1. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Timed Automata and Timed Computation Tree Logic Paul Pettersson BRICS@Aalborg

2. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. CTL Models = Kripke Structures

3. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Computation Tree Logic, CTL Clarke & Emerson 1980 Syntax

4. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. CTL, Derived Operators... p pp AF p... p EF p possible inevitable ”exists eventually” or ”reachable” ”exists globally”

5. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. CTL, Derived Operators pp p... AG p pppp p p... EG p p always potentially always for all paths next ”forall globally” or ”invariantly” ”forall eventually”

6. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Exercise 18

7. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Exercise 22

8. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Exercise 22

9. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Exercise 22

10. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Exercise 22

11. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Exercise 22

12. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Exercise 22

13. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Exercise 22

14. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Exercise 22

15. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. OffLightBright press?Press? press? Press? WANT: if press is issued twice quickly then the light will get brighter; otherwise the light is turned off. Timed Automata Intelligent Light Control

16. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Timed Automata Intelligent Light Control OffLightBright Solution: Add real-valued clock x X:=0 X<=3 X>3 press?Press? press? Press?

17. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Timed Automata n m a (Alur & Dill 1990) Clocks: x, y x 3 x := 0 Guard Boolean combination of comp with integer bounds Reset Action perfumed on clocks Transitions ( n, x=2.4, y=3.1415 ) ( n, x=3.5, y=4.2415 ) e(1.1) ( n, x=2.4, y=3.1415 ) ( m, x=0, y=3.1415 ) a State ( location, x=v, y=u ) where v,u are in R Action used for synchronization

18. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. n m a Clocks: x, y x 3 x := 0 Transitions ( n, x=2.4, y=3.1415 ) ( n, x=3.5, y=4.2415 ) e(1.1) ( n, x=2.4, y=3.1415 ) e(3.2) x<=5 y<=10 Location Invariants g1 g2 g3 g4 Invariants ensure progress!! Timed Safety Automata = Timed Automata + Invariants (Henzinger et al, 1992)

19. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Clock Constraints

20. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Timed (Safety) Automata

21. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Timed Automata: Example guard reset location

22. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Timed Automata: Example guard reset location

23. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Timed Automata: Example

24. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Timed Automata: Example

25. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Timed Automata: Example

26. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Timed Automata: Example

27. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Light Switch push click

28. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Light Switch Switch may be turned on whenever at least 2 time units has elapsed since last “turn off” push click

29. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Light Switch Switch may be turned on whenever at least 2 time units has elapsed since last “turn off” Light automatically switches off after 9 time units. push click

30. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Semantics clock valuations: state: Semantics of timed automata is a labeled transition system where action transition delay Transition g a r ll’

31. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Semantics: Example push click

32. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Networks of Timed Automata + Integer Variables + arrays …. l1 l2 a! x>=2 i==3 x := 0 i:=i+4 m1 m2 a? y<=4 …………. Two-way synchronization on complementary actions. Closed Systems! (l1, m1,………, x=2, y=3.5, i=3,…..) (l2,m2,……..,x=0, y=3.5, i=7,…..) (l1,m1,………,x=2.2, y=3.7, I=3,…..) 0.2 tau Example transitions If a URGENT CHANNEL

33. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Timed Automata in U PPAAL Timed Safety Automata + urgent actions + urgent locations (i.e. zero-delay locations) + committed locations (i.e. zero-delay and atomic locations) + data-variables (integers with bounded domains) + arrays of data-variables + guards and assignments over data-variables and arrays...

34. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Urgent and Committed Locations m n o p q r 2.5 a committed urgent d d

35. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. TCTL = CTL + Time constraints over formula clocks and automata clocks “freeze operator” introduces new formula clock z E[  U  ], A[  U  ] - like in CTL No EX 

36. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Derived Operators Along any path  holds continuously until within 7 time units  becomes valid. = = The property  becomes valid within 5 time units.

37. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Paths Example: push click

38. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Elapsed time in path Example:  

39. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. TCTL Semantics s - location w - formula clock valuation P M (s) - set of paths from s Pos(  ) - positions in  ,i) - elapsed time  (i,d) <<(i’,d’) iff (i<j) or ((i=j) and (d<d’))

40. Real-Time Systems, DTU, Feb 15, 2000 Paul Pettersson, BRICS, Aalborg, Denmark. Timeliness Properties receive(m) occurs within 5 time units after send(m) receive(m) occurs exactly 11 time units after send(m) putbox occurs periodically (exactly) every 25 time units (note: other putbox’s may occur in between)