combines long-term strategic play with highly tactical move sequences. The game has two unique features that make it stand out from other games. The first feature is the creation of phalanxes, which are groups of pieces that can move as a whole unit. As the number of pieces in a phalanx increases, the mobility and capturing power of the phalanx also increases. The second feature differs from many other strategy games: when a player makes a crossing, a winning move in the game, the second player has an opportunity to respond. This paper presents strategies and heuristics used in a Min-Max Alpha-Beta agent that plays at a novice level. Furthermore, it defines the state-space and game-tree complexities for Epaminondas. Finally, a new version of MCTS is implemented that uses the Alpha-Beta heuristic function during node selection to guide MCTS to more promising areas of the search tree. Additionally, in an effort to overcome the MCTS tactical weakness, the MCTS player implements the Alpha-Beta search once the game reaches 15 turns. Results show that the added heuristic value and the switch to Alpha-Beta for endgame play, positively impact the performance of MCTS, surpassing novice Alpha-Beta win ratios at certain time intervals.

Large-Scale Optimization for Evaluation Functions with Minimax Search

This paper presents a new method, Minimax Tree Optimization (MMTO), to learn a heuristic evaluation function of a practical alpha-beta search program. The evaluation function may be a linear or non-linear combination of weighted features, and the weights are the parameters to be optimized. To control the search results so that the move decisions agree with the game records of human experts, a well-modeled objective function to be minimized is designed. Moreover, a numerical iterative method is used to nd local minima of the objective function, and more than forty million parameters are adjusted by using a small number of hyper parameters. This method was applied to shogi, a major variant of chess in which the evaluation function must handle a larger state space than in chess. Experimental results show that the large-scale optimization of the evaluation function improves the playing strength of shogi programs, and the new method performs signicantly better than other methods. Implementation of the new method in our shogi program Bonanza made substantial contributions to the program's rst-place nish in the 2013 World Computer Shogi Championship. Additionally, we present preliminary evidence of broader applicability of our method to other two-player games such as chess.

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.

Trap<sup>A</sup><sub>B</sub>: A New Position Search Algorithm in Chinese Chess Computer Game

In this paper, we propose an improved alpha-beta search algorithm, named trappy alpha-beta (simply), for game-tree in order to identify and set the potential traps in the game playing. can be regarded as an extension of the traditional alpha-beta search algorithm which ties to predict when the opponent might make a mistake and select such moves that can most likely lead the an opponent into the trap by comparing the various scores returned through iterative deepening technology. In our experiment, we test the performance of in comparison with three game-tree search algorithms, i.e., min-max, trappy minimax, and alpha-beta, by playing with four testing opponents, which are obtained form a typical Chinese chess computer game program-Xqwizard (http://www.xqbase.com). The comparative results show that our designedcan effectively find and set the traps in the playing with opponents.

Temporal-Difference Search in Computer Go

Temporal-difference (TD) learning is one of the most successful and broadly applied solutions to the reinforcement learning problem; it has been used to achieve master-level play in chess, checkers and backgammon. Monte-Carlo tree search is a recent algorithm for simulation-based search, which has been used to achieve master-level play in Go. We have introduced a new approach to high-performance planning. Our method, TD search, combines TD learning with simulation-based search. Like Monte-Carlo tree search, value estimates are updated by learning online from simulated experience. Like TD learning, it uses value function approximation and bootstrapping to efficiently generalise between related states. We applied TD search to the game of 9x9 Go, using a million binary features matching simple patterns of stones. Without any explicit search tree, our approach outperformed a vanilla Monte-Carlo tree search with the same number of simulations. When combined with a simple alpha-beta search, our program also outperformed all traditional (pre-Monte-Carlo) search and machine learning programs on the 9x9 Computer Go Server.

Two-level Task Scheduling for Parallel Game Tree Search Based on Necessity

It is difficult to fully utilize the parallelism of large-scale computing environments in alpha-beta search. The naive parallel execution of subtrees would result in much less task pruning than may have been possible in sequential execution. This may even degrade total performance. To overcome this difficulty, we propose a two-level task scheduling policy in which all tasks are classified into two priority levels based on the necessity for their results. Low priority level tasks are only executed after all high priority level tasks currently executable have started. When new high priority level tasks are generated, the execution of low priority level tasks is suspended so that high level tasks can be executed. We suggest tasks be classified into the two levels based on the Young Brothers Wait Concept, which is widely used in parallel alpha-beta search. The experimental results revealed that the scheduling policy suppresses the degradation in performance caused by executing tasks whose results are eventually found to be unnecessary. We found the new policy improved performance when task granularity was sufficiently large.

Alpha-Beta Pruning and Althöfer’s Pathology-Free Negamax Algorithm

The minimax algorithm, also called the negamax algorithm, remains today the most widely used search technique for two-player perfect-information games. However, minimaxing has been shown to be susceptible to game tree pathology, a paradoxical situation in which the accuracy of the search can decrease as the height of the tree increases. Althöfer’s alternative minimax algorithm has been proven to be invulnerable to pathology. However, it has not been clear whether alpha-beta pruning, a crucial component of practical game programs, could be applied in the context of Alhöfer’s algorithm. In this brief paper, we show how alpha-beta pruning can be adapted to Althöfer’s algorithm.

Temporal-difference search in computer Go

Temporal-difference learning is one of the most successful and broadly applied solutions to the reinforcement learning problem; it has been used to achieve master-level play in chess, checkers and backgammon. The key idea is to update a value function from episodes of real experience, by bootstrapping from future value estimates, and using value function approximation to generalise between related states. Monte-Carlo tree search is a recent algorithm for high-performance search, which has been used to achieve master-level play in Go. The key idea is to use the mean outcome of simulated episodes of experience to evaluate each state in a search tree. We introduce a new approach to high-performance search in Markov decision processes and two-player games. Our method, temporal-difference search, combines temporal-difference learning with simulation-based search. Like Monte-Carlo tree search, the value function is updated from simulated experience; but like temporal-difference learning, it uses value function approximation and bootstrapping to efficiently generalise between related states. We apply temporal-difference search to the game of 9?9 Go, using a million binary features matching simple patterns of stones. Without any explicit search tree, our approach outperformed an unenhanced Monte-Carlo tree search with the same number of simulations. When combined with a simple alpha-beta search, our program also outperformed all traditional (pre-Monte-Carlo) search and machine learning programs on the 9?9 Computer Go Server.

Solving 4x5 Dots-And-Boxes

Dots-And-Boxes is a well-known and widely-played combinatorial game. While the rules of play are very simple, the state space for even small games is extremely large, and finding the outcome under optimal play is correspondingly hard. In this paper we introduce a Dots-And-Boxes solver which is significantly faster than the current state-of-the-art: over an order-of-magnitude faster on several large problems. We describe our approach, which uses Alpha-Beta search and applies a number of techniques—both problem-specific and general—to reduce the number of duplicate states explored and reduce the search space to a manageable size. Using these techniques, we have determined for the first time that Dots- And-Boxes on a board of 4x5 boxes is a tie given optimal play. This is the largest game solved to date.