Energy Parity Games

1. Energy Parity Games Laurent DoyenLSV, ENS Cachan & CNRS Krishnendu ChatterjeeIST Austria

2. Analysis - Synthesis Synthesis System Implementation Algorithm “How” Specification Properties Model “What” Verification Analysis

3. Analysis - Synthesis Synthesis System Implementation Algorithm “How” Specification Properties Model “What” Verification Analysis • Reactive Systems (e.g. servers, hardware,…) • Interact with environment • Non-terminating • Finite data (or data abstractions) • Control-oriented

4. Two-player games on graphs • Games for synthesis • - Reactive system synthesis = finding a winning strategy in a two-player game • - ω-regular spec : safety, reactivity, … • - quantitative spec : resource constraints • Game played on a finite graph • - Infinite number of rounds • - Player’s moves determine successor state • Outcome = infinite path in the graph

5. Two-player games

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 on graphs Qualitative Quantitative Parity games Energy games ω-regular specifications (reactivity, liveness,…) Resource-constrainedspecifications

19. Two-player games on graphs Qualitative Quantitative Parity games Energy games ω-regular specifications (reactivity, liveness,…) Resource-constrainedspecifications Mixed qualitative-quantitative Energy parity games

20. Energy games

21. Energy games Energy game: Positive and negative weights

22. Energy games Energy game: Positive and negative weights Play: Energy level: 3, 3, 4, 4, 3, 2, 1,…(sum of weights)

23. Energy games Energy game: Positive and negative weights Play: 3 Energy level: 3, 3, 4, 4, 3, 2, 1,… Initial credit A play is winning if the energy level is always nonnegative. “Never exhaust the resource (memory, battery, …)”

24. 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)

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. For energy games, memoryless strategies suffice. Player 1 (good guy)

26. 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.

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. See [CdAHS03, BFLMS08]

28. Energy games A memoryless strategy is winning if all cycles are nonnegative when is fixed. c0=0 c0=3 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=2 c0=0 Minimum initial credit can be computed in c0=1

29. Parity games

30. Parity games Parity game: integer priority on states

31. 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])

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

33. 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)

34. 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.

35. 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]

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

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

38. Energy parity games

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

40. 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 - ?

41. A story of cycles A good cycle ?

42. A story of cycles A good cycle ? A bad cycle ? - energy is (strictly) negative - or, least priority is odd

43. A story of cycles A good cycle ? A bad cycle ? - energy is (strictly) negative - or, least priority is odd Player 1 looses energy parity game iff the opponent can force a bad cycle. The opponent can force bad cycles without memory.

44. A story of cycles In energy parity games, memoryless strategies are sufficient for Player 2. Proof

45. A story of cycles In energy parity games, memoryless strategies are sufficient for Player 2. Proof Preliminary fact: under optimal strategy, energy in q is always greater than on first visit to q.

46. A story of cycles In energy parity games, memoryless strategies are sufficient for Player 2. cr Proof Assume player 1 looses with initial credit cr, then show that player 1 looses also against one of the “memoryless strategies in q”: cl ≤ cr

47. A story of cycles In energy parity games, memoryless strategies are sufficient for Player 2. cr Proof Fix winning strategy of Player 2, all outcomes are loosing for Player 1:

48. A story of cycles In energy parity games, memoryless strategies are sufficient for Player 2. cr Proof Fix winning strategy of Player 2, all outcomes are loosing for Player 1: Energy < 0 ? Then also in since

49. A story of cycles In energy parity games, memoryless strategies are sufficient for Player 2. cr Proof Fix winning strategy of Player 2, all outcomes are loosing for Player 1: Least ∞-visited priority is odd ? Then also in or

50. Complexity