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Metric Temporal Logic With Counting S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya February 1, 2016 S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting Introduction S.N.Krishna, Khushraj Madnani,

  1. Time Logic with counting: CTMTL We introduce two new modal operators for counting C and UT. CTMTL Syntax φ ::= AP | φ ∧ φ | φ ∨ φ | ¬ φ | φ U I , # φ ∼ n φ | C n I φ where I is interval of the form � x , y � , x ∈ N ∪ { 0 } , y , x ∈ N ∪ { 0 , ∞} , � ... � ∈ { [ ... ] , ( ... ) , [ ... ) , ( ... ] } , ∼ = {≥ , ≤} and n ∈ N ∪ { 0 } S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  2. CTMTL: Semantics S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  3. CTMTL: Semantics = C ∼ n ρ, i | � l , u � φ ⇐ ⇒ N φ ( � τ i + l , τ i + u � ) ∼ n S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  4. CTMTL: Semantics = C ∼ n ρ, i | � l , u � φ ⇐ ⇒ N φ ( � τ i + l , τ i + u � ) ∼ n ρ, i | = φ 1 U I , # η ∼ n φ 2 ⇐ ⇒ ∃ j > i ρ, j | = φ 2 ∧ τ j − τ i ∈ = φ 1 ∧ N ′ φ ( i , j ) ∼ n I ∧ ∀ ∧ i < k < j ρ, k | S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  5. Subclasses of CTMTL C (0 , 1) MTL: Counting of the form C ∼ n (0 , 1) . S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  6. Subclasses of CTMTL C (0 , 1) MTL: Counting of the form C ∼ n (0 , 1) . C (0 , u) MTL: Counting of the form C ∼ n (0 , u ) . S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  7. Subclasses of CTMTL C (0 , 1) MTL: Counting of the form C ∼ n (0 , 1) . C (0 , u) MTL: Counting of the form C ∼ n (0 , u ) . CMTL: Counting with C modality only. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  8. Subclasses of CTMTL C (0 , 1) MTL: Counting of the form C ∼ n (0 , 1) . C (0 , u) MTL: Counting of the form C ∼ n (0 , u ) . CMTL: Counting with C modality only. TMTL: Counting with UT Modality only. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  9. Scheduling HVAC in Demand Response: An Example In Demand Response system an important requirement is to reduce the Peak Power Demand. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  10. Scheduling HVAC in Demand Response: An Example In Demand Response system an important requirement is to reduce the Peak Power Demand. Scheduling of HVAC to limit peak power demand below threshold. HVAC are more flexible as compared to devices like microwave oven. Constant mode switching (OFF-¿ON) causes wear and tear and more power consumption due to transient currents. No. of Switch yet another important parameter to grade such scheduling algorithms. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  11. Scheduling HVAC in Demand Response: An Example AC1 AC2 AC3 30 29 28 Temperature 27 Upper Bound 26.5 26 25 Lower Bound 24.5 24 0 2 4 6 8 10 Time S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  12. Scheduling HVAC in Demand Response: An Example ♦ (0 , 3) , # Switch − ON − AC ≤ 3 ( Comfort − AC 1 ∧ Comfort − AC 2 ∧ Comfort − AC 3 ) AC1 AC2 AC3 30 29 28 Temperature 27 Upper Bound 26.5 26 25 Lower Bound 24.5 24 0 2 4 6 8 10 Time S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  13. Presentation Flow Model : Timed Words Timed Logic with Counting : Syntax and Semantics Temporal Projections : Simple and Oversampled Expressiveness Relations with Counting Extensions Decidability : Satisfiability Checking Conclusion Future Work S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  14. Definitions: Simple Projection S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  15. Definitions: Simple Projection Let Σ , X be finite disjoint set. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  16. Definitions: Simple Projection Let Σ , X be finite disjoint set. Simple Extension A (Σ , X )-simple extension is a timed word ρ over 2 X ∪ Σ such that at any point i ∈ dom ( ρ ), σ i ∩ Σ � = ∅ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  17. Definitions: Simple Projection Let Σ , X be finite disjoint set. Simple Extension A (Σ , X )-simple extension is a timed word ρ over 2 X ∪ Σ such that at any point i ∈ dom ( ρ ), σ i ∩ Σ � = ∅ Simple Projection A timed word ρ over Σ obtained by deleting symbols in X from (Σ , X ) extension ρ ′ is called its Simple Projection. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  18. Definitions: Simple Projection Let Σ , X be finite disjoint set. Simple Extension A (Σ , X )-simple extension is a timed word ρ over 2 X ∪ Σ such that at any point i ∈ dom ( ρ ), σ i ∩ Σ � = ∅ Simple Projection A timed word ρ over Σ obtained by deleting symbols in X from (Σ , X ) extension ρ ′ is called its Simple Projection. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  19. Definitions: Simple Projection Let Σ , X be finite disjoint set. Simple Extension A (Σ , X )-simple extension is a timed word ρ over 2 X ∪ Σ such that at any point i ∈ dom ( ρ ), σ i ∩ Σ � = ∅ Simple Projection A timed word ρ over Σ obtained by deleting symbols in X from (Σ , X ) extension ρ ′ is called its Simple Projection. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  20. Definitions: Oversampled Projection S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  21. Definitions: Oversampled Projection Let Σ , X be finite disjoint set. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  22. Definitions: Oversampled Projection Let Σ , X be finite disjoint set. Oversampled Behaviour A (Σ , X )-oversampled behaviour is a timed word over 2 X ∪ Σ, such that σ ′ 1 ∩ Σ � = ∅ and σ ′ | dom ( ρ ′ ) | ∩ Σ � = ∅ . S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  23. Definitions: Oversampled Projection Let Σ , X be finite disjoint set. Oversampled Behaviour A (Σ , X )-oversampled behaviour is a timed word over 2 X ∪ Σ, such that σ ′ 1 ∩ Σ � = ∅ and σ ′ | dom ( ρ ′ ) | ∩ Σ � = ∅ . Oversampled Projection A timed word ρ over Σ obtained by deleting symbols in X (and thus deleting the points containing only X )from (Σ , X ) oversampled behaviour ρ ′ is called its Oversampled Projection. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  24. Definitions: Oversampled Projection Let Σ , X be finite disjoint set. Oversampled Behaviour A (Σ , X )-oversampled behaviour is a timed word over 2 X ∪ Σ, such that σ ′ 1 ∩ Σ � = ∅ and σ ′ | dom ( ρ ′ ) | ∩ Σ � = ∅ . Oversampled Projection A timed word ρ over Σ obtained by deleting symbols in X (and thus deleting the points containing only X )from (Σ , X ) oversampled behaviour ρ ′ is called its Oversampled Projection. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  25. Definitions: Oversampled Projection Let Σ , X be finite disjoint set. Oversampled Behaviour A (Σ , X )-oversampled behaviour is a timed word over 2 X ∪ Σ, such that σ ′ 1 ∩ Σ � = ∅ and σ ′ | dom ( ρ ′ ) | ∩ Σ � = ∅ . Oversampled Projection A timed word ρ over Σ obtained by deleting symbols in X (and thus deleting the points containing only X )from (Σ , X ) oversampled behaviour ρ ′ is called its Oversampled Projection. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  26. Definitions: Equisaitisfiability modulo Temporal Projection We say that ϕ over Σ is equisatisfiable modulo temporal projection ψ over Σ ∪ 2 X iff: Figure: Figure Illustrating ϕ is equisatisfiable to ψ . Arrow represents the temporal(simple or oversampled) projection function S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  27. Flattening: An example S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  28. Flattening: An example Flattening is a technique to reduce the modal depth of the formula preserving satisfiability. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  29. Flattening: An example Flattening is a technique to reduce the modal depth of the formula preserving satisfiability. Example Let φ = a U [2 , 5] ( b U [2 , 3] c ) S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  30. Flattening: An example Flattening is a technique to reduce the modal depth of the formula preserving satisfiability. Example Let φ = a U [2 , 5] ( b U [2 , 3] c ) φ flat = a U [2 , 5] d ∧ � ( d ↔ ( b U [2 , 3] c )) ∧ � ( d → a ∨ b ∨ c ) S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  31. Flattening: An example Flattening is a technique to reduce the modal depth of the formula preserving satisfiability. Example Let φ = a U [2 , 5] ( b U [2 , 3] c ) φ flat = a U [2 , 5] d ∧ � ( d ↔ ( b U [2 , 3] c )) ∧ � ( d → a ∨ b ∨ c ) S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  32. Flattening: An example Flattening is a technique to reduce the modal depth of the formula preserving satisfiability. Example Let φ = a U [2 , 5] ( b U [2 , 3] c ) φ flat = a U [2 , 5] d ∧ � ( d ↔ ( b U [2 , 3] c )) ∧ � ( d → a ∨ b ∨ c ) S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  33. Flattening: An example Flattening is a technique to reduce the modal depth of the formula preserving satisfiability. Example Let φ = a U [2 , 5] ( b U [2 , 3] c ) φ flat = a U [2 , 5] d ∧ � ( d ↔ ( b U [2 , 3] c )) ∧ � ( d → a ∨ b ∨ c ) Thus flattening is an example of a reduction preserving satisfiability modulo simple projections. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  34. Related Work S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  35. Related Work Satisfiability Checking of MITL is decidable with EXPSPACE complexity. [Alur et al . J . ACM 1996] S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  36. Related Work Satisfiability Checking of MITL is decidable with EXPSPACE complexity. [Alur et al . J . ACM 1996] Satisfiability problem for MTL[ U I ] is decidable by reducing it to reachability problem for 1-clock Timed Alternating Automata.[Ouaknine et al . LICS 2005] S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  37. Related Work Satisfiability Checking of MITL is decidable with EXPSPACE complexity. [Alur et al . J . ACM 1996] Satisfiability problem for MTL[ U I ] is decidable by reducing it to reachability problem for 1-clock Timed Alternating Automata.[Ouaknine et al . LICS 2005] Satisfiability Checking of QTL with counting is decidable with EXPSPACE complexity. [Rabinovich et . al . FORMATS 2008] S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  38. Related Work Satisfiability Checking of MITL is decidable with EXPSPACE complexity. [Alur et al . J . ACM 1996] Satisfiability problem for MTL[ U I ] is decidable by reducing it to reachability problem for 1-clock Timed Alternating Automata.[Ouaknine et al . LICS 2005] Satisfiability Checking of QTL with counting is decidable with EXPSPACE complexity. [Rabinovich et . al . FORMATS 2008] Counting adds expressiveness to MITL over signals [Rabinovich FORMATS 2008]. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  39. Related Work Satisfiability Checking of MITL is decidable with EXPSPACE complexity. [Alur et al . J . ACM 1996] Satisfiability problem for MTL[ U I ] is decidable by reducing it to reachability problem for 1-clock Timed Alternating Automata.[Ouaknine et al . LICS 2005] Satisfiability Checking of QTL with counting is decidable with EXPSPACE complexity. [Rabinovich et . al . FORMATS 2008] Counting adds expressiveness to MITL over signals [Rabinovich FORMATS 2008]. MTL with counting over signals is expressively complete with FO [ <, +1] over reals [Hunter CSL 2013]. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  40. Related Work Satisfiability Checking of MITL is decidable with EXPSPACE complexity. [Alur et al . J . ACM 1996] Satisfiability problem for MTL[ U I ] is decidable by reducing it to reachability problem for 1-clock Timed Alternating Automata.[Ouaknine et al . LICS 2005] Satisfiability Checking of QTL with counting is decidable with EXPSPACE complexity. [Rabinovich et . al . FORMATS 2008] Counting adds expressiveness to MITL over signals [Rabinovich FORMATS 2008]. MTL with counting over signals is expressively complete with FO [ <, +1] over reals [Hunter CSL 2013]. Counting LTL is equivalent to LTL and has EXP − SPACE complete satisfiability checking.[Laroussinie et . al . TIME 2010]. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  41. Our Results S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  42. Our Results Satisfiability Checking for CTMTL is decidable. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  43. Our Results Satisfiability Checking for CTMTL is decidable. Exploring Expressiveness relations amongst fragments of MTL with counting over timed words(Pointwise Semantics). S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  44. Presentation Flow Model : Timed Words Timed Logic with Counting : Syntax and Semantics Temporal Projections : Simple and Oversampled Expressiveness Relations with Counting Extensions Satisfiability Checking: Decidability Discussion Future Work S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  45. Expressiveness Heirarchy : Logic with counting S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  46. Expressiveness Heirarchy : Non-Punctual Fragments S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  47. TMTL − CMTL � = ∅ ϕ = ♦ (0 , 1) , # a ≥ 3 b S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  48. Presentation Flow Model : Timed Words Timed Logic with Counting : Syntax and Semantics Temporal Projections : Simple and Oversampled Expressiveness Relations with Counting Extensions Satisfiability Checking: Decidability Discussion Future Work S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  49. Satisfiability Checking : Decidability S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  50. Satisfiability Checking : Decidability Flatten the formula S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  51. Satisfiability Checking : Decidability Flatten the formula All the counting modalities are of the form � ( w ↔ C ∼ n a ) and I � ( w ↔ a U I , # x ∼ n b ) S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  52. Satisfiability Checking : Decidability Flatten the formula All the counting modalities are of the form � ( w ↔ C ∼ n a ) and I � ( w ↔ a U I , # x ∼ n b ) Next we eliminate counting modalities from the above flattened formula preserving satisfiability to show decidability. S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  53. Eliminating C ≥ n I b modality S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  54. Eliminating C ≥ n I b modality Given a word ρ over Σ we construct a simple extension ρ ′ over Σ ∪ { b 0 , b 1 , . . . , b n − 1 } S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  55. Eliminating C ≥ n I b modality Given a word ρ over Σ we construct a simple extension ρ ′ over Σ ∪ { b 0 , b 1 , . . . , b n − 1 } { b 0 , b 1 , . . . , b n − 1 } works as a counter. Using their behaviour we precisely mark a as the witness for C ≥ n b. I S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  56. Construction of ρ ′ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  57. Construction of ρ ′ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  58. Construction of ρ ′ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  59. Construction of ρ ′ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  60. Construction of ρ ′ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  61. Construction of ρ ′ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  62. Construction of ρ ′ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  63. Construction of ρ ′ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  64. Construction of ρ ′ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  65. Construction of ρ ′ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  66. Construction of ρ ′ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  67. Construction of ρ ′ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  68. Construction of ρ ′ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  69. Construction of ρ ′ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

  70. Construction of ρ ′ S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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