Quantitative Reductions and Vertex-Ranked Games Alexander Weinert - - PowerPoint PPT Presentation

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quantitative reductions and vertex ranked games

Quantitative Reductions and Vertex-Ranked Games Alexander Weinert - - PowerPoint PPT Presentation

Feb 10, 2023 237 likes •637 views

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

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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

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Reachability Games

✓ Winning condition: Play reaches either

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Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 2/9

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SLIDE 3

Reachability Games

✓ Winning condition: Play reaches either

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Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 2/9

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SLIDE 4

Reachability Games

✓ Winning condition: Play reaches either

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Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 2/9

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SLIDE 5

Reachability Games

✓ Winning condition: Play reaches either

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Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 2/9

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SLIDE 6

Reachability Games

✓ Winning condition: Play reaches either

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r

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 2/9

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SLIDE 7

The Big Picture

Reachability ✓

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 3/9

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SLIDE 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

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SLIDE 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

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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

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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

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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

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 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

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SLIDE 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

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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

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The Big Picture

Reachability ✓ Generalized Reachability Qualitative Quantitative Quantitative Generalized Reachability

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 6/9

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The Big Picture

Reachability ✓ Generalized Reachability Qualitative Quantitative Quantitative Generalized Reachability

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 6/9

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SLIDE 21

The Big Picture

Reachability ✓ Generalized Reachability Qualitative Quantitative Quantitative Generalized Reachability

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 6/9

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The Big Picture

Reachability ✓ Generalized Reachability Qualitative Quantitative Quantitative Generalized Reachability

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 6/9

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The Big Picture

Reachability ✓ Generalized Reachability Qualitative Quantitative Quantitative Generalized Reachability

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 6/9

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Quantitative Generalized Reachability

Assign cost to each play. Cst(ρ) =      if { , } and { , } are visited 1 if one of them is visited 2 if neither is visited

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 7/9

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Quantitative Generalized Reachability

Assign cost to each play. Cst(ρ) =      if { , } and { , } are visited 1 if one of them is visited 2 if neither is visited

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 7/9

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Quantitative Generalized Reachability

Assign cost to each play. Cst(ρ) =      if { , } and { , } are visited 1 if one of them is visited 2 if neither is visited

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 7/9

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The Big Picture

Reachability ✓ Generalized Reachability Qualitative Quantitative Quantitative Generalized Reachability C s t = C s t = 1 C s t = 2 Vertex-Ranked Reachability

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 8/9

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The Big Picture

Reachability ✓ Generalized Reachability Qualitative Quantitative Quantitative Generalized Reachability C s t = C s t = 1 C s t = 2 Vertex-Ranked Reachability

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 8/9

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The Big Picture

Reachability ✓ Generalized Reachability Qualitative Quantitative Quantitative Generalized Reachability C s t = C s t = 1 C s t = 2 Vertex-Ranked Reachability

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 8/9

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The Big Picture

Reachability ✓ Generalized Reachability Qualitative Quantitative Quantitative Generalized Reachability C s t = C s t = 1 C s t = 2 Vertex-Ranked Reachability

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 8/9

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The Big Picture

Reachability ✓ Generalized Reachability Qualitative Quantitative Quantitative Generalized Reachability C s t = C s t = 1 C s t = 2 Vertex-Ranked Reachability

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 8/9

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The Big Picture

Reachability ✓ Generalized Reachability Qualitative Quantitative Quantitative Generalized Reachability C s t = C s t = 1 C s t = 2 Vertex-Ranked Reachability ✓

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 8/9

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The Big Picture

Reachability ✓ Generalized Reachability Qualitative Quantitative Quantitative Generalized Reachability C s t = C s t = 1 C s t = 2 Vertex-Ranked Reachability ✓

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 8/9

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Conclusion

Contribution Lifted reductions to quantitative games Solved wide range of general-purpose quantitative games Next Steps Qualitative Quantitative ✓ ✓

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 9/9

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Conclusion

Contribution Lifted reductions to quantitative games Solved wide range of general-purpose quantitative games Next Steps Qualitative Quantitative ✓ ✓

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 9/9

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Conclusion

Contribution Lifted reductions to quantitative games Solved wide range of general-purpose quantitative games Next Steps Qualitative Quantitative ✓ ✓

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 9/9

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Conclusion

Contribution Lifted reductions to quantitative games Solved wide range of general-purpose quantitative games Next Steps Qualitative Quantitative ✓ ✓

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 9/9

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Conclusion

Contribution Lifted reductions to quantitative games Solved wide range of general-purpose quantitative games Next Steps Qualitative Quantitative ✓ ✓

Alexander Weinert Saarland University Quantitative Reductions and Vertex-Ranked Games 9/9