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Pruned Search Strategies CS344 : AI - Seminar 20 th January 2011 TL Nishant Totla, RM Pritish Kamath M1 Garvit Juniwal, M2 Vivek Madan guided by Prof. Pushpak Bhattacharya

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Outline ● Two player games ● Game trees ● MiniMax algorithm ● α-β pruning ● A demonstration : Chess ● Iterative Deepening A*

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A Brief History ● Computer considers possible lines of play (Babbage, 1846) ● Minimax theorem (von Neumann, 1928) ● First chess program (Turing, 1951) ● Machine learning to improve evaluation accuracy (Samuel, 1952–57) ● Pruning to allow deeper search (McCarthy, 1956) ● Deep Blue wins 6-game chess match against Kasparov (Hsu et al, 1997) ● Checkers solved (Schae ﬀ er et al, 2007)

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Two player games ● The game is played by two players, who take alternate turns to change the state of the game. ● The game has a starting state S. ● The game ends when a player does not have a legal move. ● Both players end up with a score at an end state.

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Classification of 2-player games ● Sequential : players move one-at-a-time ● Zero-Sum game : sum of scores assigned to the players at any end state equals 0. deterministicchance Perfect informationchess, checkers, go, Othello backgammon, monopoly, roulette Imperfect informationbattleship, kriegspiel, rock-paper-scissors bridge, poker

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Game Tree ● A move changes the state of the game. ● This naturally induces a graph with the set of states as the vertices, and moves represented by the edges. ● A game tree is a graphical representation of a finite, sequential, deterministic, perfect-information game.

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Tic-Tac-Toe Game Tree

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Strategy for 2-player games? ● How does one go about playing 2-player games? Choose a move. Look at all possibile moves that the opponent can play. Choose a move for each of the opponent's possible move and so on... ● Consider an instance of Tic-Tac-Toe game played between Max and Min. ● The following two images describe strategies for each player.

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Best Strategy? The MiniMax Algorithm -- taken from Wikipedia (http://en.wikipedia.org/wiki/Minimax)

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But what about the heuristics? ● The heuristics used at the leaves of the MiniMax tree depend on the rules of the game and our understanding of the same. ● A heuristic is an objective way to quantify the “goodness” of a particular state. ● For example, in chess you can use the weighted sum of pieces remaining on the board.

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Properties of the minimax algorithm ● Space complexity: O(bh), where b is the average fanout, and h is the maximum search depth. ● Time complexity: O(b h ) ● For chess, b≈35, h≈100 for 'reasonable' games. ● ≈ nodes. ● This is about times the number of particles in the Universe (about ) ⇨ no way to examine every node! ● But do we really need to examine every node? Let's now see an improved idea.

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Improvements? ● α-β Pruning : “Stop exploring unfavourable moves if you have already found a more favourable one.”

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α-β pruning (execution) - taken from Wikipedia (http://en.wikipedia.org/wiki/Alpha-beta_pruning)

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d-depth α-β p runing (pseudocode)

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Resource Limits ● Even after pruning, Chess has too large a state space, which hence search depths are restricted. ● Fact: Deep Blue defeated human world champion Garry Kasparov in a six-game match in Deep Blue searches 200 million positions per second, uses very sophisticated evaluation, and undisclosed methods for extending some lines of search up to 40 ply.

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Demonstration! ● We shall now demonstrate a chess program that uses the Minimax algorithm with α-β pruning. ● The code is written in Scheme (functional programming language). ● After this, we move to a different pruned search strategy for general graphs.

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Iterative Deepening A*

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Search Strategies ● Two types of search algorithms: – Brute force (breadth-first, depth-first, etc.) – Heuristic (A*, heuristic depth-first, etc.)

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Definitions ● Node branching factor (b): Maximum fan-out of of the nodes of the search tree. ● Depth (d): Length of the shortest path from the initial state to a goal state ● Maximum depth(m): Maximum depth of the tree.

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Breadth First Search

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Depth First Search

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Iterative Deepening DFS Source:

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Iterative Deepening DFS Source:

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IDA* Algorithm ● IDA*, like depth-first search, except based on increasing values of total cost (f=g+h) rather than increasing depths. ● At each iteration perform a depth first search cutting off a branch when its total cost exceeds a given threshold. ● Threshold is initially set to h(start). ● Threshold used for the next iteration is the minimum cost of all values that exceeded the current threshold. ● IDA* always finds a cheapest solution if the heuristic is admissible.

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Monotonocity ● For any admissible cost function f, we can construct a monotone admissible function f ' which is at least as informed as f. ● We restrict our attention to cost functions which are monotonically non-decreasing along any path in the problem space, without loss of generality.

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Correctness ● Since the cost cutoff for each succeeding iteration is the minimum value which exceeded the previous cutoff, no paths can have a cost which lies in a gap between two successive cutoffs. ● IDA* examines nodes in order of increasing f-cost. ● Hence, IDA* finds the optimal path. ● Source:

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Why IDA* over A*? ● Uses far less space than A* ● Expands asymptotically, the same number of nodes as A* in a tree search. ● Simpler to implement since there are no open or closed lists to be managed.

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Optimality Given an admissible monotone heuristic with constant relative error, then IDA* is optimal in terms of solution cost, time, and space, over the class of admissible best- first searches on a tree.

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An Empirical Test ● Both IDA* and A* were implemented for the Fifteen Puzzle. Manhattan distance heuristic. ● A* couldn't solve most cases. It ran out of space. ● IDA* generated more nodes than A*, still ran faster than A*, due to less overhead per node. ● Also refer:

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Application to Game Trees ● We want to maximize search depth subject to fixed time and space constraints. ● Since IDA* minimizes, at least asymptotically, time and space for any given search depth, it maximizes the depth of search possible for any fixed time and space restrictions as well.

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Thank You. Questions??

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References list ● ● ve_deepening__an_opti_93341.pdf ve_deepening__an_opti_93341.pdf ● arches/intro-to-iterative-deepening.ppt arches/intro-to-iterative-deepening.ppt ● es/ids.ppt es/ids.ppt ●

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