Logica voor Informatica - 12 – Normaalvormen · Vervulbaarequivalenteformules...
Transcript of Logica voor Informatica - 12 – Normaalvormen · Vervulbaarequivalenteformules...
Modelling Computing Systems
The book is called Modelling Computing Systems - but so far we have studied:
• sets• functions• relations• propositional and predicate logic• induction
... but these are all very static concepts. There's hardly any computation or interaction.
Games
Today we will study how to apply these ideas to model a series of interactions.
Interactions can be:
• computers interacting
• brokers buying/selling shares, etc.
We focus on a more restricted form of interactions, namely games.
Games
Not computer games. Instead focus on well-specified interactions between players.
The players can:
• win the game• lose the game• potentially draw the game
Chance in games
We can distinguish between different kinds of games:
• Games of chance - like roulette - where there is no good strategy for winning.
• Games of no chance - like tic-tac-toe - where it is clear how to win.
These are two extremes: the most fun games have some element of luck (e.g. monopoly).
Games of no chance
We'll study the games without any chance element.
Furthermore, we focus on games of perfect information (e.g. chess).
Finite games - if the game could continue indefinitely call it e.g. “a draw”
Rules according to which the player makes a move on each turn make a strategy.
Strategies
Winning strategy - guarantees a victory regardless the moves of the opponent
Drawing strategy - guarantees that the opponent will not win regardless the moves of the opponent
A position in a game can be:
• winning - there is a winning strategy for the current player• losing - there is a winning strategy for the non-current player• drawing - neither player has a winning strategy
Strategies
Theorem In a two-player game-of-no-chance of perfect information, either one the two players has a winning strategy, or they both have drawing strategies.
Proof
Case 1: Player X has a winning strategy
Claim: The other one cannot have a winning strategy
If we fix the two strategies only one player will win and only the strategy of that player was the winning one
Proof
Case 2: Neither player has a winning strategy
Claim: Both players have a drawing strategy
First player (X) doesn’t have a winning strategy, so player O can always respond in order to win or draw. Ensuring that player X doesn’t win.
Similarly, player X can ensure that the player O doesn’t win.
Thus, they both have a drawing strategies.
Proof
Case 2: Neither player has a winning strategy
Claim: Both players have a drawing strategy
First player (X) doesn’t have a winning strategy, so player O can always respond in order to win or draw. Ensuring that player X doesn’t win.
Similarly, player X can ensure that the player O doesn’t win.
Thus, they both have a drawing strategies.
Strategies
Theorem In a two-player game-of-no-chance of perfect information, either of the players has a winning strategy or they both have drawing strategies.
Corollary If a game cannot end in a draw, one of the two players has a winning strategy.
Example game
Game: Starting with a pile of 10 coins, 2 players take turns removing either 2 or 3 coins from the pile. The winner is the one that takes the last pile; if one coin remains, then the game is a draw.
Example game
Game: Starting with a pile of 10 coins, 2 players take turns removing either 2 or 3 coins from the pile. The winner is the one that takes the last pile; if one coin remains, then the game is a draw.
Nim
Game:
• Board consists of an arbitrary number of piles, where each has arbitrary number of coins
• 2 players take turns removing each time 1 or more coins from exactly one pile
• Goal is to take the last coin
Nim
Game:
• Board consists of an arbitrary number of piles, where each has arbitrary number of coins
• 2 players take turns removing each time 1 or more coins from exactly one pile
• Goal is to take the last coin
Nim
Game:
• Board consists of an arbitrary number of piles, where each has arbitrary number of coins
• 2 players take turns removing each time 1 or more coins from exactly one pile
• Goal is to take the last coin
Nim
Game:
• Board consists of an arbitrary number of piles, where each has arbitrary number of coins
• 2 players take turns removing each time 1 or more coins from exactly one pile
• Goal is to take the last coin
Nim
Game:
• Board consists of an arbitrary number of piles, where each has arbitrary number of coins
• 2 players take turns removing each time 1 or more coins from exactly one pile
• Goal is to take the last coin
Nim
Game:
• Board consists of an arbitrary number of piles, where each has arbitrary number of coins
• 2 players take turns removing each time 1 or more coins from exactly one pile
• Goal is to take the last coin
Nim
Winning strategies
• 1 pile game (W)• 2 pile game
• piles are equal (L)• piles are not equal (W)
Nim
Winning strategies
• 1 pile game (W)• 2 pile game
• piles are equal (L)• piles are not equal (W)
• 3 pile game• 2 piles are equal
Nim
Winning strategies
• 1 pile game (W)• 2 pile game
• piles are equal (L)• piles are not equal (W)
• 3 pile game• 2 piles are equal (W)
Nim
Winning strategies
• 1 pile game (W)• 2 pile game
• piles are equal (L)• piles are not equal (W)
• 3 pile game• 2 piles are equal (W)• piles contain 1, 2 and 3 coins
Nim
Winning strategies
• 1 pile game (W)• 2 pile game
• piles are equal (L)• piles are not equal (W)
• 3 pile game• 2 piles are equal (W)• piles contain 1, 2 and 3 coins
Nim
Winning strategies
• 1 pile game (W)• 2 pile game
• piles are equal (L)• piles are not equal (W)
• 3 pile game• 2 piles are equal (W)• piles contain 1, 2 and 3 coins
Nim
Winning strategies
• 1 pile game (W)• 2 pile game
• piles are equal (L)• piles are not equal (W)
• 3 pile game• 2 piles are equal (W)• piles contain 1, 2 and 3 coins
Nim
Winning strategies
• 1 pile game (W)• 2 pile game
• piles are equal (L)• piles are not equal (W)
• 3 pile game• 2 piles are equal (W)• piles contain 1, 2 and 3 coins
Nim
Winning strategies
• 1 pile game (W)• 2 pile game
• piles are equal (L)• piles are not equal (W)
• 3 pile game• 2 piles are equal (W)• piles contain 1, 2 and 3 coins
Nim
Winning strategies
• 1 pile game (W)• 2 pile game
• piles are equal (L)• piles are not equal (W)
• 3 pile game• 2 piles are equal (W)• piles contain 1, 2 and 3 coins (L)
Nim
Winning strategies
• 1 pile game (W)• 2 pile game
• piles are equal (L)• piles are not equal (W)
• 3 pile game• 2 piles are equal (W)• piles contain 1, 2 and 3 coins (L)• arbitrary number of coins (?)
Nim
Winning strategies
• 1 pile game (W)• 2 pile game
• piles are equal (L)• piles are not equal (W)
• 3 pile game• 2 piles are equal (W)• piles contain 1, 2 and 3 coins (L)• arbitrary number of coins (?)
Nim
Universal winning strategy:
If all of the columns have even parity, the position is balanced, otherwise it’s unbalanced
Nim
Observation:
1. If the position is balanced, then every move will lead to an unbalanced position
2. If the position is unbalanced, then there exists a move that will lead to a balanced position
3. Empty board is a balanced position
Nim
Universal winning strategy:
3) If the position is unbalanced (W), player should make a move to make it balanced each time and he is insured to win
Nim
Universal winning strategy:
4) If the position is balanced (L), the opponent has a chance to use the same strategy to win
Chomp
Game:
• we have a n x m bar where leftmost-topmost square is poisonous• two players take turns to bite of the bar, where each player has to
choose a remaining square and eat all the squares below it and to the right from it
• the player that eats the poisonous square lost the game
Chomp
Game:
• we have a n x m bar where leftmost-topmost square is poisonous• two players take turns to bite of the bar, where each player has to
choose a remaining square and eat all the squares below it and to the right from it
• the player that eats the poisonous square lost the game
Chomp
Strategies:
• 1 x 1 (L)• 1 x n (W) - take all the squares apart from the poisonous one• 2 x n (W) - take the bottom-right square (and keep that structure)
Chomp
Strategies:
• 1 x 1 (L)• 1 x n (W) - take all the squares apart from the poisonous one• 2 x n (W) - take the bottom-right square (and keep that structure)• n x n
Chomp
Strategies:
• 1 x 1 (L)• 1 x n (W) - take all the squares apart from the poisonous one• 2 x n (W) - take the bottom-right square (and keep that structure)• n x n (W) - remove (n-1) x (n-1) squares and mimic the payer
response in the following moves
Chomp
Strategies:
• 1 x 1 (L)• 1 x n (W) - take all the squares apart from the poisonous one• 2 x n (W) - take the bottom-right square (and keep that structure)• n x n (W) - remove (n-1) x (n-1) squares and mimic the payer
response in the following moves• n x m (?)
Chomp
Theorem: Except for the 1 x 1 case, the first player always has a winning strategy.
Proof:
Suppose that the second player has a winning strategy.
Draw a contradiction.
Hex
Game:
• played on a board consisting of n x n hexagons• territories in NE and SW belong to player 1• territories in SE and NW belong to player 2• players take turns marking empty cells with their respective
symbol• the winner is the player that connects the two disconnected
territories first with an unbroken chain of their symbols
Hex
Game:
• played on a board consisting of n x n hexagons• territories in NE and SW belong to player 1• territories in SE and NW belong to player 2• players take turns marking empty cells with their respective
symbol• the winner is the player that connects the two disconnected
territories first with an unbroken chain of their symbols
Bridg-it
Game:
• played on a staggered n x n board• a pair of opposing borders belongs to each player (S&N and W&E)• players take turns connecting neighbouring dots of their respective
color, horizontally or vertically• the winner is the player that connects their the
two opposite borders• 2 lines are not allowed to cross
Conclusion
We have learned:
• How to model player interactions• How to formalise a game• How to evaluate the quality of a position• How mathematical theory can be used to make a winning strategy
These concepts are quite useful in Game theory - application of agent interactions in economics, social sciences, computer science, etc. (e.g. auction, bargaining)
Agents might have different goals and potentially cooperate