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Syntax Matters

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Syntax Matters

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  1. Syntax Matters Tom Henzinger (IST Austria) with Barbara Jobstmann (Verimag) Maria Mateescu (EPFL) Verena Wolf (Saarbruecken)

  2. Science Experiment Theory

  3. Mathematics 14

  4. Mathematics Semantics Syntax 14

  5. Mathematics Semantics Syntax 14 1110 XIV

  6. Syntax Matters • Expressiveness 0

  7. Syntax Matters • Expressiveness 0 • Succinctness IIII IIII IIII

  8. Syntax Matters • Expressiveness 0 • Succinctness IIII IIII IIII • Operations +1 addition multiplication 14 x 34 = 42 56 476

  9. Syntax is More than Notation • Expressiveness 0 • Succinctness IIII IIII IIII • Operations +1 addition multiplication 14 x 34 = 42 56 476 (* 14 34) =

  10. Syntax is More than Notation • Expressiveness 0 • Succinctness IIII IIII IIII • Operations +1 addition multiplication Two languages are equivalent if there is a linear translation from each to the other (e.g. prefix – infix, binary – decimal).

  11. Computer Science Semantics Syntax 1 1 2 6 24 ... 0 1 2 3 4 ... (defun f (n) (if (<= n 1) 1 (* n (f (- n 1)))))

  12. Computer Science Semantics Syntax 1 1 2 6 24 ... 0 1 2 3 4 ... (defun f (n) (if (<= n 1) 1 (* n (f (- n 1))))) finite infinite

  13. Computer Science Semantics Syntax 1 1 2 6 24 ... 0 1 2 3 4 ... (defun f (n) (if (<= n 1) 1 (* n (f (- n 1))))) finite recursively enumerable infinite

  14. Computer Science Semantics Syntax 1 1 2 6 24 ... 0 1 2 3 4 ... (defun f (n) (if (<= n 1) 1 (* n (f (- n 1))))) Composition: (defun h (n) (f (g n)))

  15. Computer Science Semantics Syntax x := 0; y := 0; while true do x := x + 1; y := y + 1 end

  16. Computer Science Semantics Syntax x := 0; y := 0; while true do x := x + 1; y := y + 1 end 2,2 3,2 1,1 2,1 0,0 1,0

  17. Computer Science Semantics Syntax (inc x)* 0,0 1,0 2,0

  18. Computer Science Semantics Syntax (inc x)* (inc y) * 0,2 0,1 0,0 1,0 2,0

  19. Computer Science Semantics Syntax (inc x)* (inc y) * 1,2 0,2 2,2 0,1 1,1 2,1 Composition: (inc x)* || (inc y)* 0,0 1,0 2,0

  20. Markovian Population Models

  21. Markovian Population Models +

  22. Markovian Population Models +

  23. Markovian Population Models +

  24. Markovian Population Models State: ( 8 , 6 , 6 )

  25. Markovian Population Models State: ( 9 , 7 , 5 ) Transition: State: ( 8 , 6 , 6 )

  26. Markovian Population Models -discrete state -location unaware

  27. Markovian Population Models -discrete state -location unaware -stochastic transition -continuous time

  28. Markovian Population Models 8, 6, 6 9, 7, 5 deterministic

  29. Markovian Population Models 8, 6, 6 9, 7, 5 deterministic

  30. Markovian Population Models 8, 6, 6 9, 7, 5 deterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time

  31. Markovian Population Models 8, 6, 6 9, 7, 5 deterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time 0 0 1

  32. Markovian Population Models 8, 6, 6 9, 7, 5 deterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time 1 0.4 0.6

  33. Markovian Population Models 8, 6, 6 9, 7, 5 deterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time 2 0.64 0.36

  34. Markovian Population Models 8, 6, 6 9, 7, 5 deterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time 3 0.784 0.216 (9,7,5): 0.36 (8,6,6): 0.64 (9,7,5): 0.216 (8,6,6): 0.784

  35. Markovian Population Models 8, 6, 6 9, 7, 5 deterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time 0.5 8, 6, 6 9, 7, 5 continuous time exit rate 0.5 exp residence time 2

  36. Markovian Population Models 0.5 8, 6, 6 9, 7, 5 continuous time 0exit rate 0.5 exp residence time 2 0 1

  37. Markovian Population Models 0.5 8, 6, 6 9, 7, 5 continuous time 1exit rate 0.5 exp residence time 2 0.4 0.6

  38. Markovian Population Models 0.5 8, 6, 6 9, 7, 5 continuous time 1.8exit rate 0.5 exp residence time 2 0.6 0.4

  39. Markovian Population Models 8, 6, 6 9, 7, 5 nondeterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time 0.5 8, 6, 6 9, 7, 5 continuous time

  40. Markovian Population Models 8, 6, 6 9, 7, 5 nondeterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time MDP 0.3 0.7 0.5 8, 6, 6 9, 7, 5 continuous time

  41. Markovian Population Models 8, 6, 6 9, 7, 5 nondeterministic 0.6 8, 6, 6 0.4 9, 7, 5 discrete time MDP 0.3 0.7 0.5 8, 6, 6 9, 7, 5 continuous time exit rate 2 exp residence time 0.5 1.5

  42. Markovian Population Models 0.1 0.1 0.1 0.2 0.2 0.2 7, 5, 5 8, 6, 4 9, 7, 3 0.1 0.1 0.1 0.2 0.2 0.2 7, 5, 6 8, 6, 5 9, 7, 4 0.1 0.1 0.1 0.2 0.2 0.2 7, 5, 7 8, 6, 6 9, 7, 5 CTMC

  43. Markovian Population Models 0.2 + 0.1 Syntax: set of transition classes (finite object)

  44. Markovian Population Models 0.2 + 0.1 Syntax: set of transition classes (finite object) 0.1 0.1 0.2 0.2 8, 6, 6 9, 7, 5 Semantics: CTMC (infinite object)

  45. Markovian Population Models 0.2 + 0.1 Syntax: set of transition classes (finite object) 0.6 0.5 8, 6, 6 9, 7, 5 Semantics: CTMC (infinite object)

  46. Markovian Population Models 0.2 + 0.1 Syntax: set of transition classes (finite object) 0.6 0.5 9.6 12.6 8, 6, 6 9, 7, 5 Semantics: CTMC (infinite object)

  47. Syntax: Transition Class Model (TCM) Dimension n: state (x1, ..., xn) 2 S state space S = Nn Finite set of transition classes: each transition class consists of 1. guard G µ S 2. injective update function u: G ! S 3. rate function : G !R+

  48. Syntax: Transition Class Model (TCM) Dimension n: state (x1, ..., xn) 2 S state space S = Nn Finite set of transition classes: each transition class consists of 1. guard G µ S 2. injective update function u: G ! S 3. rate function : G !R+ n = 3 G1: x1¸ 1 Æ x2¸ 1 u1(x1,x2,x3) = (x1-1, x2-1, x3+1) 1(x1,x2,x3) = 0.2 ¢ x1 ¢ x2 G2: x3¸ 1 u2(x1,x2,x3) = (x1, x2, x3-1) 2(x1,x2,x3) = 0.1 ¢ x3

  49. Semantics: Continuous-Time Markov Chain (CTMC) For all times t 2R+, a random variable X(t) 2 S.

  50. Semantics: Continuous-Time Markov Chain (CTMC) For all times t 2R+, a random variable X(t) 2 S. Syntax ! Semantics: TCM ! CTMC For each transition class (Gi,ui,i) and all times t 2R+ and ! 0, Pr( X(t+) = ui(x) | X(t) = x ) = i(x) ¢. In addition, Pr( X(0) = x0 ) = 1 for some given initial state x02 S.