How i taught a computer to play chess
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My attempt to build a chess engine. This is exploratory project I continue to develop in my free time. I have collected information from various sources books, blogs and YouTube courses which are cross referenced in the presentation.
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How i taught a computer to play chess
- 1. HOW I TAUGHT A COMPUTER TO PLAY CHESS The Algorithms and Techniques• March 18, 2017
- 2. Claude Shannon predicted the number of chess games Number of chess games are 10120 for 40 pair moves Number of atoms in Universe 1078 𝑡𝑜 1082 Number of seconds in a year 𝜋 𝑥 107 Number of nanoseconds in year 109 Number of years in the universe 1010 Shannon's Number Complexity of the Game
- 3. The Objective is to check mate the opponent King One may also want to draw a game after some evaluation Seek draw by repeated moves if the game is not making progress Objective of game of Chess Objective
- 4. Is it a solved problem? Is it a computationally solvable problem? Can we make a computer play a chess game? It is a game with perfect information? What do we know about Chess NO Answers YES Answers
- 6. Develop Knights before Bishops Kc3 Kf3 Kc6 Kf6 Develop minor pieces first Control the center D4 D5 E4 E5 Castle and Protect the King 0-0-0 or 0-0 No double pawns Knights at the rim is dim. A and H file Use standard openings – Ruy Lopez, Kings Pawn Opening, Queens Gamit, Burlin etc. Centralize your King during the end game General rule to improve your chances of winning General Chess Theory
- 7. Position evaluation Searching for the next move and choosing the next best move Building a Chess Engine Two components
- 8. • 𝐸𝑣𝑎𝑙 𝑠 = 𝑤𝑖 𝑓1 𝑠 + 𝑤2 𝑓2 𝑠 + ⋯ + 𝑤 𝑛 𝑓𝑛 𝑠 = 𝑖=1 𝑛 𝑤𝑖 𝑓𝑖 𝑆 • 𝑓𝑖 : Number of each kind of piece • 𝑤𝑖 : Is the weight of piece Position Evaluation in Game of Chess Linear Function (1*9)+(2*3.25)*+(2*3)+(1*6)+(2*5)+(1*20) = 57.5 (1*9)+(2*3.25)+(1*3)+(1*8)+(2*5)+(1*20) = 56.5
- 9. By searching. But its exhaustive By using a heuristics. Decision Tree Min Max Algorithm Pruning based approach - Heuristics Alpha beta Pruning Solution Approaches : Searching for Next Move Algorithm Advanced Techniques – Adversarial Search Algorithms
- 10. Mini - Max algorithms Adversarial Search
- 11. Min- Max Algorithm on Chess board Adversarial Search
- 12. Alpha Beta - Pruning Adversarial Search 1 2 3 4 5 6
- 13. • Examine capturing moves before development moves • Killer heuristics Alpha Beta - Pruning with Heuristics Adversarial Search
- 14. 1 2 3 4 Solution Approaches • Min-Max and Alpha Beta Pruning • Adversarial Search better models a zero sum game where either party wins • Alpha Beta Pruning helps you remove entire branches and work with interesting branches only • Alpha Beta can be applied to trees of any depth • Opening Book • End Game • Chess openings are well developed in chess theory • End games are completely solved because its manageable with limited number of pieces left on the board • Progressive Deepening • Iterative Deepening is DFS with benefits of BFS. • A Depth limited version of DFS with progressive increments • It’s a time management strategy in DFS • Proves to be beneficial with Move ordering techniques • Uneven tree development • Some situation at the bottom of the tree may be dynamic • You may capture a queen • You may want to double check by developing the tree
- 15. Uneven Tree Development Classic Chess Puzzle
- 16. Alpha Go: Notice patterns which humans cannot see AI are now correcting novels AI are now improving music by adding some notes Creativity !!! Creativity
- 17. Research in AI -> Algorithms have numerous applications in health care Adversarial Search helps in building strategies. Business decisions Smarter Software: Will it unveil patterns? Like in Biology Parkinson's Cancer Research in Math has benefited numerous fields of sciences. AI can help us find some cool solutions Benefits Research
- 19. • https://www.youtube.com/watch?v=STjW3eH0Cik MIT OCW • https://chessprogramming.wikispaces.com/ Resources Web Resources
- 20. Machine Learning – Tom Mitchell Artificial Intelligence A Modern Approach – Peter Norvig, Stuart Russell Pattern Classification – Richard Duda Introduction to Algorithms – Thomas Cormen Resources Books
- 21. YOU THANK
Editor's Notes
- Lets just try to understand what we are dealing with here. The number of possible chess games is 10^120. This is a conservative lower bound. This bound is an assumption for games which lasts 40 moves The solution space is simply huge. The number is bigger than number of atom in the observable universe. The number is Ten quadrillion vigintillion and one-hundred thousand quadrillion vigintillion atoms. Lets add up Number of atoms + seconds in a year + nanoseconds +age of universe .Even then 10^106 its 14 orders of magnitude short Reference: https://www.youtube.com/watch?v=Km024eldY1A Reference: http://www.universetoday.com/36302/atoms-in-the-universe/ Reference: http://vision.unipv.it/IA1/ProgrammingaComputerforPlayingChess.pdf
- Chess is in that sense very formal The Objective function is to check mate an opponent or at-least draw if your position is not very good
- Chess by no means is a solved problem There are computers that can beat humans at chess but that does not mean, its solved So what is a computationally solved game? A game is computationally solved if a computer can explore every possible move in that game. This will help computer generate the best move and its certain computer knows that the game tree will eventually result in a win, assuming its not playing against itself or at least playing with a mere mortal(human) Its practically impossible to do that. Because combinatorically it’s a huge problem. In-fact its popularly termed has combinatorial explosion. Chess is a game with a fully observable environment. This is different from game of cards like black jack, solatire, pocker, bridge and backgammon. Partially observable and imperfect information. TODO: Statistics on how much time it will take
- We should be able to formally represent the problem so that we can solve it with a computer For example if you are solving a graph you will come up with representation like edge list, adjacency list or adjacency matrix etc Chess games are documented in a standard format called algebraic notation Reference: https://en.wikipedia.org/wiki/Portable_Game_Notation
- These are some general guidelines which chess grandmasters have for amateur and novice players They are no rules which guarantee success but are sure to improve your chances of winning You can easily build heuristics around these rules. And if you want to teach computer how to play chess, you must certainly have these rules. But what can we do with these rules and how will they help us?
- Position Evaluation: It’s a process of evaluating any position and gives it a numerically value Positive means white is better, negative means black is better, while a evaluation value of 0.0 means the position is equal This leads us to a natural question on how a position is evaluated Adding the pieces on the board. 1.0 for a pawn, 3.0 for knight and bishop, 5.0 for a Rook, 9.0 for a Queen - negative values are used for black. We will discuss more in the next slide Searching the next best move is a difficult problem The Shannon’s number clearly indicates that we cannot explore every branch as the tree grows exponentially So we have to cut out a few branches and only look at branches which are promising We humans also do the same. We look at 1 or 2 branches to evaluate our next move But what algorithmic techniques can we use to narrow down these branches. Lets discuss them in detail in the following presentation Resources: https://www.chess.com/blog/zaifrun/creating-a-chess-engine-from-scratch-part-1 Resources: https://en.lichess.org
- Performance of chess playing program depends on the evaluation function. An inaccurate evaluation function will guide an agent towards positions that turn out to be loss Also the computation should not take long. The whole point is to search faster What we are looking at is a weighted linear function F are the features and w is the weight associated with the feature. So the natural question is what is f? Lets evaluate for white Number of White Pawns Number of White Major pieces (Knight + Bishop) Number of Queens King safety Pawn structure And what is W? Weight is the weight associated with every feature we are considering Pawn – 1 Knight – 3 Bishop - 3.25 R – 5 Q – 9 Good pawn structure and King safety are ½ pawn each Lets follow the order Queen Bishop Night Pawn Rook King in the linear equation expansion As you can see Black is slightly better in this case As you can see the Stockfish, the best chess engine is also is in blacks favor. But evaluation is just one step and we still need to find the best move. Reference: https://en.lichess.org
- We can use Decision Trees. It’s like if then rules in programming. But how big will that tree be. Remember Shannons number? Lets put things in perspective. On average each player has 30 legal moves For two moves. Which is 1st player makes a move then then 2nd player makes a move you have 30*30= 900 moves. And 3rd move 900*30 =2700. and so on. Can we have a decision tree that big which could have explored the entire possibilities. No. Neural Networks. Well they can learn anything. From speech recognition, hand gesture recognition, face recognition. Neural Networks are good at classification problems. They learn and they can fire based on what function they are trained to learn Min Max search is the good fit in adversarial problems where there is a zero sum game. Each player tries to improve his/her chances of winning. Reference: https://www.youtube.com/watch?v=Km024eldY1A (Numberphile YouTube channel)
- Lets image a simple game where one player makes a move and the second player makes his move and the game is over. In this game each player has 3 moves to choose from. Max moves to the maximum value. Min moves to the minimum value Pink triangle player goes first. He is the maximizer. He has three choices and he chooses a1 leading to game state B. In game state B Minimizer chooses the state which goes in his favor. It’s 3 in this case. Optimal play for Max assumes Min also plays optimally. If MIN does not play optimally. MAX will do even better. If in state B Min chooses to go with 8. Well Max has already won. Now with a Branching factor of 3. The maximum depth of 2. The number of nodes to be evaluated is b^d. In this case its 9 Reference: Image made with draw.io
- We are looking at a portion of the min max tree. Every node is the game state. At each level the evaluation function is used. This evaluation function determines how the game is going to pan out for Maximizer and Minimizer The problem with Min-Max is the number of games sates it has to examine is exponential in the depth of the tree 30^10 is 590,490,000,000,000. And Stockfish managed to analyze till the depth of 22!!! How did it do that? Pruning comes to your rescue which we will discuss in the next slides
- Class of branch and bound algorithms. Starts with α been negative infinity and β been positive infinity. Both players are starting with lowest possible score Alpha Beta Pruning: This slides shows stages of calculation of the optimal decision tree for the game tree. The first leaf below B has value of 3. Hence B is a min node and has a value of at-most 3. The second leaf below B has 12. B will avoid this move. So B is still at-most 3. The third leaf below B has a value of 8. We have seen all B successors and hence value is 3 Leaf below C has value 2. Hence C been min node has value 2. But we know B is 3. So Max will never choose C. There is no point looking at other successors of B First leaf below D has value 14. Now D is at-most 14. This is higher than the max alternative 3. So explore D successors. The second successor of D is 5 which is bigger than min 3. Hence explore more and we find 2. So D is 2. Max decision at the root is to move to D Reference: Artificial Intelligence a Modern Approach –Peter Norvig
- Capture moves may be more interesting. Evaluating them and exploring the tree first may lead to cutoff or development Alpha beta Pruning time complexity is again exponential. Hence a killer heuristics will help examine the interesting moves first. It only needs to consider the other player's possible responses to that best move And can skip evaluation of responses to (worse) moves it will not make. There are some other concepts like Refutation tables and transposition tables. Reference: https://en.wikipedia.org/wiki/Alpha%E2%80%93beta_pruning Reference: https://en.wikipedia.org/wiki/Killer_heuristic
- Is it model of our intelligence?
- White to play Rook to A8. Q x A8. BF3+. King has to move. And the queen is taken. Well interesting situations like these need to be explored in greater depth. Rather than relying on simple rule of capturing the rook blindly. Reference: https://twitter.com/SusanPolgar Puzzle by Susan on Twitter timeline
- Systems are learning patterns in from the data set without knowing the data Earlier AI would have rules books based learning (decision trees). So it knows what decision AI has recently been able to come up with the concept of Cat by just studying YouTube videos. This learning was un-supervised Reference: https://www.youtube.com/watch?v=5lur9fY86xM
- This slide is more philosophical in nature Knowing how we can play certain complex games the field of Algorithms can explore and learn new way of doing things