Developing Gomoku chess based on Minimax algorithm 1st Pham Dang Khoa Department of Computer Science Vietnam National University – Ho Chi Minh City University of information Technology 18520930@gm.uit.edu.vn 2nd Nguyen Hoang Quan Department of Computer Science Vietnam National University – Ho Chi Minh City University of information Technology 17520936@gm.uit.edu.vn Abstract— We proposed to develop a new version of Gomoku chess game based on the idea of Minimax algorithm as well as Alpha-Beta Pruning Minimax is one of backtracking algorithms that is widely applied in making decision to search the optimal move for player, in case that its enemy makes the optimal step too Alpha-beta pruning is the optimization algorithm reduces computation time by seeking to decrease the number of nodes by cutting off branches which are not necessary to search because there has already been a better move in next steps In this project, we will implement it by C# with Visual Studio tool, NET framework, Windows Forms library In this version, we are planning to build a truly-sized game board and integrate some functions depending on player's demand: give the right to choose difficulty, undo your moves, etc Keywords—Minimax, Alpha-Beta Pruning, optimization, backtracking algorithm (key words) I INTRODUCTION Long time ago, searching was always a fundamental manipulation for several tasks The main purpose of searching is to find to optimal way to collect the necessary information for an important decision Generally, searching can be implied that it is the method to find one or more objects which are satisfactory at any requirements in a large set of objects We can see many problems of searching in real life For example, some games such as Gomoku chess – also called Caro, among many probable moves, we must find the best one that can lead to winning position Minimax is the specialized search algorithm to return a series of optimal moves for a player in a zero-sum game [1] A zero-sum game is a game where the total value of the outcome of winner is fixed Whichever side wins (+1) finally makes the other components lose (-1), which corresponds to a pure competitive situation, eventually leading to a total (+ 1- 1) = Our game, Gomoku chess, is also a zero-sum game because there is never a situation that both of two players have a draw result Consequently, Minimax is a technique whose purpose is to minimize the possible loss in a worst case, and to maximize the minimum gain when dealing with gains The Minimax algorithm helps find the best move, by going backwards from the end of the game At each step, it will estimate that Player A is XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE 3rd Le Viet Trung Department of Computer Science Vietnam National University – Ho Chi Minh City University of information Technology 17521173@gm.uit.edu.vn trying to maximize A's winning chances when it is his turn, and in the next move, Player B tries to minimize the chances of Player A winning (meaning maximizing B's chances of winning) II MINIMAX ALGORITHM A Ideas Two players in that game are MIN and MAX MAX is supposed to be a player who tries to obtain the winning chances In contrary, MIN is the opponent who tries to minimize MAX’s score at each move We suppose that MAX also uses the same information as MIN First, we represent it by a game tree Each node in the tree represents a specific status The root node denotes the beginning state of the game While the leaf nodes represent an end state (win or lose) of the game in each case If the state x is represented by the node n then the children of n represent all the resulting states of the possible moves from state x Because the two players alternate their moves with each other, the levels (classes) on the game tree also alternate as MAX and MIN Therefore, this game tree is also called the MINMAX tree In the game tree, the nodes which are respective to the state where the MAX player chooses the move will belong to the MAX class, the nodes corresponding to the state from which the MIN player selects the move will belong to the MIN class The minimax strategy is represented by the rule as follows: If this node is leaf node, then we assign it a value to denote the winning or losing state of the players After that, we take their values to determine nodes’ values in the upper levels in the game tree followed by the rule:  The nodes belonging to MAX class will be assigned the max value of their children  The nodes belonging to MIN class will be assigned the value of their children These values assigned to each state based on the above rule will indicate the value of the best state which each player expects to achieve And players will use them to choose their own proper moves When MAX player is in turn, it will choose the move corresponding to the state with the highest value in the sub-states, and in contrast, when MIN player is in turn, it will choose the move corresponding to the state of the price smallest value in the sub-states Figure 1: Illustration of Minimax applied in Tic Tac Toe game – a short version of Gomoku chess B Minimax Illustration From the ideas mentioned above, we can implement the Minimax algorithm as follows: First of all, the Minimax function takes a position named pos and returns the value of that position If the pos position corresponds to the leaf node in the game tree, the value which was assigned to the leaf node is returned In contrast, we will give pos a temporary value of -∞ or ∞ depending on the situation that whether the pos is the MAX or MIN node and consider the sub-positions of pos After a child of pos has value V, we reset value = max (value, V) if n is node MAX and value = (value, V) if n is node MIN When all the children of n are considered, the temporary value of pos becomes its value We introduce you a pseudo-code for this algorithm: begin {Initialize temporary value for best variable} if pos is MAX node then best:= -INFINITY else best := INFINITY; {genPos function generates all possible moves from the current position named pos} genPos(pos); {Check all children of pos, each time that we determine one child node’s value, we must reassign the temporary value for it.} while (we can still take a move) begin pos := (The new value computed by the previous move); value = Minimax(pos); if pos is MAX node then if (value > best) then best := value; if pos is MIN node then if (value < best) then best := value; end; Minimax := best; end; end; In the program, the chess board was represented by global variables Therefore, instead of passing parameter which is a new chess called pos into Minimax parameter, we transform the global variable by making a “trial” move After computing based on the chess board saved by global variables, the algorithm will use some procedures to remove this move C Minimax evaluation Minimax algorithm traverses the entire game tree by using Depth-First Search Therefore, this algorithmic complexity algorithm is directly proportional to the size of the search space bd, where b is the branching factor of the tree or is the legal move at each point, d is the maximum depth of the tree The number of traversed nodes is b(bd-1)/(b-1) However, the evaluation function is the most important method and only works with leaf nodes, so it is unnecessary to traverse to nodes which are not leaf nodes [8] Hence the time complexity is O (bd) The essence of this algorithm seems to be Depth-first Search, so its memory space requirements are linear only with d and b So the spatial complexity is O (bd) If the branching factor of tree is b = 40, and the maximum depth is d = 4, then the number of nodes need evaluating is 40 = 2560000, it is an enormous figure To solve this problem, we consider using another more effective algorithm such as AlphaBeta Pruning III ALPHA-BETA PRUNING function Minimax(pos): integer; value, best : integer; begin if pos is leaf node then return eval(pos) else In this part, we will present an algorithm decreasing the number of nodes that are evaluated It stops evaluating one move when that move is proven to be worse than a previously examined move Therefore, it is unnecessary to evaluate those moves anymore The aim of this algorithm is to save time execution but does not influence much on the search results, especially final decision A Idea The idea of an Alpha-beta search is simple: Instead of searching the entire space until the specific depth, Alpha-beta search proceeds in a depth-first fashion There are two values, called alpha and beta, that are generated during a search: The alpha value is an initial or temporary value associated with MAX nodes and never decreases, it can only go up In contrast, beta value is associated with MIN nodes and never increases, it can only go down Initially, both players start the game with their own worst case The searching process win end when beta value of MIN player becomes less than or equal to alpha value of its MAX parent nodes, the MIN will not need to consider descendants of this node since it is not important The same rule is also applied for MAX player B Alpha-Beta Illustration Here are some core steps that demonstrate how the algorithm works:  If the current level is root, it sets the values of    alpha and beta to - and +, respectively If it has reached search limit (leaf nodes), then computes the static value of current state corresponding current player and record result For MIN player: it applies Alpha-Beta procedure with current alpha and beta values for one child node, then store its result Next, it compares this stored value to beta, if that result is smaller then it sets beta to this new value These works are repeated until all descendants have been considered or alpha is greater or equal to beta For MAX player: it applies Alpha-Beta procedure with current alpha and beta values for one child node, then store its result Next, it compares this stored value to alpha, if that result is larger then it sets alpha to this new value These works are also repeated until all descendants have been considered or alpha is greater or equal to beta Figure 2: Alpha-Beta Algorithm illustration We also introduce pseudo-code for this algorithm as it is convenient to compare with Minimax function AlphaBeta(alpha, beta, depth): integer; begin if (depth = 0) or (pos is leaf node) then AlphaBeta := Eval { Calculate position’s value } else begin best := -INFINITY; Gen; { Generate all possible moves from the position called pos } while (Still can get a move) and (best < beta) begin if best > Alpha then Alpha := best; Make the move; value := -AlphaBeta(-beta, -Alpha, depth-1); Cancel making the move; if value > best then best := value; end; AlphaBeta := best; end; end; C Alpha-Beta evaluation Generally, alpha-beta algorithm helps us save a lot of time compared to Minimax while making sure that search results have high accuracy However, this amount of savings is not stable since it depends on the number of nodes cut In the worst case the algorithm could not cut a branch and must consider the exact number of nodes by the Minimax algorithm We need to accelerate the removal by accelerating the shrinking of the Alpha-beta search interval This interval is gradually narrowed when there is a new value that is better than the old value When facing the best fit value, this interval is narrowed the most Therefore, we need to make the leaf nodes arranged in order from high to low The better this order is, the more rapid the algorithm executes D Comparsion between Minimax and Alpha-Beta As you can see, the increase in the number of nodes when increasing the depth of Minimax is always the branching factor b, in this case is 40 In contrary, The number of Alpha-beta increases is much less: only 1.7 times when increasing from d odd to even 23.2 times when d is from even to odd, the average increase is only about times when d increases The formula for calculating the number of nodes displays that the number of nodes to consider when using Alpha-beta is much less than that of Minimax but it is still an exponential function and still leads to combinatorial explosion Alpha-beta algorithm does not completely avoid combinatorial explosion, only lowers the explosion rate Due to this problem, both players cannot search all possibilities Therefore, the only method is to search only to a limited depth and choose the move which leads to the best state for us In the light of the enemy's ability to fight back, we cannot use normal search algorithms A separate search algorithm for the game tree must be used They are the Minimax algorithm and its improvement called Alpha-beta algorithm While both algorithms cannot avoid combinatorial explosion, the Alpha-beta algorithm just slows down it and; therefore; it is used more often in chess games IV DEMO APPLICATION After presenting and concluding all the theories researched, we will introduce some works that we have made in our project In our User Interfaces, there are game board with the size of 20x20, UIT logo and three main function buttons: New Game, Undo and Exit When you press the button “New Game”, it will ask whether you or Computer play first and the difficulty you want to choose The “Undo” button helps you reverse your previous move, even you can reverse more than one moves Finally, the “Exit” button will quit the game if you don’t want to play anymore Here are some pictures of (UI) in our application: V CONCLUSION AND FUTURE WORK In summary, we have created a Caro game for one player competing with the AI The algorithms that our group implements are Minmax and AlphaBeta Pruning, we have evaluated the efficiency of each of them as well as given a general comparison between both of them In spite of our efforts, it is certain that our program does not avoid some limitations, hopefully some potential problem will be further researched and developed with improved algorithms Our group considers installing Expectimax algorithm in the future and compare different evaluation functions to assess the states REFERENCES [1] Jessica Billings (2008), The Minimax Algorithm, CS 330 [2] Alpha-Beta pruning - Wikipedia [3] CPSC 352 Artificial Intelligence Notes: Minimax and Alpha Beta Pruning [4] Đỗ Trung Tuấn (1997), Trí tuệ nhân tạo, NXB Giáo dục [5] Heylighen (1993), Zero sum games – Principia Cybernetica Web ... Notes: Minimax and Alpha Beta Pruning [4] Đỗ Trung Tuấn (1997), Trí tuệ nhân tạo, NXB Giáo dục [5] Heylighen (1993), Zero sum games – Principia Cybernetica Web  ... button will quit the game if you don’t want to play anymore Here are some pictures of (UI) in our application: V CONCLUSION AND FUTURE WORK In summary, we have created a Caro game for one player... chess games IV DEMO APPLICATION After presenting and concluding all the theories researched, we will introduce some works that we have made in our project In our User Interfaces, there are game