 Download Presentation Syntax Matters

# Syntax Matters

Download Presentation ## 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 Semantics Syntax 14

4. Mathematics Semantics Syntax 14 1110 XIV

5. Syntax Matters • Expressiveness 0

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

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

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

9. 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).

10. 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)))))

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))))) finite infinite

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 recursively enumerable 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))))) Composition: (defun h (n) (f (g n)))

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

15. 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

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

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

18. 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

19. Markovian Population Models

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

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

22. Markovian Population Models -discrete state -location unaware

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

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

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

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

27. 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

28. 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

29. 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

30. 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

31. 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

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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

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

40. 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)

41. 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)

42. 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)

43. 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+

44. 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

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

46. 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.