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Winning concurrent reachability games requires doubly-exponential patience Michal Koucký IM AS CR, Prague Kristoffer Arnsfelt Hansen, Peter Bro Miltersen Aarhus U., Denmark

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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

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3 Entrance fee: $15 Win: $20 Observation: To break even, you need at least ¾ 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 ¾ is 10 25 years (one move per day). … the age of universe: 10 11 years

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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. … … … …

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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) – π : states actions. Memory-less (non-adaptive) – π : states actions. Adaptive – π : history actions. Adaptive – π : history actions. Probabilistic strategy: π gives a probability distribution of possible actions. Patience of a memory-less strategy π = 1/min non-zero prob. in π … [Everett ’57]

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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 – .

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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

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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.

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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 Ω( 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 ¾ is 10 25 years (one move per day).

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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

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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

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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?

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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 – .

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