FUNDAMENTALS OF ARTIFICIAL INTELLIGENCE USING HEURISTICS IN GAMES

Transcription

1 Riga Technical University Faculty of Computer Science and Information Technology Department of Systems Theory and Design FUNDAMENTALS OF ARTIFICIAL INTELLIGENCE Lecture 6 USING HEURISTICS IN GAMES Dr.habil.sc.ing., professor Janis Grundspenkis, Dr.sc.ing., lecturer Alla Anohina-Naumeca Department of Systems Theory and Design Faculty of Computer Science and Information Technology Riga Technical University { janis.grundspenkis, alla.anohina-naumeca}@rtu.lv Address: Meza street 1/ 4- {55, 545}, Riga, Latvia, LV-148 Phone: (+371) 6789{581, 595}

2 Heuristics and games Games are one of the oldest field of the application of Artificial Intelligence, because any game demands intelligence. Implementing games two methods of artificial intelligence are used: search and heuristics. In this course the simplest kind of games, so called two-person games with perfect information, will be considered. 2/25

3 Two-person games Two-person games are characterized by the following features: Two players participate in a game, for example, one is a person, other is a computer Players make moves one by one up to the end of the game There is a conflict because each player wishes to win the game, so stupid moves are avoided Players use heuristics to guide the game along a path to a winning state 3/25

4 Two-person games with perfect information (1) In two-person games with perfect information: Both players have access to complete information about the state of the game. No information is hidden from either player No chance (e.g., using dice) involved 4/25

5 Two-person games with perfect information (2) Examples of two-person games with perfect information are: Checkers Tic-tac-toe Chess Non-examples: Bridge Solitaire Backgammon 5/25

6 Game tree Two-person games with perfect information are represented using a game tree (a special kind of a state space). A game tree consists of nodes and arcs connecting nodes. This is a visual representation of a game tree: 6/25

7 Nodes of a game tree Nodes of a game tree display possible discrete states of the game. For example, in Tic-tac-toe a state is a number of X and O. A game tree includes the following nodes: A root Dead-ends Internal nodes 7/25

8 Levels of a game tree Typically a game tree consists of several levels. One or several nodes are located at each level and they display states of the game from which only one of two players makes moves. Player 1 Player 2 Player 1 Player 2 8/25

9 A root A root is located at the highest level of a game tree and displays an initial state of the game. For example, this is an initial state of Tic-tac-toe: 9/25

10 Dead-ends Dead-ends of a game tree do not have outgoing arcs. They correspond to final states of the game. Some final states of Tic-tac-toe are the following: X X X X X X X X X X 1/25

11 Internal nodes Internal nodes represent intermediate states of the game. Some intermediate states of Tic-tac-toe are the following: X X X X X 11/25

12 Arcs of a game tree Arcs of a game tree display moves allowed by the rules of the game. A move is s transition from one state to another: X A move X X A state before a move A state after the move Two restrictions are imposed on arcs displaying two-person game with perfect information with a game tree. 12/25

13 First restriction of arcs Arcs between nodes on the same level are not allowed Forbidden 13/25

14 Second restriction of arcs Arcs over some levels are not allowed. Forbidden 14/25

15 Outgoing arcs It is possible to perform several moves from the same state. This situation is displayed by several outgoing arcs. Example: X X X X 15/25

16 It is possible to reach a state form several different states. This situation is displayed by several ingoing arcs. Ingoing arcs Example: X X X X X X X X 16/25

17 Minimax algorithm Minimax is one of the algorithms for the implementation of two-person games. In the mentioned algorithm players are referred as a maximizer and a minimizer. Each of players has a particular strategy in the game: The maximizer tries to win or to maximize his/her result The minimizer attempts to minimize maximizer s score 17/25

18 Maximizer and minimizer Levels of a game tree are labeled as or alternately. It is not important which level is the first: 18/25

19 Steps of the Minimax algorithm (1) 1. Create a game tree 2. Label levels of the game tree as and 3. Acquire heuristic values of dead-ends. Heuristic values are assigned in the following way: a state acquires 1 if it is a win for the maximizer, if it is a win for the minimizer, or +1 if it is a win for the maximizer, it is a drawn game, -1 if it is a win for the minimizer 19/25

20 Steps of the Minimax algorithm (2) 4. Propagate heuristic values of dead-ends up the tree according to the following rules: If a state is a node on level it acquires the maximum value among its direct descendants If a state is a node on level it acquires the minimal value among its direct descendants 5. Determine a result of the game and winning paths 2/25

21 Result of a game and winning paths The result of the game is a value of the root node. It points out who wins the game. Winning paths are determined on the basis of the result of a game: A winning path starts from a root node and reaches a dead-end: 1. All nodes in a winning path are closely related 2. All nodes in a winning path have the same heuristic value equal to the result of the game: If the result of the game is 1, then the maximizer wins the game and all nodes in winning paths have the heuristic value 1 If the result of the game is (or -1 if we use three heuristic values: +1,, -1), then the minimizer wins the game and all nodes in winning paths have the heuristic value (or -1) 21/25

22 Example (1) As an example a game called Nim will be considered. There are 7 safety-matches. In each move they must be divided in two piles with different number of safety-matches. A player who cannot perform a move loses a game. Step 1 and 2. Creation of a game tree and labeling of its levels by and /25

23 Example (2) Step 3. Acquiring of heuristic values of dead-ends , because from this state the minimizer must perform its move but it is impossible, so the minimizer loses the game but the maximizer wins, because from this state the maximizer must perform its move but it is impossible, so the maximizer loses the game, because from this state the maximizer must perform its move but it is impossible, so the maximizer loses the game 23/25

24 Example (3) Step 4. Propagation of heuristic values from the lowest level up to the root of the game tree /25

25 Example (4) Step 5. Determination of winning paths 7 1, this is the result of the game. So the maximizer wins. Winning paths are in red /25