COMP4031/COMP4631 2006-7 Artificial Intelligence for Games and Puzzles Dr. Arthur Cater

1. COMP4031/COMP4631 2006-7Artificial Intelligence for Games and PuzzlesDr. Arthur Cater http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

2. Course Web Page - http://csiweb.ucd.ie/Staff/acater/comp4031.html • As time progresses, Lecture Notes and Assignments will be added. Watch for them! • Assessment will be partly by examination (60%) and partly by programming assignment (2x 20%). • Assignment 1 (20% of unit) : set in week 4 and due in week 8. • Assignment 2 (20% of unit) : set in week 7 and due in week 11. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

3. What sort of Games and Puzzles? • There are two broad categories of multi-player games: • Parlour Games of a primarily intellectual character • Chess, Poker, Draughts, Backgammon, Connect-Four, … • Physical attributes of a player (dexterity, strength, steadiness, speed) have no real bearing on the game • Games of a sports character • Soccer, Tennis, “finger-twitching” computer games, … • Physical attributes are important (too) • We will deal almost exclusively with the “primarily intellectual” games, and their close kin, combinatorial puzzles. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

4. Is this just frivolous? • Certainly it should be some fun: interesting, entertaining, challenging. Further, • Studies of game playing have led historically to valuable spin-offs: • In mid-17th Century, mathematicians Pascal, Fermat, and others laid the foundations of probability theory, and hence statistics, arising from a study of gambling games (though beaten by Cardan by a century) • In 20th Century, von Neumann and others formulated game theory, which now has applications in economics and commerce - being used to design auctions for telecoms bandwidth for example, and guiding corporate takeover strategy • AI for games has direct commercialisation possibilities • chess machines, in-flight entertainment consoles, computer games with AI-driven opposition http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

5. Is this just frivolous? (2) • Games provide a proving ground for AI (and cognitive science) theories of mentality: • perception, representation, reasoning, learning, modelling, risk assessment, … • There may be prizes to be won • $1m “Ing prize” for Go sadly no longer available • Prize for beating Taiwanese junior Go champion • Prize for Arimaa • There is fame and prestige in beating champions, winning tournaments • There are still scientific open problems to be solved http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

6. Parlour Games: “The Three Games” • Three particular traditional games show up a further significant division among competitive parlour games: • Chess - “last man standing” type of game • There are many games with this flavour. They are characterised by a race to achieve some goal, often involving capture or destruction or immobilisation of the opponent. • Backgammon - “the gods help those who help themselves” type of game • In games with this flavour, there is a chance element, depending usually on dice or cards. More skilled players will nevertheless usually win. • Go - “take the lion’s share” type of game • In games with this flavour, players compete for shares of a resource of some kind. To win you need not win everything, but through give-and-take, just win a greater share than the opponent. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

7. Last-Man-Standing game: Chess • Chess is the dominant intellectual game in the west. In AI it is the most heavily researched game by far. After about 50 years work, a chess machine was developed which beat the reigning world champion, Garry Kasparov. • Two players each have six kinds of piece, each with its own movement rules. Players alternate play, moving one piece at a time, sometimes capturing and removing an opponent’s piece. • Win by “checkmate”, where the opponent has legal moves but none of them will prevent immediate capture of the king. • Note a mirror-image symmetry, top-to-bottom with “colrev” - colour reversal of pieces. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

8. Last-Man-Standing game: Draughts / Checkers • Draughts (Checkers) is played on a chessboard, or a similar board 10x10. • A program beat the then world champion, Tinsley, in ill health, in the 1990s. • There is initially just one kind of piece, which can move one square along diagonals in a forward direction. Upon reaching the far edge they are promoted to “kings”, able to go backward too. • A piece may capture an opponent piece by hopping over it to a vacant square beyond. Many captures in one move are possible. • Win by capturing all the opponent’s pieces. • Note a rotational symmetry, with colrev. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

9. Puzzle: Eternity • There are 209 playing pieces, all of different shapes but covering the same area. They have jagged edges with a small number of angles and straight-line lengths. • They are assemblies of six “tridrafters” - 30o-60o-90o triangles. The pieces are in reality all the same colour and can be used either way up. Each piece therefore has 12 possible manifestations. • The puzzle is to fit them together to fill perfectly a particular shape of board - shown here as the lined blue dodecahedron. The board has mirror symmetry along 2 axes and 6-way rotational symmetry. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

10. Game with chance element: Backgammon • White tries to move pieces anticlockwise to end in the bottom right, Red tries to move clockwise to top right. (Logically, it is just a straight line, wrapped over to be compact) • Players in turn roll two dice. Each die roll allows one piece to be advanced the given number of points, to land on a point that is empty, occupied by friendly pieces, or occupied by only one enemy piece - a ‘blot’. The blot is removed and must begin from an imaginary off-board point before the starting point. This can cost moves. • Win by getting all pieces, first into the final 6 points, then to (or beyond) another imaginary off-board point beyond the end. • Gambling for money is usual. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

11. Lion’s-Share game: Go • Players take turns to place one stone on any unoccupied intersection of a (usually) 19x19 board, which is initially empty. • Blocks of same-colour strongly-connected stones are captured and removed if they are surrounded so that they have no adjacent empty intersection. • Capturing stones is part of the game, but not its object. One wins mainly by surrounding empty space in which opponent stones could not survive. • Rules are very simple, good play is very hard. • Note 4-rotation + mirror symmetry. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

12. If you are not already familiar with it, spend a few minutes on the 9-dot problem: • Draw four straight lines, joined end-to-end, to pass through all nine dots. • (Do not ask me questions about this now! ) http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

13. Finger-twitching or not Number of players: 1, 2, many Chance element or not Racing To Finish or Sharing Out Zero-Sum or not Mathematically Solved or not Kinds of symmetry Perfect Information or not Past moves, or current state Options of other players Simultaneous move or not Impartial rules or not Points of difference between various games (& puzzles) http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

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15. Last-Man-Standing game: Nine Men’s Morris • Beginning with an empty board, in phase 1 players alternately put a new peg (man) onto an empty intersection. If they get 3 in a row - a mill - they capture an opponent’s man. • When both players have placed all 9 men, phase 2 begins. A player may slide a man to an adjacent connected empty intersection, capturing an enemy man if making a new mill. • A player with exactly 3 men left may jump a man to any empty intersection, capturing if making a new mill. Win by reducing opponent to 2 men. • Note 4-way rotational symmetry, combined with two forms of mirror symmetry, without requiring colrev. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

16. A very simple example of a game where there is no capturing, merely a race to achieve an objective. There is often no winner: a draw occurs when no player may make a move. 4-way rotational symmetry with mirror symmetry. Last-Man-Standing race game: Tic-Tac-Toe (Noughts&Crosses) http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

17. Last-Man-Standing race game: Connect-Four • Players take turns to place one piece of their colour on the topmost empty cell of one of seven columns. • There is no capturing. Win by being first to get four pieces of your colour in a row, either vertically horizontally or diagonally. • There is only one mirror symmetry, left-to-right. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

18. Last-Man-Standing race game: Go-Moku / Five In A Row • Players take turns to put a piece of their colour onto (almost) any empty point in a large grid: sometimes infinite, sometimes a 19x19 Go board, … • The winner is the first to get five pieces in a row. • Note 4-rotational + mirror symmetry. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

19. Last-Man-Standing race game: Fox and Hounds / Fox and Geese • Played on an 8x8 chessboard. • One player controls the four hounds, which can move one square along diagonals in a forwards-only direction. The other player controls the fox, which moves one square along diagonals either forwards or backwards. No two pieces can occupy the same square. • Hounds win by trapping the fox. Fox wins by slipping through the line of hounds. • Note there is no symmetry. • Also the players are bound by different rules of movement. The rules are partisan not impartial. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

20. Last-Man-Standing game: Roshambo / Rock-Paper-Scissors • Two players simultaneously pick one of three options, • Rock wins over scisssors • Scissors wins over paper • Paper wins over Rock • There is arguably a 3-rotational cyclic symmetry here. • This is the first multi-player game we have seen to not feature turn-taking. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

21. Last-Man-Standing race game: Nim • Players take turns to remove 1, 2, or 3 items all from the same row. The one to take the last item is the winner. • (Alternative rule: the one to take the last item is the loser.) • There is some symmetry, since all rows are interchangeable, even though not all positions generated by symmetry are reachable. • The game is solved: there is an algorithm that can be followed for winning. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

22. Puzzles: Last-Man-Standing games versus “the rules”: Solitaire • Start with pieces occupying all pits except the central one. You may jump a piece over another into an empty pit, removing the piece you jumped. • You succeed (win) if you end with just one piece remaining, in the central pit. • You lose if you cannot move. • Note 4-way rotational + mirror symmetry, and also a symmetry between the two ends of the game: the finish is like the start, with empty and filled pits interchanged. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

23. You are given pieces of information from which you should be easily able to deduce pairs of attributes that could not go together, as well as pairs that do go together. Ultimately, you will be left with a few alternatives that must be enumerated in order to find the one consistent solution. There is typically a concealed symmetry, which can be made explicit with a tabular representation. Five men went to five separate cities on five different days by five modes of transport. Mr. Brown went on the train.. Mr. Green went to Galway. The bus went on Saturday. …. Who went where, how, and when? Puzzle: Logic puzzles http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

24. Puzzle: Kakuro • There is a grid like a crossword. • Each “word” across or down must be filled with digits 1 - 9, summing to a specified total, without duplicates. • Some “words” can be filled only by certain combinations of digits: eg • 7 in 3 cells: 1+2+4 • 34 in 5 cells: 4+6+7+8+9 • In “easy” kakuro puzzles there are several cells where highly-constrained words meet: so you can get started by fixing a few cell values. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

25. Game of Chance: Yahtzee • Any number can play. • Roll 5 dice per turn, up to three times in a turn, keeping as many as you like from previous roll that turn. • Add the pips on qualifying dice to make a score in one of the non-bonus categories. You may ultimately be forced score zero for some categories. • At the end, add your scores and bonuses, player with highest score wins. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

26. Game of chance and bluff: Liar Dice • For two or more players. • The first rolls the five dice, keeping them hidden, and passes them to the next player with a description of their goodness. Each player may then secretly re-roll all some or none, and pass them on with an ever more impressive description. • A player may choose not to accept the description given them. If the dice are at least as good as described, they lose; otherwise the player caught lying loses. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

27. Game of chance: Russian Roulette • Not recommended. • There is no winner, only a loser. • It is not, in game-theoretic terms, a “zero-sum game”. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

28. Lion’s-Share game: Mancala / Awari / Kalah • Each player controls the pits on one side of the board. On your turn, pick up the stones from one of your pits and add one of them to each succeeding pit in an anticlockwise direction. If the last pit you reach now has 2 or 3 stones, remove them, and then do likewise for the preceding pit and so on. • The winner is the one who captures the majority of the stones. • Note rotational 2-symmetry; all pieces alike. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

29. Lion’s-Share game: Othello / Reversi • Players begin with two pieces each as shown. They take turns, adding one new piece of their colour. All enemy pieces in a vertical/diagonal/horizontal line between the new piece and another friendly piece are replaced with friendly pieces. • In practice, 2-sided pieces are used. • A player who cannot move misses a turn. • Win by having more pieces than the opponent. • 4-way rotational + mirror symmetry. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

30. Lion’s-share games of chance: Poker, Bridge, Whist, Piquet, … • The use of playing cards, shuffled and dealt in secret, introduces an element of chance. • Some are strictly many-player games (poker, bridge), some are 2-player (Piquet, Nomination Whist). • I’d call Poker a lion’s-share game because although there is an outright winner of a single deal, there are usually many deals in a session. The winner is the one who wins most money overall. • In Bridge, Whist, etc, there are several tricks to be shared out among the players or teams; and winning is determined in terms of the share won - perhaps set against a “contract”. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

31. Lion’s-share game: Diplomacy • 2 to 7 players control Army and Navy pieces for seven countries. Every other turn, players may lose pieces or acquire pieces depending on their being the last to occupy certain “provinces”. • Attacks may be made on pieces occupying provinces, with both offence and defence fortified by neighbouring pieces. Moves are written secretly and revealed simultaneously. Conspiracy and treachery are both encouraged. • The winner is the first to control more than half of the salient provinces. • No symmetry; alliances are important. http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

32. Many real-world situations and problems can be viewed as games of a sort: Players have a choice of actions, Players have conflicting goals, Players may move sequentially or simultaneously, Alliances may prosper, Treachery may occur, Understanding of the goals of others may be useful in predicting their actions and planning actions of one’s own. Parlour games offer environments in which various kinds of simplification can be made in order to focus attention on particular AI issues: perception, representation, reasoning, learning, opponent modelling, and risk assessment. Stock Market War Passing legislation Hustling Cartels Fight for market share Biological evolution Industrial relations Democratic elections Takeover negotiations Games in the real world http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

33. Finger-twitching or not Number of players: 1, 2, many Chance element or not Racing To Finish or Sharing Out Zero-Sum or not Mathematically Solved or not Kinds of symmetry Perfect Information or not Past moves, or current state Options of other players Simultaneous move or not Impartial rules or not Points of difference between various games (& puzzles) http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles

34. The End http://csiweb.ucd.ie/Staff/acater/comp4031.htmlArtificial Intelligence for Games and Puzzles