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Counting Problems for Parikh Images Christoph Haase Stefan Kiefer Markus Lohrey MFCS 2017, Aalborg 25 August 2017 Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 1 The Cost Problem 2 10 : 5 3 8 10 : 15

  1. Counting Problems for Parikh Images Christoph Haase Stefan Kiefer Markus Lohrey MFCS 2017, Aalborg 25 August 2017 Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 1

  2. The Cost Problem 2 10 : 5 3 8 10 : 15 10 : 10 7 10 : 20 What is the probability to reach the gate in 25–45min? Quantiles? Input: Cost Markov chain cost formula ϕ 25 ≤ cost ≤ 45 Cost Problem := Output: Pr ( ϕ ) Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 2

  3. The Cost Problem 2 10 : 5 3 8 10 : 15 10 : 10 7 10 : 20 What is the probability to reach the gate in 25–45min? Quantiles? Input: Cost Markov chain cost formula ϕ 25 ≤ cost ≤ 45 Cost Problem := threshold τ ∈ [ 0 , 1 ] Output: Is Pr ( ϕ ) ≥ τ ? Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 2

  4. Complexity of the Cost Problem Theorem (Laroussinie, Sproston, FoSSaCS’05) The cost problem is in EXPTIME. The cost problem is NP-hard. Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 3

  5. Complexity of the Cost Problem Theorem (Laroussinie, Sproston, FoSSaCS’05) The cost problem is in EXPTIME. The cost problem is NP-hard NP-hard. by reduction from the K th largest subset problem Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 3

  6. Complexity of the Cost Problem Theorem (Laroussinie, Sproston, FoSSaCS’05) The cost problem is in EXPTIME. The cost problem is NP-hard NP-hard. by reduction from the K th largest subset problem Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 3

  7. Complexity of the Cost Problem Theorem (Laroussinie, Sproston, FoSSaCS’05) The cost problem is in EXPTIME. The cost problem is NP-hard NP-hard. by reduction from the K th largest subset problem Theorem (HK, IPL ’16) The Kth largest subset problem is PP-complete. a superset of NP Corollary The cost problem is PP-hard. Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 3

  8. Complexity of the Cost Problem EXPTIME PSPACE Cost PP NP Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 4

  9. Complexity of the Cost Problem circuit value ∗ EXPTIME ∗ − PSPACE + + Cost 1 0 Input: arithmetic circuit := PP PosSLP Output: Is the value > 0 ? NP Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 4

  10. Complexity of the Cost Problem EXPTIME Theorem (HK, ICALP’15) The cost problem is PosSLP-hard. PSPACE Cost Input: arithmetic circuit := PP PosSLP Output: Is the value > 0 ? NP Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 4

  11. Complexity of the Cost Problem EXPTIME Theorem (HK, ICALP’15) The cost problem is PosSLP-hard. The cost problem is in PSPACE. PSPACE Cost Input: arithmetic circuit := PP PosSLP Output: Is the value > 0 ? NP Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 4

  12. Complexity of the Cost Problem EXPTIME Theorem (HK, ICALP’15) The cost problem is PosSLP-hard. The cost problem is in PSPACE. PSPACE CH Theorem (HKL, LICS’17) The cost problem is in CH. Cost PP PosSLP NP Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 4

  13. Solving the Cost Problem Input: Cost Markov chain cost formula ϕ Cost Problem := Output: Pr ( ϕ ) 1 2 : 15 1 : 10 7 3 10 : 5 10 : 20 1 2 : 25 1 : 10 Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 5

  14. Solving the Cost Problem Input: Cost Markov chain cost formula ϕ Cost Problem := Output: Pr ( ϕ ) 1 2 : 15 1 : 10 1 1 7 3 10 : 5 10 : 20 2 1 2 2 1 2 : 25 1 : 10 Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 5

  15. Solving the Cost Problem Input: Cost Markov chain cost formula ϕ Cost Problem := Output: Pr ( ϕ ) 1 2 : 15 1 : 10 1 1 7 3 10 : 5 10 : 20 2 1 2 2 1 2 : 25 1 : 10 Enumerate the Parikh images . Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 5

  16. Solving the Cost Problem Input: Cost Markov chain cost formula ϕ Cost Problem := Output: Pr ( ϕ ) 1 2 : 15 1 : 10 1 1 7 3 10 : 5 10 : 20 2 1 2 2 1 2 : 25 1 : 10 Enumerate the Parikh images . Problem: there might be multiple paths per Parikh image. Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 5

  17. Solving the Cost Problem Input: Cost Markov chain cost formula ϕ Cost Problem := Output: Pr ( ϕ ) Enumerate the Parikh images . Problem: there might be multiple paths per Parikh image. Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 5

  18. Solving the Cost Problem Input: Cost Markov chain cost formula ϕ Cost Problem := Output: Pr ( ϕ ) Enumerate the Parikh images . Problem: there might be multiple paths per Parikh image. Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 5

  19. The BEST Theorem Theorem (de Bruijn, van Aardenne-Ehrenfest, Smith, Tutte) The number of Eulerian cycles in an Eulerian graph G equals � � � t ( G ) · d ( v ) − 1 ! v ∈ V number of directed spanning trees in G Theorem (Tutte’s matrix-tree theorem) t ( G ) = det( L ( G ) 11 ) Laplacian of G Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 6

  20. Q UANT : A Tool for the Cost Problem Theorem ([HK, ICALP’15], [HK, IPL ’16], [HKL, LICS’17]) The cost problem is hard for PP and PosSLP . The cost problem is in CH. Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 7

  21. Q UANT : A Tool for the Cost Problem 0 . 45 : ( 1 , 0 ) 0 . 1 : ( 0 , 0 ) 0 . 45 : ( 0 , 1 ) What is Pr ( cost ∈ [ 4 , 6 ] 2 ) ? Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 8

  22. Q UANT : A Tool for the Cost Problem 0 . 45 : ( 1 , 0 ) 0 . 1 : ( 0 , 0 ) 0 . 45 : ( 0 , 1 ) What is Pr ( cost ∈ [ 4 , 6 ] 2 ) ? Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 8

  23. Q UANT : A Tool for the Cost Problem 0 . 45 : ( 1 , 0 ) 0 . 1 : ( 0 , 0 ) 0 . 45 : ( 0 , 1 ) What is Pr ( cost ∈ [ 4 , 6 ] 2 ) ? Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 8

  24. Q UANT : A Tool for the Cost Problem 0 . 45 : ( 1 , 0 ) 0 . 1 : ( 0 , 0 ) 0 . 45 : ( 0 , 1 ) What is Pr ( cost ∈ [ 4 , 6 ] 2 ) ? Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 8

  25. Q UANT : A Tool for the Cost Problem 0 . 45 : ( 1 , 0 ) 0 . 1 : ( 0 , 0 ) 0 . 45 : ( 0 , 1 ) What is Pr ( cost ∈ [ 4 , 6 ] 2 ) ? Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 8

  26. Q UANT : A Tool for the Cost Problem 0 . 45 : ( 1 , 0 ) 0 . 1 : ( 0 , 0 ) 0 . 45 : ( 0 , 1 ) What is Pr ( cost ∈ [ 4 , 6 ] 2 ) ? Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 8

  27. Q UANT : A Tool for the Cost Problem 0 . 45 : ( 1 , 0 ) 0 . 1 : ( 0 , 0 ) 0 . 45 : ( 0 , 1 ) What is Pr ( cost ∈ [ 4 , 6 ] 2 ) ? Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 8

  28. Q UANT : A Tool for the Cost Problem 0 . 45 : ( 1 , 0 ) 0 . 1 : ( 0 , 0 ) 0 . 45 : ( 0 , 1 ) What is Pr ( cost ∈ [ 4 , 6 ] 2 ) ? Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 8

  29. Q UANT : A Tool for the Cost Problem time in sec 200 Q UANT [ 18 , 20 ] d Q UANT [ 13 , 15 ] d 150 Q UANT [ 8 , 10 ] d 100 50 0 dimension d 3 4 5 6 7 8 Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 9

  30. Q UANT : A Tool for the Cost Problem time in sec 200 Q UANT [ 18 , 20 ] d Q UANT [ 13 , 15 ] d 150 Q UANT [ 8 , 10 ] d P RISM [ 8 , 10 ] d 100 P RISM [ 13 , 15 ] d 50 P RISM [ 18 , 20 ] d 0 dimension d 3 4 5 6 7 8 Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 9

  31. Q UANT : A Tool for the Cost Problem time in sec 200 Q UANT [ 18 , 20 ] d Q UANT [ 13 , 15 ] d 150 Q UANT [ 8 , 10 ] d P RISM [ 8 , 10 ] d 100 P RISM [ 13 , 15 ] d 50 P RISM [ 18 , 20 ] d 0 dimension d 3 4 5 6 7 8 Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 9

  32. Q UANT : A Tool for the Cost Problem Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 10

  33. Q UANT : A Tool for the Cost Problem Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 10

  34. Counting Parikh Images Σ : finite alphabet p ∈ N Σ : vector A : language generator (DFA, NFA, CFG) N ( A , p ) : number of words accepted by A with Parikh image p Example: for A = a ∗ ba ∗ and p = ( 2 , 1 ) : N ( A , p ) = 3 Input: Language generators A , B vector p ∈ N Σ PosParikh := Output: Is N ( A , p ) > N ( B , p ) ? Different variants: language generator: DFA, NFA, CFG unary or binary encoding of p fixed or variable alphabet Σ Christoph Haase, Stefan Kiefer , Markus Lohrey Counting Problems for Parikh Images 11

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