Foundations of Artificial Intelligence May 11, 2020 — 40. Board Games: Introduction and State of the Art Foundations of Artificial Intelligence 40. Board Games: Introduction and State of the Art 40.1 Introduction Malte Helmert and Thomas Keller 40.2 State of the Art University of Basel 40.3 Summary May 11, 2020 M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 1 / 22 M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 2 / 22 Classification Board Games: Overview classification: Board Games chapter overview: environment: ◮ 40. Introduction and State of the Art ◮ static vs. dynamic ◮ 41. Minimax Search and Evaluation Functions ◮ deterministic vs. non-deterministic vs. stochastic ◮ 42. Alpha-Beta Search ◮ fully vs. partially vs. not observable ◮ 43. Monte-Carlo Tree Search: Introduction ◮ discrete vs. continuous ◮ 44. Monte-Carlo Tree Search: Advanced Topics ◮ single-agent vs. multi-agent (opponents) ◮ 45. AlphaGo and Outlook problem solving method: ◮ problem-specific vs. general vs. learning M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 3 / 22 M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 4 / 22
40. Board Games: Introduction and State of the Art Introduction 40. Board Games: Introduction and State of the Art Introduction Why Board Games? Board games are one of the oldest areas of AI (Shannon 1950; Turing 1950). 40.1 Introduction ◮ abstract class of problems, easy to formalize ◮ obviously “intelligence” is needed (really?) ◮ dream of an intelligent machine capable of playing chess is older than electronic computers ◮ cf. von Kempelen’s “Schacht¨ urke” (1769), Torres y Quevedo’s “El Ajedrecista” (1912) German: Brettspiele M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 5 / 22 M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 6 / 22 40. Board Games: Introduction and State of the Art Introduction 40. Board Games: Introduction and State of the Art Introduction Games Considered in This Course Example: Chess We consider board games with the following properties: ◮ current situation representable by finite set of positions Example (Chess) ◮ changes of situations representable by finite set of moves ◮ positions described by: ◮ there are two players ◮ configuration of pieces ◮ whose turn it is ◮ in each position, it is the turn of one player, ◮ en-passant and castling rights or it is a terminal position ◮ turns alternate ◮ terminal positions have a utility ◮ terminal positions: checkmate and stalemate positions ◮ utility for player 2 always opposite of utility for player 1 ◮ utility of terminal position for first player (white): (zero-sum game) ◮ +1 if black is checkmated ◮ “infinite” game progressions count as draw (utility 0) ◮ 0 if stalemate position ◮ no randomness, no hidden information ◮ − 1 if white is checkmated German: Positionen, Z¨ uge, am Zug sein, Endposition, Nutzen, Nullsummenspiel M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 7 / 22 M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 8 / 22
40. Board Games: Introduction and State of the Art Introduction 40. Board Games: Introduction and State of the Art Introduction Other Game Classes Terminology Compared to State-Space Search important classes of games that we do not consider: ◮ with randomness (e.g., backgammon) Many concepts for board games are similar to state-space search. ◮ with more than two players (e.g., chinese checkers) Terminology differs, but is often in close correspondence: ◮ with hidden information (e.g., bridge) ◮ state � position ◮ goal state � terminal position ◮ with simultaneous moves (e.g., rock-paper-scissors) ◮ without zero-sum property (“games” from game theory ◮ action � move � auctions, elections, economic markets, politics, . . . ) ◮ search tree � game tree ◮ . . . and many further generalizations Many of these can be handled with similar/generalized algorithms. M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 9 / 22 M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 10 / 22 40. Board Games: Introduction and State of the Art Introduction 40. Board Games: Introduction and State of the Art Introduction Formalization Specific vs. General Algorithms Board games are given as state spaces S = � S , A , cost , T , s 0 , S ⋆ � with two extensions: ◮ We consider approaches that must be tailored ◮ player function player : S \ S ⋆ → { 1 , 2 } to a specific board game for good performance, indicates whose turn it is e.g., by using a suitable evaluation function. ◮ utility function u : S ⋆ → R indicates utility of terminal position � see chapters on informed search methods for player 1 ◮ Analogously to the generalization of search methods to declaratively described problems (automated planning), other differences: board games can be considered in a more general setting, ◮ action costs cost not needed where game rules (state spaces) are part of the input. ◮ non-terminal positions must have at least one successor � general game playing: annual competitions since 2005 We do not go into more detail here as we have previously seen sufficiently many similar definitions. M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 11 / 22 M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 12 / 22
40. Board Games: Introduction and State of the Art Introduction 40. Board Games: Introduction and State of the Art Introduction Why are Board Games Difficult? Algorithms for Board Games As in classical search problems, the number of positions of (interesting) board games is huge: ◮ Chess: roughly 10 40 reachable positions; properties of good algorithms for board games: game with 50 moves/player and branching factor 35: ◮ look ahead as far as possible (deep search) tree size roughly 35 100 ≈ 10 154 ◮ consider only interesting parts of the game tree ◮ Go: more than 10 100 positions; (selective search, analogously to heuristic search algorithms) game with roughly 300 moves and branching factor 200: ◮ evaluate current position as accurately as possible tree size roughly 200 300 ≈ 10 690 (evaluation functions, analogously to heuristics) In addition, it is not sufficient to find a solution path: ◮ We need a strategy reacting to all possible opponent moves. ◮ Usually, such a strategy is implemented as an algorithm that provides the next move on the fly (i.e., not precomputed). M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 13 / 22 M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 14 / 22 40. Board Games: Introduction and State of the Art State of the Art 40. Board Games: Introduction and State of the Art State of the Art State of the Art some well-known board games: ◮ Chess, Go: � next slides 40.2 State of the Art ◮ Othello: Logistello defeated human world champion in 1997; best computer players significantly stronger than best humans ◮ Checkers: Chinook official world champion (since 1994); proved in 2007 that it cannot be defeated and perfect game play results in a draw (game “solved”) German: Schach, Go, Othello/Reversi, Dame M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 15 / 22 M. Helmert, T. Keller (University of Basel) Foundations of Artificial Intelligence May 11, 2020 16 / 22
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