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Exploring Reachability Objectives in Graph Games: Chess, Soccer, and Poker Mix

This workshop explores the intersection of Chess, Soccer, and Poker in graph games with reachability objectives, discussing Zermelo's theorem, player strategies, reachability concepts, and game models. Learn about deterministic and memoryless strategies, objectives, and applications in system verification and control. Discover how reachability games and winning sets determine player outcomes. Delve into chess theorems and the theory of simultaneous games, including examples like the Prisoner's Dilemma and penalty shoot-outs in soccer.

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Exploring Reachability Objectives in Graph Games: Chess, Soccer, and Poker Mix

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  1. Graph Games with Reachabillity Objectives: Mixing Chess, Soccer and Poker KrishnenduChatterjee 5th Workshop on Reachability Problems, Genova, Sept 30, 2011 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

  2. Games on Graphs • Games on graphs. • History • Zermelo’s theorem about Chess in 1913 • From every configuration • Either player 1 can enforce a win. • Or player 2 can enforce a win. • Or both players can enforce a draw.

  3. Chess: Games on Graph • Chess is a game on graph. • Configuration graph.

  4. Graphs vs. Games Two interacting players in games: Player 1 (Box) vs Player 2 (Diamond).

  5. Game Graph

  6. Game Graphs • A game graph G= ((S,E), (S1, S2)) • Player 1 states (or vertices) S1 and similarly player 2 states S2, and (S1, S2) partitions S. • E is the set of edges. • E(s) out-going edges from s, and assume E(s) non-empty for all s. • Game played by moving tokens: when player 1 state, then player 1 chooses the out-going edge, and if player 2 state, player 2 chooses the outgoing edge.

  7. Game Example

  8. Game Example

  9. Game Example

  10. Strategies • Strategies are recipe how to move tokens or how to extend plays. Formally, given a history of play (or finite sequence of states), it chooses a probability distribution over out-going edges. • ¾: S*S1 D(S). • ¼: S*S2! D(S).

  11. Strategies • Strategies are recipe how to move tokens or how to extend plays. Formally, given a history of play (or finite sequence of states), it chooses a probability distribution over out-going edges. • ¾: S*S1! D(S). • History dependent and randomized. • History independent: depends only current state (memoryless or positional). • ¾: S1! D(S) • Deterministic: no randomization (pure strategies). • ¾: S*S1! S • Deterministic and memoryless: no memory and no randomization (pure and memoryless and is the simplest class). • ¾: S1! S • Same notations for player 2 strategies ¼.

  12. Objectives • Objectives are subsets of infinite paths, i.e., õ S!. • Reachability: there is a set of good vertices (example check-mate) and goal is to reach them. Formally, for a set T if vertices or states, the objective is the set of paths that visit the target T at least once.

  13. Applications: Verification and Control of Systems • Verification and control of systems • Environment • Controller M satisfies property (Ã) E C

  14. Applications: Verification and Control of Systems • Verification and control of systems • Question: does there exists a controller that against all environment ensures the property. C M E satisfies property (Ã) || ||

  15. Game Models Applications -synthesis [Church, Ramadge/Wonham, Pnueli/Rosner] -model checking of open systems -receptiveness [Dill, Abadi/Lamport] -semantics of interaction [Abramsky] -non-emptiness of tree automata [Rabin, Gurevich/ Harrington] -behavioral type systems and interface automata[deAlfaro/ Henzinger] -model-based testing [Gurevich/Veanes et al.] -etc.

  16. Reachability Games • Pre(X): given a set X of states, Pre(X) is the set of states such that player 1 can ensure next state in X. X T

  17. Reachability Games • Pre(X): given a set X of states, Pre(X) is the set of states such that player 1 can ensure next state in X. X T

  18. Reachability Games • Pre(X): given a set X of states, Pre(X) is the set of states such that player 1 can ensure next state in X. X T

  19. Reachability Games • Pre(X): given a set X of states, Pre(X) is the set of states such that player 1 can ensure next state in X. • Fix-point X T

  20. Reachability Games • Winning set for a partition: Determinacy • Player 1 wins: then no matter what player 2 does, certainly reach the target. • Player 2 wins: then no matter what player 1 does, the target is never reached. • Memoryless winning strategies. • Can be computed in linear time [Beeri 81, Immerman 81].

  21. Chess Theorem • Zermelo’s Theorem Win1 Win2 Both draw

  22. Game Graphs Till Now • Game graphs we have seen till now • Many rounds (possibly infinite). • Turn-based.

  23. Simultaneous Games • Theory of rational behavior as game theory • von Neumann- Morgenstern games • Matrix zero-sum games R P S R (0,0) (-1,1) (1,-1) P (1,-1) (0,0) (-1,1) S (-1,1) (1,-1) (0,0)

  24. Simultaneous Games • Theory of rational behavior as game theory • von Neumann- Morgenstern games • Matrix zero-sum games R P S R (0,0) (-1,1) (1,-1) P (1,-1) (0,0) (-1,1) S (-1,1) (1,-1) (0,0)

  25. Simultaneous Games • Example: Prisoners dilemma. • Another example. R L C R (1,-1) (-1,1) (-1,1) L (-1,1) (1,-1) (-1,1) C (-1,1) (-1,1) (1,-1)

  26. Simultaneous Games • Example: Prisoners dilemma. • Another example. R L C R (1,-1) (-1,1) (-1,1) L (-1,1) (1,-1) (-1,1) C (-1,1) (-1,1) (1,-1)

  27. Simultaneous Games • Another example: Penalty shoot-out (Soccer) R L C R (1,-1) (-1,1) (-1,1) L (-1,1) (1,-1) (-1,1) C (-1,1) (-1,1) (1,-1)

  28. Chess Vs. Soccer (Penalty) • Chess: • Turn-based • Possibly infinite rounds • Theory of simultaneous games (Soccer) • Concurrent • One-shot (one-round) • Mix chess and soccer • Concurrent games on graphs

  29. Mixing Chess and Soccer: Concurrent Graph Games

  30. Concurrent Game Graphs • A concurrent game graph is a tuple G =(S,M,¡1,¡2,±) • S is a finite set of states. • M is a finite set of moves or actions. • ¡i: S !2Mn; is an action assignment function that assigns the non-empty set ¡i(s) of actions to player i at s, where i2 {1,2}. • ±: S £ M £ M ! S, is a transition function that given a state and actions of both players gives the next state.

  31. An Example: Snow-ball Game Hide Run Throw Wait run, wait hide, throw [Everett 57] run, throw R s hide, wait

  32. New Solution Concepts • Sure winning for turn-based. • New solution concepts • Almost-sure winning. • Limit-sure winning.

  33. Almost-sure Winning Example head, head tail, tail R s head, tail tail, head Almost-sure winning strategy: say head and tail with probability ½. Randomization is necessary.

  34. Concurrent reachabilitygames: limit-sure Hide Run Throw Wait run, wait hide, throw [Everett 57] run, throw R s hide, wait MoveProbability runq hide1-q (q>0) Win at s with probability 1-q, for all q> 0.

  35. Concurrent reachability games: limit-sure Hide MoveProbability runq hide1-q (q>0) Run Throw Wait Win at s with probability 1-q, for all q > 0. w = 0 1 1 run, wait hide, throw [Everett 57] run, throw R s hide, wait Player 1 cannot achieve w(s) = 1, only w(s) = 1-qfor all q > 0.

  36. Results for Concurrent Reachability Games • Sure winning: • Deterministic memoryless sufficient. • Linear time. • Almost-sure winning: • Randomization is necessary. • Randomized memoryless is sufficient. • Quadratic time algorithm. • Limit-sure winning: • Randomization is necessary. • Randomized memoryless is sufficient. • Quadratic time algorithm. • Results from [dAHK98, CdAH06, CdAH09]

  37. Games Till Now • Turn-based graph games • Concurrent graph games • Applications: again verification and synthesis with synchronous interaction. • Both these games are perfect-information games. Players know the precise state of the game. • The game of Poker: players play but do not know the perfect state of the game.

  38. Summary: Theory of Graph Games

  39. Mixing Chess and Poker: Partial-information Graph Games

  40. Why Partial-information • Perfect-information: controller knows everything about the system. • This is often unrealistic in the design of reactive systems because • systems have internal state not visible to controller (private variables) • noisy sensors entail uncertainties on the state of the game • Partial-observation Hidden variables = imperfect information. Sensor uncertainty = imperfect information.

  41. Partial-information Games • A PIG G =(L, A, , O) is as follows • L is a finite set of locations (or states). • A is a finite set of input letters (or actions). • µ L £ A £ L non-deterministic transition relation that for a state and an action gives the possible next states. • O is the set of observations and is a partition of the state space. The observation represents what is observable. • Perfect-information: O={{l} | l 2 L}.

  42. PIG: Example b a a,b b a

  43. New Solution Concepts • Sure winning: winning with certainty (in perfect information setting determinacy). • Almost-sure winning: win with probability 1. • Limit-sure winning: win with probability arbitrary close to 1. • We will illustrate the solution concepts with card games.

  44. Card Game 1 • Step 1: Player 2 selects a card from the deck of 52 cards and moves it from the deck (player 1 does not know the card). • Step 2: • Step 2 a: Player 2 shuffles the deck. • Step 2 b: Player 1 selects a card and view it. • Step 2 c: Player 1 makes a guess of the secret card or goes back to Step 2 a. • Player 1 wins if the guess is correct.

  45. Card Game 1 • Player 1 can win with probability 1: goes back to Step 2 a until all 51 cards are seen. • Player 1 cannot win with certainty: there are cases (though with probability 0) such that all cards are not seen. Then player 1 either never makes a guess or makes a wrong guess with positive probability.

  46. Card Game 2 • Step 1: Player 2 selects a new card from an exactly same deck and puts is in the deck of 52 cards (player 1 does not know the new card). So the deck has 53 cards with one duplicate. • Step 2: • Step 2 a: Player 2 shuffles the deck. • Step 2 b: Player 1 selects a card and view it. • Step 2 c: Player 1 makes a guess of the secret duplicate card or goes back to Step 2 a. • Player 1 wins if the guess is correct.

  47. Card Game 2 • Player 1 can win with probability arbitrary close to 1: goes back to Step 2 a for a long time and then choose the card with highest frequency. • Player 1 cannot win probability 1, there is a tiny chance that not the duplicate card has the highest frequency, but can win with probability arbitrary close to 1, (i.e., for all ² >0, player 1 can win with probability 1- ², in other words the limit is 1).

  48. Sure winning for Reachability • Result from [Reif 79] • Memory is required. • Exponential memory required. • Subset construction: what subsets of states player 1 can be. Reduction to exponential size turn-based games. • EXPTIME-complete.

  49. Almost-sure winning for Reachability • Result from [CDHR 06, CHDR 07] • Standard subset construction fails: as it captures only sure winning, and not same as almost-sure winning. • More involved subset construction is required. • EXPTIME-complete.

  50. Summary: Theory of Graph Games

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