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# CS 236501 Introduction to AI

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1. CS 236501Introduction to AI Tutorial 5 Adversary Search

2. Agenda • Introduction: • Why games? • Assumptions • Minimax algorithm • General idea • Minimax with limited depth • Alpha-Beta search • Pruning • Search routine • Example • Enhancements to the algorithm Intro. to AI – Tutorial 5 – By Nela Gurevich

3. Why Games? • Games are fun • Easy to measure results • Simple moves • Big search spaces Examples: Chess: Deep Junior Checkers: Chinook, Nemesis Othello: Bill Backgammon: TDGammon Intro. to AI – Tutorial 5 – By Nela Gurevich

4. Assumptions • Two-players game • Perfect information The knowledge available to each player is the same • Zero Sum The move good for a player is bad for his adversary Intro. to AI – Tutorial 5 – By Nela Gurevich

5. Game Trees MIN MAX Win Win Loss Win Loss Draw Draw Intro. to AI – Tutorial 5 – By Nela Gurevich

6. The Minimax Algorithm • e(v) if v is a terminal node • MM(v) = max{MM(succ)} v is a max node • min{MM(succ)} v is a min node • Where succ = successors(v) and • 1 if v is a WIN node • e(v) = 0 if v is a DRAW node • -1 if v is a LOSS node A problem: big branching factor, deep trees Intro. to AI – Tutorial 5 – By Nela Gurevich

7. Minimax search to limited depth • Search the game tree to some search frontier d. • Compute a static evaluation function f to assess the strength values of nodes at that frontier. • Use the minimax rule to compute approximations of the strength values of the shallower nodes. • f(v) if d=0 or v is terminal MM(v,d) = max{MM(succ, d-1)} v is a max node • min{MM(succ, d-1)} v is a min node Intro. to AI – Tutorial 5 – By Nela Gurevich

8. 10 αβSearch Shallow pruning Deep pruning MM 10≤ MM 10≤ 10 MM ≤ 5 5 MM ≤ 5 The node will not contribute to the max value of the father The node will not contribute to the max value of the ancestor 5 Intro. to AI – Tutorial 5 – By Nela Gurevich

9. αβProcedure α : highest max among ancestors of a node β : lowest min among ancestors of a node First call: αβ(v, d, min/max, -∞, ∞) // The αβ procedure αβ(v, d, node-type, α, β) { If v is terminal, or d = 0 return f(v) Intro. to AI – Tutorial 5 – By Nela Gurevich

10. αβProcedure: MAX node • if node-type = MAX • { • curr-max ← -infinity • loop for viє Succ(v) • { • board-val ← αβ(vi, d-1, min, α, β) • curr-max ← max(board-val, curr-max) • α ← max(curr-max, α) • if (curr-max ≥ β) // Bigger than lowest min • end loop • } • return curr-max • } Intro. to AI – Tutorial 5 – By Nela Gurevich

11. αβProcedure: MIN node • if node-type = MIN • { • curr-min ← infinity • loop for viє Succ(v) • { • board-val ← αβ(vi, d-1, max, α, β) • curr-min ← min(board-val, curr-min) • β ← min(curr-min, β) • if (curr-min ≤ α) // Smaller than highest max • end loop • } • return curr-min • } • } // end of αβ Intro. to AI – Tutorial 5 – By Nela Gurevich

12. 10 11 9 12 14 15 13 14 5 2 4 1 3 22 20 21 Game Tree Example MIN MAX Intro. to AI – Tutorial 5 – By Nela Gurevich

13. ? 10 11 9 14 15 13 5 2 4 1 3 22 20 Stage 1 α = -∞ β = ∞ α = -∞ β = ∞ α = -∞ β = ∞ α = -∞ β = ∞ 12 14 21 Intro. to AI – Tutorial 5 – By Nela Gurevich

14. ? 10 11 9 14 15 13 5 2 4 1 3 22 20 Stage 1 α = 10 β = ∞ 10 α = -∞ β = 10 α = 10 β = ∞ 10 12 14 21 Intro. to AI – Tutorial 5 – By Nela Gurevich

15. 10 11 9 Stage 2 – Shallow Pruning α = -∞ β = 10 10 α = -∞ β = 10 α = 10 β = ∞ 10 9 α = -∞ β = 10 α = 10 β = 9 12 14 15 13 14 5 2 4 1 3 22 20 21 Intro. to AI – Tutorial 5 – By Nela Gurevich

16. 10 11 9 12 14 15 13 14 5 2 4 1 3 22 20 21 Game Tree example contd. α = 10 β = ∞ 10 10 α = -∞ β = 10 14 α = 14 β = 10 14 α = -∞ β = 10 Intro. to AI – Tutorial 5 – By Nela Gurevich

17. 10 11 9 12 14 15 13 14 5 2 4 1 3 22 20 21 Game Tree example contd. α = 10 β = ∞ α = 10 β = ∞ α = 10 β = ∞ α = 10 β = ∞ Intro. to AI – Tutorial 5 – By Nela Gurevich

18. 10 11 9 12 14 15 13 14 5 2 4 1 3 22 20 21 Game Tree example contd. α = 10 β = ∞ 5 α = 10 β = 5 α = 10 β = ∞ Intro. to AI – Tutorial 5 – By Nela Gurevich

19. 10 11 9 12 14 15 13 14 5 2 4 1 3 22 20 21 Game Tree example contd. 10 5 α = 10 β = 5 5 α = 10 β = ∞ 4 α = 10 β = 4 Intro. to AI – Tutorial 5 – By Nela Gurevich

20. αβFeatures • Correctness: Minimax(v, d) = αβ(v, d, -∞, ∞) • Pruning: The values of the tree leaves and the search ordering determine the amount of pruning • For any given tree and any search ordering, there exists a sequence of values for the leaves, such that αβ prunes no leaf. • For any given tree there exists such ordering that αβ prunes the maximal possible number of leaves. • For randomly ordered trees αβ expands Θ(b(3/4d)) leaves • Pruning decreases the effective branching factor, and thus allows us to search game tree for greater depth Intro. to AI – Tutorial 5 – By Nela Gurevich

21. Iterative αβ Perform αβ search to increasing depth beginning from some initial depth. • Useful when the time is bounded – when the time is over, the value computed during the previous iteration can be returned • The values computed during the previous iterations may be used to perform a heuristic ordering on the nodes of the tree to increase the pruning rate Intro. to AI – Tutorial 5 – By Nela Gurevich

22. The End Intro. to AI – Tutorial 5 – By Nela Gurevich