Group 1 : Ashutosh Pushkar Ameya Sudhir From

Motivation  Game playing was one of the first tasks undertaken in AI  Study of games brings us closer to :  Machines capable of logical deduction  Machines for making strategic decisions  Analyze the limitations of machines to human thought process  Games are an idealization of worlds  World state is fully accessible  Actions & outcomes are well-defined

Outline of the presentation  Evaluation Functions  Algorithms  Deep Blue  Conclusions from Deep Blue  Conclusion  References

Chess  Neither too simple  Nor too difficult for satisfactory solution  Requires “thinking” for a skilled player  Designing a chess playing program  Perfect chess playing : INTRACTABLE  Legal Chess : TRIVIAL  Play tolerably good game : SKILLFULLY

Evaluation Function

 Utility function  Whole game tree is explored  computationally expensive task !!  Estimates the expected utility of a state  Evaluation functions  cut off the exploration depth by estimating whether a state will lead to a win or loss

Evaluation Function (cont.)  A good evaluation function should  not take too long  Preserve ordering of the terminal states otherwise it will lead to bad decision making  Consider strategic moves that lead to long term advantages

Evaluation Function (cont.)  Typically includes :  Material Advantage (difference in total material of both sides) f (P) = 200(k – k’) + 9(q – q’) + 5(r – r’) + 3(b – b’) + (p – p’) + g (P) + h (P) + …  Positions of pieces  Rook on open file  double rooks  rook on seventh rank etc. and their relative positions.  Pawn Formation  Mobility Evaluation function is an attempt to write a mathematical formula for intelligence

Algorithms

Games as Search Problems  Initial states  Where game starts  Initial position in chess  Successor function  List of all legal moves from current position  Terminal State  Where the game is concluded  Utility function  Numeric value for all terminal states

Minimax Strategy

 Optimal strategy  Assumption : opponent plays his best possible move.  An option is picked which  Minimizes damage done by opponent  Does most damage to the opponent  Idea:  For each node find minimum minimax value  Choose the node with maximum of such values  This will ensure best value against most damage done by opponent

Strategy of Minimax  Opponent tries to reduce utility function’s value  For any move made by opponent in reply of computer’s move, choose minimum reduced value by opponent  Find the move with maximum such value

Analysis  Algorithm is complete for complete tree only  Not best strategy against irrational opponent  According to definition  Time complexity :O(b m )  b = max. no. of possible moves  m = max. depth of tree  In chess even in average case, b = 35 and m = 100 => time exceeds practical limits  # of states grows exponentially as per number of moves played

α-β Pruning

 The problem of minimax search  # of state to examine: exponential in number of moves  Returns same moves as minimax does  Prunes away branches that can’t influence final decision.  α: the value of the best (highest) choice so far in search of MAX  β: the value of the best (lowest) choice so far in search of MIN  Order of considering successor

Algorithm for α-β Pruning  Current highest β is found and assigned as α  β is current lowest for α’s from that move  For next possible node, while finding β, if some α is found lower than current highest β:  It will only give lesser value of final β  S0, other α’s are not found for that node  After calculating β for this node, α is replaced by max(α,β for this node)  In this way after all possible set of moves final value of α is found

α-β Pruning (2) If m is better than n for Player, we will never get to n in play and just prune it.

Analysis of α-β Pruning:  Does not affect final results  Worthwhile to examine that successor first which is likely to be best  Time complexity: O(b (m/2) )  Effective branching factor = √b  i.e. 6 rather than 35  In case of random ordering :  Total number of nodes examined is of the order O(b^ (3m/4) ) Improvement over minimax algorithm

Transposition table  Dynamic programming  Multiple paths to the same position  Savings through memorization  Use a hash table of evaluated positions

Iterative Deepening  Sometime chess is played under a strict time  Depth of search depend on time  Use of Breadth first Search  Advantage : program know which move was best at previous level

Horizon Effect  Problem with fixed depth search  Positive Horizon Effect  Negative horizon effect

Quiescence Search  Search till “quiet” position  Quiet Position  Doesn’t affect the current position so much  Example : no capture of any piece, no check, no pawn promotions/threats

State of the art : DEEP BLUE

Defeats Gary Kasparov  Won a match in 1997  Brute force computing power  Massive, parallel architecture  Special purpose hardware for chess  Parameters of the evaluation function  Learnt by studying many master games  Different evaluation function for different positions  Utilized heavily loaded endgame databases

Humans vs. Computers HumansComputers Lower Computational SpeedHigher Errors PossibleError Free Tend to be instinctiveNo instincts ImaginativeNone High Learning CapabilitiesLimited InductiveNot Inductive

Some intricacies of a chess playing system  Should not play the same sequence of moves again  A player wins a match against the computer  Starts playing the same sequence of moves  Hence, a statistical element is required  Opponent can learn the algorithm used by computer  Hence, again the need for a statistical element  Different game play during different phases  Start Game  Mid Game  End Game

Conclusion  Computer chess as a search problem  Good enough decisions  Simulation of “skill” by “knowledge”  Limitations of computers to humans  Future work :  Better evaluation functions through learning  Need for different AI techniques to play chess

References  Claude E. Shannon: Programming a Computer for Playing Chess, Philosophical Magazine, Ser.7, Vol. 41, No. 314, March  S.Russel & P. Norvig: Artificial Intelligence: A Modern Approach 2/E, Prentice Hall, ISBN-10:  Wikipedia

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