# Krishnendu Chatterjee1 Graph Games with Reachabillity Objectives: Mixing Chess, Soccer and Poker Krishnendu Chatterjee 5 th Workshop on Reachability Problems,

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<ul><li> Slide 1 </li> <li> Krishnendu Chatterjee1 Graph Games with Reachabillity Objectives: Mixing Chess, Soccer and Poker Krishnendu Chatterjee 5 th Workshop on Reachability Problems, Genova, Sept 30, 2011 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A </li> <li> Slide 2 </li> <li> Krishnendu Chatterjee2 Games on Graphs Games on graphs. History Zermelos 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. </li> <li> Slide 3 </li> <li> Krishnendu Chatterjee3 Chess: Games on Graph Chess is a game on graph. Configuration graph. </li> <li> Slide 4 </li> <li> Krishnendu Chatterjee4 Graphs vs. Games Two interacting players in games: Player 1 (Box) vs Player 2 (Diamond). </li> <li> Slide 5 </li> <li> Krishnendu Chatterjee5 Game Graph </li> <li> Slide 6 </li> <li> Krishnendu Chatterjee6 Game Graphs A game graph G= ((S,E), (S 1, S 2 )) Player 1 states (or vertices) S 1 and similarly player 2 states S 2, and (S 1, S 2 ) 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. </li> <li> Slide 7 </li> <li> Krishnendu Chatterjee7 Game Example </li> <li> Slide 8 </li> <li> Krishnendu Chatterjee8 Game Example </li> <li> Slide 9 </li> <li> Krishnendu Chatterjee9 Game Example </li> <li> Slide 10 </li> <li> Krishnendu Chatterjee10 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. : S * S 1 D(S). : S * S 2 ! D(S). </li> <li> Slide 11 </li> <li> Krishnendu Chatterjee11 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. : S * S 1 ! D(S). History dependent and randomized. History independent: depends only current state (memoryless or positional). : S 1 ! D(S) Deterministic: no randomization (pure strategies). : S * S 1 ! S Deterministic and memoryless: no memory and no randomization (pure and memoryless and is the simplest class). : S 1 ! S Same notations for player 2 strategies . </li> <li> Slide 12 </li> <li> Krishnendu Chatterjee12 Objectives Objectives are subsets of infinite paths, i.e., 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. </li> <li> Slide 13 </li> <li> Krishnendu Chatterjee13 Applications: Verification and Control of Systems Verification and control of systems Environment Controller M satisfies property ( ) E C </li> <li> Slide 14 </li> <li> Krishnendu Chatterjee14 Applications: Verification and Control of Systems Verification and control of systems Question: does there exists a controller that against all environment ensures the property. M satisfies property ( ) EC || </li> <li> Slide 15 </li> <li> Krishnendu Chatterjee15 Applications: Systems for Specification Synthesis of systems from specification Input/Output signals. Automata over I/O that specifies the desired set of behaviors. Can the input player present input such that no matter how the output player plays the generated sequence of I/O signals is accepted by automata ? Deterministic automata: Games. </li> <li> Slide 16 </li> <li> Krishnendu Chatterjee16 -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. Game Models Applications </li> <li> Slide 17 </li> <li> Krishnendu Chatterjee17 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 </li> <li> Slide 18 </li> <li> Krishnendu Chatterjee18 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 </li> <li> Slide 19 </li> <li> Krishnendu Chatterjee19 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 </li> <li> Slide 20 </li> <li> Krishnendu Chatterjee20 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 </li> <li> Slide 21 </li> <li> Krishnendu Chatterjee21 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]. </li> <li> Slide 22 </li> <li> Krishnendu Chatterjee22 Chess Theorem Zermelos Theorem Win 1 Win 2 Both draw </li> <li> Slide 23 </li> <li> Krishnendu Chatterjee23 Game Graphs Till Now Game graphs we have seen till now Many rounds (possibly infinite). Turn-based. </li> <li> Slide 24 </li> <li> Krishnendu Chatterjee24 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) </li> <li> Slide 25 </li> <li> Krishnendu Chatterjee25 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) </li> <li> Slide 26 </li> <li> Krishnendu Chatterjee26 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) </li> <li> Slide 27 </li> <li> Krishnendu Chatterjee27 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) </li> <li> Slide 28 </li> <li> Krishnendu Chatterjee28 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) </li> <li> Slide 29 </li> <li> Krishnendu Chatterjee29 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 </li> <li> Slide 30 </li> <li> Krishnendu Chatterjee30 Mixing Chess and Soccer: Concurrent Graph Games </li> <li> Slide 31 </li> <li> Krishnendu Chatterjee31 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 ! 2 M n ; is an action assignment function that assigns the non-empty set i (s) of actions to player i at s, where i 2 {1,2}. : S M M ! S, is a transition function that given a state and actions of both players gives the next state. </li> <li> Slide 32 </li> <li> Krishnendu Chatterjee32 An Example: Snow-ball Game s R run, wait hide, throw hide, wait run, throw [Everett 57] Run Hide Throw Wait </li> <li> Slide 33 </li> <li> Krishnendu Chatterjee33 New Solution Concepts Sure winning for turn-based. New solution concepts Almost-sure winning. Limit-sure winning. </li> <li> Slide 34 </li> <li> Krishnendu Chatterjee34 Almost-sure Winning Example s R head, head tail, tail head, tail tail, head Almost-sure winning strategy: say head and tail with probability . Randomization is necessary. </li> <li> Slide 35 </li> <li> Krishnendu Chatterjee35 Concurrent reachability games: limit-sure s R run, wait hide, throw hide, wait run, throw [Everett 57] Move Probability run q hide 1-q (q>0) Win at s with probability 1-q, for all q > 0. Run Hide Throw Wait </li> <li> Slide 36 </li> <li> Krishnendu Chatterjee36 Concurrent reachability games: limit-sure s R run, wait hide, throw hide, wait run, throw Run Hide Throw Wait [Everett 57] Move Probability run q hide 1-q (q>0) Win at s with probability 1-q, for all q > 0. w = 011 Player 1 cannot achieve w(s) = 1, only w(s) = 1-q for all q > 0. </li> <li> Slide 37 </li> <li> Krishnendu Chatterjee37 Concurrent reachability games Almost-sure winning: requires alternation of mu-calculus formula. Positive to lower rank and with probability 1 in the almost-sure winning set. R(1) R R(0) </li> <li> Slide 38 </li> <li> Krishnendu Chatterjee38 Concurrent reachability games Limit-sure winning: requires alternation of mu-calculus formula. Positive to lower rank and can allow to escape but with vanishing probability. Let be green prob, and be red prob, then it can be ensured that , for all >0. R(1) R R(0) </li> <li> Slide 39 </li> <li> Krishnendu Chatterjee39 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] </li> <li> Slide 40 </li> <li> Krishnendu Chatterjee40 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. </li> <li> Slide 41 </li> <li> Krishnendu Chatterjee41 Summary: Theory of Graph Games Winning Mode/ Game Graphs SureAlmost-sureLimit-sure Turn-based Games (CHESS) Linear time (PTIME-complete) Linear-time (PTIME-complete) Linear-time (PTIME-complete) Concurrent Games (CHESS+ SOCCER) Linear time (PTIME-complete) Quadratic time (PTIME-complete) Quadratic time (PTIME-complete) Partial-information Games (CHESS + SOCCER+ POKER) </li> <li> Slide 42 </li> <li> Krishnendu Chatterjee42 Mixing Chess and Poker: Partial-information Graph Games </li> <li> Slide 43 </li> <li> Krishnendu Chatterjee43 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. </li> <li> Slide 44 </li> <li> Krishnendu Chatterjee44 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). 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}. </li> <li> Slide 45 </li> <li> Krishnendu Chatterjee45 PIG: Example a,b a ba b </li> <li> Slide 46 </li> <li> Krishnendu Chatterjee46 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. </li> <li> Slide 47 </li> <li> Krishnendu Chatterjee47 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. </li> <li> Slide 48 </li> <li> Krishnendu Chatterjee48 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. </li> <li> Slide 49 </li> <li> Krishnendu Chatterjee49 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. </li> <li> Slide 50 </li> <li> Krishnendu Chatterjee50 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 >0, player 1 can win with probability 1- , in other words the limit is 1). </li> <li> Slide 51 </li> <li> Krishnendu Chatterjee51 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. </li> <li> Slide 52 </li> <li> Krishnendu Chatterjee52 Partial-information Games a a a a b b a b a a b b In starting play a. In yellow play a and b at random. In purple: if last was yellow then a if last was starting, then b. Requires both randomization and memory </li> <li> Slide 53 </li> <li> Krishnendu Chatterjee53 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. </li> <li> Slide 54 </li> <li> Krishnendu Chatterjee54 Summary: Theory of Graph Games Winning Mode/ Game Graphs SureAlmost-sureLimit-sure Turn-based Games (CHESS) Linear time (PTIME-complete) Linear-time (PTIME-complete) Linear-time (PTIME-complete) Concurrent Games (CHESS+ SOCCER) Linear time (PTIME-complete) Quadratic time (PTIME-complete) Quadratic time (PTIME-complete) Partial-information Games (CHESS + SOCCER+ POKER) EXPTIME-complete </li> <li> Slide 55 </li> <li> Krishnendu Chatterjee55 Limit-sure winning for Reachability Limit-sure winning for reachability is undecidable [GO 10, CH 10]. Reduction from the Post-correspondence problem (PCP). </li> <li> Slide 56 </li> <li> Krishnendu Chatterjee56 Mixing Ch...</li></ul>

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