1. Energy Parity Games Laurent Doyen LSV, ENS Cachan & CNRS Krishnendu ChatterjeeIST Austria ICALP’10 & GT-Jeux’10

2. Games for system design input output Spec: φ(input,output) System • Verification: check if a given system is correct •  reduces to graph searching

3. Games for system design ? input output Spec: φ(input,output) • Verification: check if a given system is correct •  reduces to graph searching • Synthesis : construct a correct system •  reduces to game solving – finding a winning strategy

4. Games for system design ? input output Spec: φ(input,output) • Verification: check if a given system is correct •  reduces to graph searching • Synthesis : construct a correct system •  reduces to game solving – finding a winning strategy • … in a game played on a graph: • - two players: System vs. Environment • - turn-based, infinite number of rounds • - winning condition given by the spec

5. Two-player games on graphs

6. Two-player games Player 1 (good guy) Player 2 (bad guy) • Turn-based • Infinite

7. Two-player games Player 1 (good guy) Player 2 (bad guy) • Turn-based • Infinite Play:

8. Two-player games Player 1 (good guy) Player 2 (bad guy) • Turn-based • Infinite Play:

9. Two-player games Player 1 (good guy) Player 2 (bad guy) • Turn-based • Infinite Play:

10. Two-player games Player 1 (good guy) Player 2 (bad guy) • Turn-based • Infinite Play:

11. Two-player games Player 1 (good guy) Player 2 (bad guy) • Turn-based • Infinite Play:

12. Two-player games Player 1 (good guy) Player 2 (bad guy) • Turn-based • Infinite Play:

13. Two-player games Player 1 (good guy) Player 2 (bad guy) • Turn-based • Infinite Play:

14. Two-player games Player 1 (good guy) Player 2 (bad guy) • Turn-based • Infinite Play:

15. Two-player games Player 1 (good guy) Player 2 (bad guy) • Turn-based • Infinite Play:

16. Two-player games Player 1 (good guy) Player 2 (bad guy) • Turn-based • Infinite Play:

17. Two-player games Player 1 (good guy) Player 2 (bad guy) • Turn-based • Infinite Strategies = recipe to extend the play prefix Player 1: outcome of two strategies is a play Player 2:

18. Two-player games Player 1 (good guy) Player 2 (bad guy) • Turn-based • Infinite • Winning condition ? Strategies = recipe to extend the play prefix Player 1: outcome of two strategies is a play Player 2:

19. Two-player games on graphs Who wins ? Qualitative Parity games ω-regular specifications (reactivity, liveness,…)

20. Two-player games on graphs Who wins ? Qualitative Quantitative Parity games Energy games ω-regular specifications (reactivity, liveness,…) Mean-payoff games Resource-constrainedspecifications

21. Energy games

22. Energy games Energy game: Positive and negative weights(encoded in binary)

23. Energy games Energy game: Positive and negative weights (encoded in binary) Play: 3 3 4 4 -1 … Energy level: 3, 3, 4, 4, -1,…(sum of weights)

24. Energy games Energy game: Positive and negative weights (encoded in binary) Play: 3 3 4 4 -1 … 3 Energy level: 3, 3, 4, 4, -1,… Initial credit A play is winning if the energy level is always nonnegative. “Never exhaust the resource (memory, battery, …)”(equivalent to a game on a one-counter automaton)

25. Energy games Initial credit problem: Decide if there exist an initial credit c0 and a strategy of player 1 to maintain the energy level always nonnegative. Player 1 (good guy)

26. Energy games Initial credit problem: Decide if there exist an initial credit c0 and a strategy of player 1 to maintain the energy level always nonnegative. For energy games, memoryless strategies suffice. Player 1 (good guy)

27. Energy games A memoryless strategy is winning if all cycles are nonnegative when is fixed. Initial credit problem: Decide if there exist an initial credit c0 and a strategy of player 1 to maintain the energy level always nonnegative.

28. Energy games A memoryless strategy is winning if all cycles are nonnegative when is fixed. Initial credit problem: Decide if there exist an initial credit c0 and a strategy of player 1 to maintain the energy level always nonnegative. See [CdAHS03, BFLMS08]

29. Energy games A memoryless strategy is winning if all cycles are nonnegative when is fixed. Initial credit problem: Decide if there exist an initial credit c0 and a strategy of player 1 to maintain the energy level always nonnegative. See [CdAHS03, BFLMS08]

30. Energy games A memoryless strategy is winning if all cycles are nonnegative when is fixed. c0=0 c0=15 Initial credit problem: Decide if there exist an initial credit c0 and a strategy of player 1 to maintain the energy level always nonnegative. c0=0 c0=0 c0=10 c0=0 Minimum initial credit can be computed in c0=5

31. Mean-payoff games Mean-payoff value of a play = limit-average of the visited weights Optimal mean-payoff value can be achieved with a memoryless strategy. Decision problem: Decide if there exists a strategy for player 1 to ensure mean-payoff value ≥ 0.

32. Mean-payoff games Mean-payoff value of a play = limit-average of the visited weights Optimal mean-payoff value can be achieved with a memoryless strategy. Decision problem: Decide if there exists a strategy for player 1 to ensure mean-payoff value ≥ 0.

33. Mean-payoff games Memoryless strategy σ ensures nonnegative mean-payoff value iff all cycles are nonnegative in Gσ iff memoryless strategy σ is winning in energy game Mean-payoff games with threshold 0 are equivalent to energy games. [BFLMS08]

34. Parity games

35. Parity games Parity game: integer priority on states

36. Parity games Parity game: integer priority on states Play: 5, 5, 5, 1, 0, 2, 4, 5, 5, 3… A play is winning if the least priority visited infinitely often is even. “Canonical representation of ω-regular specifications ” (e.g. all requests are eventually granted – G[r -> Fg])

37. Parity games Decision problem: Decide if there exists a winning strategy of player 1 for parity condition. Player 1 (good guy)

38. Parity games Decision problem: Decide if there exists a winning strategy of player 1 for parity condition. For parity games, memoryless strategies suffice. Player 1 (good guy)

39. Parity games A memoryless strategy is winning if all cycles are even when is fixed. Decision problem: Decide if there exists a winning strategy of player 1 for parity condition.

40. Parity games A memoryless strategy is winning if all cycles are even when is fixed. Decision problem: Decide if there exists a winning strategy of player 1 for parity condition. See [EJ91]

41. Summary Energy games - “never exhaust the resource” Parity game – “always eventually do something useful”

42. Energy parity games

43. Two-player games on graphs Qualitative Quantitative Parity games Energy games ω-regular specifications (reactivity, liveness,…) Mean-payoff games Mixed qualitative-quantitative Energy parity games (Mean-payoff parity games have been studied in Lics’05)

44. Energy parity games “never exhaust the resource”Energy parity games: and “always eventually do something useful” Decision problem: Decide the existence of a finite initial credit sufficient to win.

45. Energy parity games “never exhaust the resource”Energy parity games: and “always eventually do something useful”

46. Summary Energy games - “never exhaust the resource” Parity game – “always eventually do something useful”

47. Energy parity games “never exhaust the resource”Energy parity games: and “always eventually do something useful” Decision problem: Decide the existence of a finite initial credit sufficient to win. Energy games – a story of cycles Parity games – a story of cycles Energy parity games - ?

48. A story of cycles A one-player energy Büchi game

49. A story of cycles A good cycle ? good for parity good for energy

50. A story of cycles Winning strategy: 1. reach and repeat positive cycle to increase energy; 2. visit priority 0; 3. goto 1;