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Quantitative Reductions and Vertex-Ranked Games Alexander Weinert Saarland University September 13th, 2017 Highlights 2017 - London Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 1/9 Reachability

  1. Quantitative Reductions and Vertex-Ranked Games Alexander Weinert Saarland University September 13th, 2017 Highlights 2017 - London Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 1/9

  2. Reachability Games ✓ Winning condition: Play reaches either or or Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 2/9

  3. Reachability Games ✓ Winning condition: Play reaches either or or Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 2/9

  4. Reachability Games ✓ Winning condition: Play reaches either or or Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 2/9

  5. Reachability Games ✓ Winning condition: Play reaches either or or Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 2/9

  6. Reachability Games ✓ Winning condition: Play reaches either or or Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 2/9

  7. The Big Picture Reachability ✓ Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 3/9

  8. Generalized Reachability: The Problem Winning condition: Reach one from { } and one from { } . , , Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 4/9

  9. Generalized Reachability: The Problem Winning condition: Reach one from { } and one from { } . , , Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 4/9

  10. Generalized Reachability: One Solution { } { } , · · · { } { } , · · · { } , · · · { } Winning condition: Reach some memory state S with S ∩ { } � = ∅ and with S ∩ { } � = ∅ , , Reachability Condition Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 5/9

  11. Generalized Reachability: One Solution { } { } , · · · { } { } , · · · { } , · · · { } Winning condition: Reach some memory state S with S ∩ { } � = ∅ and with S ∩ { } � = ∅ , , Reachability Condition Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 5/9

  12. Generalized Reachability: One Solution { } { } , · · · { } { } , · · · { } , · · · { } Winning condition: Reach some memory state S with S ∩ { } � = ∅ and with S ∩ { } � = ∅ , , Reachability Condition Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 5/9

  13. Generalized Reachability: One Solution { } { } , · · · { } { } , · · · { } , · · · { } Winning condition: Reach some memory state S with S ∩ { } � = ∅ and with S ∩ { } � = ∅ , , Reachability Condition Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 5/9

  14. Generalized Reachability: One Solution { } { } , · · · { } { } , · · · { } , · · · { } Winning condition: Reach some memory state S with S ∩ { } � = ∅ and with S ∩ { } � = ∅ , , Reachability Condition Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 5/9

  15. Generalized Reachability: One Solution { } { } , · · · { } { } , · · · { } , · · · { } Winning condition: Reach some memory state S with S ∩ { } � = ∅ and with S ∩ { } � = ∅ , , Reachability Condition Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 5/9

  16. Generalized Reachability: One Solution { } { } , · · · { } { } , · · · { } , · · · { } Winning condition: Reach some memory state S with S ∩ { } � = ∅ and with S ∩ { } � = ∅ , , Reachability Condition Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 5/9

  17. Generalized Reachability: One Solution { } { } , · · · { } { } , · · · { } , · · · { } Winning condition: Reach some memory state S with S ∩ { } � = ∅ and with S ∩ { } � = ∅ , , Reachability Condition Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 5/9

  18. Generalized Reachability: One Solution { } { } , · · · { } { } , · · · { } , · · · { } Winning condition: Reach some memory state S with S ∩ { } � = ∅ and with S ∩ { } � = ∅ , , Reachability Condition Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 5/9

  19. The Big Picture Quantitative Generalized Reachability Quantitative Qualitative Generalized Reachability Reachability ✓ Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 6/9

  20. The Big Picture Quantitative Generalized Reachability Quantitative Qualitative Generalized Reachability Reachability ✓ Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 6/9

  21. The Big Picture Quantitative Generalized Reachability Quantitative Qualitative Generalized Reachability Reachability ✓ Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 6/9

  22. The Big Picture Quantitative Generalized Reachability Quantitative Qualitative Generalized Reachability Reachability ✓ Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 6/9

  23. The Big Picture Quantitative Generalized Reachability Quantitative Qualitative Generalized Reachability Reachability ✓ Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 6/9

  24. Quantitative Generalized Reachability Assign cost to each play.  0 if { } and { } are visited , ,   Cst ( ρ ) = 1 if one of them is visited  2 if neither is visited  Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 7/9

  25. Quantitative Generalized Reachability Assign cost to each play.  0 if { } and { } are visited , ,   Cst ( ρ ) = 1 if one of them is visited  2 if neither is visited  Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 7/9

  26. Quantitative Generalized Reachability Assign cost to each play.  0 if { } and { } are visited , ,   Cst ( ρ ) = 1 if one of them is visited  2 if neither is visited  Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 7/9

  27. The Big Picture Quantitative Vertex-Ranked Generalized Reachability Reachability 0 = 1 t Quantitative s = 2 C = t s C Qualitative t s C Generalized Reachability Reachability ✓ Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 8/9

  28. The Big Picture Quantitative Vertex-Ranked Generalized Reachability Reachability 0 = 1 t Quantitative s = 2 C = t s C Qualitative t s C Generalized Reachability Reachability ✓ Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 8/9

  29. The Big Picture Quantitative Vertex-Ranked Generalized Reachability Reachability 0 = 1 t Quantitative s = 2 C = t s C Qualitative t s C Generalized Reachability Reachability ✓ Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 8/9

  30. The Big Picture Quantitative Vertex-Ranked Generalized Reachability Reachability 0 = 1 t Quantitative s = 2 C = t s C Qualitative t s C Generalized Reachability Reachability ✓ Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 8/9

  31. The Big Picture Quantitative Vertex-Ranked Generalized Reachability Reachability 0 = 1 t Quantitative s = 2 C = t s C Qualitative t s C Generalized Reachability Reachability ✓ Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 8/9

  32. The Big Picture Quantitative Vertex-Ranked Generalized Reachability Reachability ✓ 0 = 1 t Quantitative s = 2 C = t s C Qualitative t s C Generalized Reachability Reachability ✓ Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 8/9

  33. The Big Picture Quantitative Vertex-Ranked Generalized Reachability Reachability ✓ 0 = 1 t Quantitative s = 2 C = t s C Qualitative t s C Generalized Reachability Reachability ✓ Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 8/9

  34. Conclusion Contribution Lifted reductions to quantitative games Solved wide range of general-purpose quantitative games Next Steps ✓ Quantitative Qualitative ✓ Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 9/9

  35. Conclusion Contribution Lifted reductions to quantitative games Solved wide range of general-purpose quantitative games Next Steps ✓ Quantitative Qualitative ✓ Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 9/9

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