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Dec ecidab idability ility and Symb mbolic olic Ve Verif rification ication Kim Ki m G. . La Lars rsen Aa Aalb lborg org Univ iversity ersity, , DENMARK NMARK Dec Decid idabi ability lity Reachability chability ? a b

  1. Dec ecidab idability ility and Symb mbolic olic Ve Verif rification ication Kim Ki m G. . La Lars rsen Aa Aalb lborg org Univ iversity ersity, , DENMARK NMARK

  2. Dec Decid idabi ability lity

  3. Reachability chability ? a b OBSTACLE: Uncountably infinite state space c locations clock-valuations Reachable from initial state (L0,x=0,y=0) ? Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [3] School. l. September mber 2013.

  4. The he Regi gion on Abstr traction action Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [4] School. l. September mber 2013.

  5. Time me Abstracte tracted Bisim simulation ulation Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [5] School. l. September mber 2013.

  6. Regi gions ons – From Infinite to Finite Reset region THM [AD90] Successor Successor Successor Reachability is decidable Regions Regions regions (and PSPACE-complete) for timed automata + THM [CY90] Time-optimal reachability is decidable (and PSPACE-complete) for A region timed automata Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [6] School. l. September mber 2013.

  7. Regi gion on Graph aph Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [7] School. l. September mber 2013.

  8. Region gion Au Automaton omaton = = Finite nite Bisimulation simulation Qu Quotiont otiont Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [8] School. l. September mber 2013.

  9. An n Example ample Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [9] School. l. September mber 2013.

  10. Regi gion on Aut utoma omaton ton LARGE ARGE: : exponential in the number of clocks and in the constants (if encoded in binary). The number of regions is     | | !| 2 X (2 2) | M X x  x X Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [10 10] School. l. September mber 2013.

  11. Fun undamental mental Resul ults ts  Reachability   Model-checking  TCTL  ; MTL  ; MITL   Bisimulation, Simulation  Timed  ; Untimed   Trace-inclusion ; Untimed   Timed  Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [11 11] School. l. September mber 2013.

  12. Sym Symbol bolic ic Ve Veri rific fication ation The UPPAAL Verification Engine

  13. Regi gions ons – From om In Infinit finite e to to Fi Fini nite te + Regi gion on constru ructi ction: on: [AD94] 94] In practice: ce: Zones Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [13 13] School. l. September mber 2013.

  14. Zo Zone nes – From om Fini nite te to to Ef Effic ficienc iency A zone Z : 1 · x · 2 Æ 0 · y · 2 Æ x - y ¸ 0 Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [14 14] School. l. September mber 2013.

  15. Zo Zone nes - Op Operations ations (n, 2 · x · 4 Æ (n, 2 · x Æ (n, 2 · x Æ 1 · y · 3 Æ y-x · 0 ) 1 · y · 3 Æ y-x · 0 ) 1 · y Æ -3 · y-x · 0 ) y y y x x x Delay Delay (stopwatch) y y y (n, 2 · x · 4 Æ 1 · y ) (n, x=0 Æ 1 · y · 3 ) 2 x x x Reset Extrapolation Convex Hull Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [15 15] School. l. September mber 2013.

  16. Symbolic mbolic Transition ansitions 1<=x<=4 1<=x, 1<=y 1<=y<=3 y -2<=x-y<=3 y delays to x x x>3 y y 3<x, 1<=y conjuncts to -2<=x-y<=3 a x x y:=0 3<x, y=0 projects to Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [16 16] School. l. September mber 2013.

  17. For orwa ward rd Reachability chability Init -> Final ? INITIAL Passed := Ø; PW Waiting := {(n 0 ,Z 0 )} Waiting Final REPEAT pick (n,Z) in Waiting if (n,Z) = Final return true for all (n,Z)  ( n’,Z’): if for some ( n’,Z’’) Z’  Z’’ continue else add ( n’,Z’) to Waiting move (n,Z) to Passed UNTIL Waiting = Ø return false Init Passed Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [17 17] School. l. September mber 2013.

  18. For orwa ward rd Reachability chability Init -> Final ? INITIAL Passed := Ø; PW Waiting := {(n 0 ,Z 0 )} Waiting Final REPEAT pick (n,Z) in Waiting if (n,Z) = Final return true for all (n,Z)  (n’,Z’): if for some (n’,Z’’) Z’  Z’’ continue else add (n’,Z’) to Waiting move (n,Z) to Passed UNTIL Waiting = Ø return false Init Passed Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [18 18] School. l. September mber 2013.

  19. For orwa ward rd Reachability chability Init -> Final ? INITIAL Passed := Ø; PW Waiting := {(n 0 ,Z 0 )} Waiting Final? REPEAT pick (n,Z) in Waiting if (n,Z) = Final return true for all (n,Z)  (n’,Z’): if for some (n’,Z’’) Z’  Z’’ continue else add (n’,Z’) to Waiting move (n,Z) to Passed UNTIL Waiting = Ø return false Init Passed Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [19 19] School. l. September mber 2013.

  20. For orwa ward rd Reachability chability Init -> Final ? INITIAL Passed := Ø; PW Waiting := {(n 0 ,Z 0 )} Waiting Final REPEAT pick (n,Z) in Waiting if (n,Z) = Final return true for all (n,Z)  ( n’,Z’) : if for some ( n’,Z’’) Z’  Z’’ continue else add (n’,Z’) to Waiting move (n,Z) to Passed UNTIL Waiting = Ø return false Init Passed Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [20 20] School. l. September mber 2013.

  21. For orwa ward rd Reachability chability Init -> Final ? INITIAL Passed := Ø; PW Waiting := {(n 0 ,Z 0 )} Waiting Final REPEAT pick (n,Z) in Waiting if (n,Z) = Final return true for all (n,Z)  ( n’,Z’) : if for some ( n’,Z’’) Z’  Z’’ continue else add (n’,Z’) to Waiting move (n,Z) to Passed UNTIL Waiting = Ø return false Init Passed Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [21 21] School. l. September mber 2013.

  22. For orwa ward rd Reachability chability Init -> Final ? INITIAL Passed := Ø; PW Waiting := {(n 0 ,Z 0 )} Waiting Final REPEAT pick (n,Z) in Waiting if (n,Z) = Final return true for all (n,Z)  ( n’,Z’) : if for some ( n’,Z’’) Z’  Z’’ continue else add (n’,Z’) to Waiting move (n,Z) to Passed UNTIL Waiting = Ø return false Init Passed Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [22 22] School. l. September mber 2013.

  23. For orwa ward rd Reachability chability Init -> Final ? INITIAL Passed := Ø; PW Waiting := {(n 0 ,Z 0 )} Waiting Final REPEAT pick (n,Z) in Waiting if (n,Z) = Final return true for all (n,Z)  ( n’,Z’) : if for some ( n’,Z’’) Z’  Z’’ continue else add (n’,Z’) to Waiting move (n,Z) to Passed UNTIL Waiting = Ø return false Init Passed Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [23 23] School. l. September mber 2013.

  24. Symbolic mbolic Explora ploration tion y x Reachable? Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [24 24] School. l. September mber 2013.

  25. Symbolic mbolic Explora ploration tion y x Delay Reachable? Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [25 25] School. l. September mber 2013.

  26. Symbolic mbolic Explora ploration tion y x Left Reachable? Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [26 26] School. l. September mber 2013.

  27. Symbolic mbolic Explora ploration tion y x Left Reachable? Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [27 27] School. l. September mber 2013.

  28. Symbolic mbolic Explora ploration tion y x Delay Reachable? Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [28 28] School. l. September mber 2013.

  29. Symbolic mbolic Explora ploration tion y x Left Reachable? Verifi ifica catio ion n Theory, y, Systems s and Appli plica catio ions s Summer r Kim Larse sen [29 29] School. l. September mber 2013.

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