Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Lecture 9 of 14 Friday, 10 September 2004 William H. Hsu Department of Computing and Information Sciences, KSU Reading: Chapter 6, Russell and Norvig 2e Game Tree Search: Minimax and Alpha-Beta

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Lecture Outline Today’s Reading –Sections , Russell and Norvig 2e –Recommended references: Rich and Knight, Winston Reading for Next Class: Sections , Russell and Norvig Games as Search Problems –Frameworks: two-player, multi-player; zero-sum; perfect information –Minimax algorithm Perfect decisions Imperfect decisions (based upon static evaluation function) –Issues Quiescence Horizon effect –Need for pruning Next Lecture: Alpha-Beta Pruning, Expectiminimax, Current “Hot” Problems Next Week: Knowledge Representation – Logics and Production Systems

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Overview Perfect Play –General framework(s) –What could agent do with perfect info? Resource Limits –Search ply –Static evaluation: from heuristic search to heuristic game tree search –Examples Tic-tac-toe, connect four, checkers, connect-five / Go-Moku / wu 3 zi 3 qi 2 Chess, go Games with Uncertainty –Explicit: games of chance (e.g., backgammon, Monopoly) –Implicit: see project suggestions! Adapted from slides by S. Russell, UC Berkeley

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Games versus Search Problems Unpredictable Opponent –Solution is contingency plan –Time limits Unlikely to find goal Must approximate Plan of Attack –Algorithm for perfect play (J. von Neumann, 1944) –Finite horizon, approximate evaluation (C. Zuse, 1945; C. Shannon, 1950, A. Samuel, ) –Pruning to reduce costs (J. McCarthy, 1956) Adapted from slides by S. Russell, UC Berkeley

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Types of Games Information: Can Know (Observe) –… outcomes of actions / moves? –… moves committed by opponent? Uncertainty –Deterministic vs. nondeterministic outcomes –Thought exercise: sources of nondeterminism? Adapted from slides by S. Russell, UC Berkeley

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Minimax Adapted from slides by S. Russell, UC Berkeley Figure 5.2 p. 125 R&N

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Minimax Algorithm: Decision and Evaluation Adapted from slides by S. Russell, UC Berkeley  what’s this? Figure 5.3 p. 126 R&N

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Properties of Minimax Adapted from slides by S. Russell, UC Berkeley Complete? –… yes, provided following are finite: Number of possible legal moves (generative breadth of tree) “Length of game” (depth of tree) – more specifically? –Perfect vs. imperfect information? Q: What search is perfect minimax analogous to? A: Bottom-up breadth-first Optimal? –… yes, provided perfect info (evaluation function) and opponent is optimal! –… otherwise, guaranteed if evaluation function is correct Time Complexity? –Depth of tree: m –Legal moves at each point: b –O(b m ) – NB, m  100, b  35 for chess! Space Complexity? O(bm) – why?

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Resource Limits Adapted from slides by S. Russell, UC Berkeley

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Static Evaluation Function Example: Chess Adapted from slides by S. Russell, UC Berkeley Figure 5.4(c), (d) p. 128 R&N

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Do Exact Values Matter? Adapted from slides by S. Russell, UC Berkeley

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Cutting Off Search [1] Adapted from slides by S. Russell, UC Berkeley

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Cutting Off Search [2] Adapted from slides by S. Russell, UC Berkeley Issues –Quiescence Play has “settled down” Evaluation function unlikely to exhibit wild swings in value in near future –Horizon effect “Stalling for time” Postpones inevitable win or damaging move by opponent See: Figure 5.5 R&N Solutions? –Quiescence search: expand non-quiescent positions further –“No general solution to horizon problem at present

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Why Prune? Adapted from slides by S. Russell, UC Berkeley Figure 5.6 p. 131 R&N

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Summary Points Introduction to Games as Search Problems –Frameworks Two-player versus multi-player Zero-sum versus cooperative Perfect information versus partially-observable (hidden state) –Concepts Utility and representations (e.g., static evaluation function) Reinforcements: possible role for machine learning Game tree Family of Algorithms for Game Trees: Minimax –Propagation of credit –Imperfect decisions –Issues Quiescence Horizon effect –Need for pruning

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Terminology Game Graph Search –Frameworks Two-player versus multi-player Zero-sum versus cooperative Perfect information versus partially-observable (hidden state) –Concepts Utility and representations (e.g., static evaluation function) Reinforcements: possible role for machine learning Game tree: node/move correspondence, search ply Family of Algorithms for Game Trees: Minimax –Propagation of credit –Imperfect decisions –Issues Quiescence Horizon effect –Need for (alpha-beta) pruning