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Limit Your Consumption! Finding Bounds in Average-energy Games Joint work with Kim G. Larsen and Simon Laursen (Aalborg University) Martin Zimmermann Saarland University April, 3nd 2016 QAPL 16 Martin Zimmermann Saarland University Finding

  1. Limit Your Consumption! Finding Bounds in Average-energy Games Joint work with Kim G. Larsen and Simon Laursen (Aalborg University) Martin Zimmermann Saarland University April, 3nd 2016 QAPL 16 Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 1/14

  2. Motivation Shift from programs to reactive systems: non-terminating interacting with a possibly antagonistic environment communication-intensive Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 2/14

  3. Motivation Shift from programs to reactive systems: non-terminating interacting with a possibly antagonistic environment communication-intensive Successful approach to verification and synthesis: an infinite game between the system and its environment: two players infinite duration perfect information system player wins if specification is satisfied Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 2/14

  4. Motivation Shift from programs to reactive systems: non-terminating interacting with a possibly antagonistic environment communication-intensive Successful approach to verification and synthesis: an infinite game between the system and its environment: two players infinite duration perfect information system player wins if specification is satisfied Here: graph-based games with quantitative winning conditions modeling consumption of a ressource Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 2/14

  5. An Example v 2 v 0 v 1 Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  6. An Example v 2 v 0 v 1 A play: v 0 Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  7. An Example v 2 v 0 v 1 A play: v 0 v 2 Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  8. An Example v 2 v 0 v 1 A play: v 0 v 2 v 1 Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  9. An Example v 2 v 0 v 1 A play: v 0 v 2 v 1 v 0 Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  10. An Example v 2 v 0 v 1 A play: v 0 v 2 v 1 v 0 v 2 Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  11. An Example v 2 v 0 v 1 A play: v 0 v 2 v 1 v 0 v 2 v 0 Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  12. An Example v 2 v 0 v 1 A play: v 0 v 2 v 1 v 0 v 2 v 0 v 1 Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  13. An Example v 2 v 0 v 1 A play: · · · v 0 v 2 v 1 v 0 v 2 v 0 v 1 Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  14. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  15. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A play (with energy levels): ( v 0 , 0) Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  16. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A play (with energy levels): ( v 0 , 0) ( v 2 , 3) Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  17. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A play (with energy levels): ( v 0 , 0) ( v 2 , 3) ( v 1 , 0) Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  18. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A play (with energy levels): ( v 0 , 0) ( v 2 , 3) ( v 1 , 0) ( v 0 , 0) Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  19. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A play (with energy levels): ( v 0 , 0) ( v 2 , 3) ( v 1 , 0) ( v 0 , 0) ( v 2 , 3) Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  20. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A play (with energy levels): ( v 0 , 0) ( v 2 , 3) ( v 1 , 0) ( v 0 , 0) ( v 2 , 3) ( v 0 , 5) Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  21. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A play (with energy levels): ( v 0 , 0) ( v 2 , 3) ( v 1 , 0) ( v 0 , 0) ( v 2 , 3) ( v 0 , 5) ( v 1 , 4) Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  22. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A play (with energy levels): ( v 0 , 0) ( v 2 , 3) ( v 1 , 0) ( v 0 , 0) ( v 2 , 3) ( v 0 , 5) ( v 1 , 4) · · · Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  23. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A strategy: ( v 0 , 0) Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  24. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A strategy: ( v 0 , 0) ( v 2 , 3) Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  25. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A strategy: ( v 1 , 0) ( v 0 , 0) ( v 2 , 3) ( v 0 , 5) Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  26. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A strategy: ( v 1 , 0) ( v 0 , 0) ( v 2 , 3) ( v 0 , 5) ( v 1 , 4) Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  27. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A strategy: ( v 1 , 0) ( v 0 , 0) ( v 2 , 3) ( v 0 , 5) ( v 1 , 4) Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  28. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A strategy: ( v 1 , 0) ( v 0 , 0) ( v 2 , 3) ( v 0 , 5) ( v 1 , 4) Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  29. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A strategy: ( v 1 , 0) ( v 0 , 0) ( v 2 , 3) Energy level always between 0 and 5 ( v 0 , 5) ( v 1 , 4) Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  30. An Example 3 -1 v 2 2 -3 0 v 0 v 1 -1 A strategy: ( v 1 , 0) ( v 0 , 0) ( v 2 , 3) Energy level always between 0 and 5 ( v 0 , 5) ( v 1 , 4) Average energy level at most 4 Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 3/14

  31. Previous Work objective Complexity Memory Requirements EG L NP ∩ co-NP memoryless EG LU ExpTime -complete pseudopolynomial Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 4/14

  32. Previous Work objective Complexity Memory Requirements EG L NP ∩ co-NP memoryless EG LU ExpTime -complete pseudopolynomial AE NP ∩ co-NP memoryless AE LU ExpTime -complete pseudopolynomial Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 4/14

  33. Previous Work objective Complexity Memory Requirements EG L NP ∩ co-NP memoryless EG LU ExpTime -complete pseudopolynomial AE NP ∩ co-NP memoryless AE LU ExpTime -complete pseudopolynomial AE LU ( U − L poly) NP ∩ co-NP polynomial Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 4/14

  34. Previous Work objective Complexity Memory Requirements EG L NP ∩ co-NP memoryless EG LU ExpTime -complete pseudopolynomial AE NP ∩ co-NP memoryless AE LU ExpTime -complete pseudopolynomial AE LU ( U − L poly) NP ∩ co-NP polynomial AE L ExpTime -hard ≥ pseudopolynomial Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 4/14

  35. Previous Work objective Complexity Memory Requirements EG L NP ∩ co-NP memoryless EG LU ExpTime -complete pseudopolynomial AE NP ∩ co-NP memoryless AE LU ExpTime -complete pseudopolynomial AE LU ( U − L poly) NP ∩ co-NP polynomial AE L ExpTime -hard ≥ pseudopolynomial W.l.o.g.: fix lower bound 0 In all problems, lower and upper bounds part of the input. Here: upper bound existentially quantified. Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 4/14

  36. Objectives Capacity cap ∈ N , threshold t ∈ N EG L = { v 0 v 1 · · · | ∀ n . 0 ≤ EL ( v 0 · · · v n ) } EG LU ( cap ) = { v 0 v 1 · · · | ∀ n . 0 ≤ EL ( v 0 · · · v n ) ≤ cap } Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 5/14

  37. Objectives Capacity cap ∈ N , threshold t ∈ N EG L = { v 0 v 1 · · · | ∀ n . 0 ≤ EL ( v 0 · · · v n ) } EG LU ( cap ) = { v 0 v 1 · · · | ∀ n . 0 ≤ EL ( v 0 · · · v n ) ≤ cap } � n − 1 1 AE( t ) = { v 0 v 1 · · · | lim sup i =0 EL ( v 0 · · · v i ) ≤ t } n n →∞ Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 5/14

  38. Objectives Capacity cap ∈ N , threshold t ∈ N EG L = { v 0 v 1 · · · | ∀ n . 0 ≤ EL ( v 0 · · · v n ) } EG LU ( cap ) = { v 0 v 1 · · · | ∀ n . 0 ≤ EL ( v 0 · · · v n ) ≤ cap } � n − 1 1 AE( t ) = { v 0 v 1 · · · | lim sup i =0 EL ( v 0 · · · v i ) ≤ t } n n →∞ AE L ( t ) = EG L ∩ AE( t ) AE LU ( cap , t ) = EG LU ( cap ) ∩ AE( t ) Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 5/14

  39. Finding Bounds in Average-energy Games Input : Weighted arena A Question : Exists a threshold t ∈ N s.t. Player 0 wins ( A , AE L ( t ))? Martin Zimmermann Saarland University Finding Bounds in Average-energy Games 6/14

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