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double AlphaBeta (state, depth , alpha, beta ) begin if depth <= 0 then

double AlphaBeta (state, depth , alpha, beta ) begin if depth <= 0 then return evaluation(state) //op pov for each action “a” possible from state nextstate = performAction(a, state) rval = - AlphaBeta (nextstate, depth-1 , -beta, -alpha ); if (rval >= beta) return rval;

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double AlphaBeta (state, depth , alpha, beta ) begin if depth <= 0 then

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  1. double AlphaBeta(state, depth, alpha, beta) begin if depth <= 0 then return evaluation(state) //op pov for each action “a” possible from state nextstate = performAction(a, state) rval = -AlphaBeta(nextstate, depth-1, -beta, -alpha); if (rval >= beta)return rval; if (rval > alpha) alpha = rval; endfor return alpha; end

  2. Meta-Reasoning for Search • problem: one legal move only (or one clear favourite) alpha-beta search will still generate (possibly large) search tree • similar symmetrical situations • idea: compute utility of expanding a node before expanding it • meta-reasoning (reasoning about reasoning): reason about how to spend computing time

  3. Where We are: Chess

  4. Deep Blue • algorithm: • iterative-deepening alpha-beta search, transposition table, databases incl. openings, grandmaster games (700000), endgames (all with 5 pieces, many with 6) • hardware: • 30 IBM RS/6000 processors • software search: at high level • 480 custom chess processors for • hardware search: search deep in the tree, move generation and ordering, position evaluation (8000 features) • average performance: • 126 million nodes/sec., 30 billion position/move generated, search depth: 14 (but up to 40 plies)

  5. Samuel’s Checkers Program (1952) • learn an evaluation function by self-play(see: machine learning) • beat its creator after several days of self-play • hardware: IBM 704 • 10kHz processor • 10000 words of memory • magnetic tape for long-term storage

  6. Chinook: Checkers World Champion • simple alpha-beta search (running on PCs) • database of 444 billion positions with eight or fewer pieces • problem: Marion Tinsley • world checkers champion for over 40 years • lost three games in all this time • 1990: Tinsley vs. Chinook: 20.5-18.5 • Chinook won two games! • 1994: Tinsley retires (for health reasons)

  7. Backgammon • TD-GAMMON • search only to depth 2 or 3 • evaluation function • machine learning techniques (see Samuel’s Checkers Program) • neural network • performance • ranked amongst top three players in the world • program’s opinions have altered received wisdom

  8. Go • most popular board game in Asia • 19x19 board: initial branching factor 361 • too much for search methods • best programs: Goemate/Go4++ • pattern recognition techniques (rules) • limited search (locally) • performance: 10 kyu (weak amateur)

  9. A Dose of Reality: Chance • unpredictability: • in real life: normal; often external events that are not predictable • in games: add random element, e.g. throwing dice, shuffling of cards • games with an element of chance are less “toy problems”

  10. Example: Backgammon • move: • roll pair of dice • move pieces according to result

  11. Search Trees with Chance Nodes • problem: • MAX knows its own legal moves • MAX does not know MIN’s possible responses • solution: introduce chance nodes • between all MIN and MAX nodes • with n children if there are n possible outcomes of the random element, each labelled with • the result of the random element • the probability of this outcome

  12. Example: Search Tree for Backgammon MAX move CHANCE probability +outcome 1/361-1 1/366-6 1/181-2 1/185-6 MIN move CHANCE probability +outcome

  13. Optimal Decisions for Games with Chance Elements • aim: pick move that leads to best position • idea: calculate the expected value over all possible outcomes of the random element • expectiminimax value

  14. Example: Simple Tree 2.1 1.3 0.9 × 2 + 0.1 × 3 = 2.1 0.9 × 1 + 0.1 × 4 = 1.3 0.9 0.1 0.9 0.1 2 3 1 4 2 2 3 3 1 1 4 4

  15. Complexity of Expectiminimax • time complexity: O(bmnm) • b: maximal number of possible moves • n: number of possible outcomes for the random element • m: maximal search depth • example: backgammon • average b is around 20 (but can be up to 4000 for doubles) • n = 21 • about three ply depth is feasible

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