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*Winning concurrent reachability games requires doubly-exponential patience Michal Kouck½ IM...*

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Winning concurrent reachability games requires doubly-exponential patience Michal Kouck IM AS CR, Prague Kristoffer Arnsfelt Hansen, Peter Bro Miltersen Aarhus U., Denmark Slide 2 2 Example Player 1 chooses A {t,h} Player 1 chooses A {t,h} Player 2 chooses B {t,h} Player 2 chooses B {t,h}If A = B then move one level up, A = B then move one level up, A B = t then move to 1 st level, A B = t then move to 1 st level, A B = h then Player 1 loses. A B = h then Player 1 loses. Entrance fee: $15 Win: $20 W 7 6 5 4 3 2 1 Slide 3 3 Entrance fee: $15 Win: $20 Observation: To break even, you need at least probability to win. Good news: you can win with probability arbitrary close to 1. Bad news: the expected time to win the game with probability at least is 10 25 years (one move per day). the age of universe: 10 11 years Slide 4 4 Concurrent reachability games [de Alfaro, Henzinger, Kupferman 98, Everett 57] Two players play on a graph of states. At each step they simultaneously (independently) pick one of possible actions each and based on a transition table move to the next state. Slide 5 5 Goals:Player 1 wants to reach a specific state or states. Player 2 wants to prevent Player 1 from reaching these states. Strategy of a player: Memory-less (non-adaptive) : states actions. Memory-less (non-adaptive) : states actions. Adaptive : history actions. Adaptive : history actions. Probabilistic strategy: gives a probability distribution of possible actions. Patience of a memory-less strategy = 1/min non-zero prob. in [Everett 57] Slide 6 6 Winning starting states: Sure Player 1 has a winning strategy that never fails. Sure Player 1 has a winning strategy that never fails. Almost-Sure Player 1 has a randomized strategy that reaches goal with probability 1. Almost-Sure Player 1 has a randomized strategy that reaches goal with probability 1. Limit-Sure For every > 0, Player 1 has a strategy that reaches goal with probability at least 1 . Limit-Sure For every > 0, Player 1 has a strategy that reaches goal with probability at least 1 . Slide 7 7 Purgatory n Player 1 chooses A {t,h} Player 1 chooses A {t,h} Player 2 chooses B {t,h} Player 2 chooses B {t,h}If A = B then move one level up, A = B then move one level up, A B = t then move to 1 st level, A B = t then move to 1 st level, A B = h then move to state H. A B = h then move to state H. P n n-1 3 2 1 H Slide 8 8 Our results Thm:1) For every 0 1/ 2 n-2. 2) For every l 2 2 n-l-2. Thm:For every 0 61 actions in total, both players have -optimal strategies with patience 61 actions in total, both players have -optimal strategies with patience < 1/ 2 42m. Slide 9 9 Thm:1) For every 0 t then the expected time to win the game by any -optimal strategy of Player 1 can be forced to be ( t ). patience ~ expected time to win patience ~ expected time to win All the results essentially hold also for adaptive strategies All the results essentially hold also for adaptive strategies Recall: the expected time to win Purgatory 7 with probability at least is 10 25 years (one move per day). Slide 10 10 Algorithmic consequences Three algorithmic questions: 1. What are *-SURE states? PTIME [dAHK] 2. What are the winning probabilities of different states? PSPACE [EY] 3. What is the ( -)optimal strategy? EXP-EXP-TIME upper-bound [CdAH,] EXP-SPACE lower-bound [our results] Cor: Any algorithm that manipulates winning strategies in explicit representation must use exponential space. explicit representation: integer fractions Slide 11 11 Purgatory n p i probability of playing t in state i in -optimal strategy of Player 1. p i probability of playing t in state i in -optimal strategy of Player 1. Claim: 1) 0< p i < 1, for all i. 2) p i < , for all i. 3) p 1 p 2. p 3 p n 4) p i p i+1. p i+2 p n P n n-1 3 2 1 1\2th tlevel+1loss hlevel=1level+1 p n p n-1 p3p3p2p2p1p1p3p3p2p2p1p1 Player 2 plays h Player 2 plays t Player 2 plays h t t t t t Slide 12 12 Open problems Generic algorithm for - optimal strategy with symbolic representation? Generic algorithm for - optimal strategy with symbolic representation? How to redefine the game to be more realistic? How to redefine the game to be more realistic? Slide 13 13 Goals:Player 1 wants to reach a specific state or states. Player 2 wants to prevent Player 1 from reaching these states. Winning starting states: Sure Player 1 has a winning strategy that never fails. Sure Player 1 has a winning strategy that never fails. Almost-Sure Player 1 has a randomized strategy that reaches goal with probability 1. Almost-Sure Player 1 has a randomized strategy that reaches goal with probability 1. Limit-Sure For every > 0, Player 1 has a strategy that reaches goal with probability at least 1 . Limit-Sure For every > 0, Player 1 has a strategy that reaches goal with probability at least 1 .