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Regular Model Checking Parosh Aziz Abdulla Uppsala University Cooperation with B. Jonsson, M. Nilsson, J. d’Orso.

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Presentation on theme: "Regular Model Checking Parosh Aziz Abdulla Uppsala University Cooperation with B. Jonsson, M. Nilsson, J. d’Orso."— Presentation transcript:

1 Regular Model Checking Parosh Aziz Abdulla Uppsala University Cooperation with B. Jonsson, M. Nilsson, J. d’Orso

2 Outline  Model Checking  Infinite-State Systems  Parameterized Systems  Regular Model Checking  Column Transducer Construction  Sufficient Conditions for Exactness  Future Work

3 Model Checking S sat  ? system specification

4 Infinite State Systems 1. Unbounded Data Structures Timed Automata Push-Down Automata Communicating Finite State Automata Counter Automata 2. Unbounded Control Structures Parameterized Systems Dynamic Systems

5 Parameterized Systems Mutual exclusion protocols Cache coherence protocols Broadcast protocols Dynamic Systems Security protocols Multi-threaded programs

6 Model Checking S sat  ? Parameterized systemspecification Classification S :  Topology  Components  Communication mechanisms  Safety properties  Liveness properties 

7 Topology set array

8 Tree

9 Matrix

10 Components Simple: finite state process Extended: clocks, counters, buffers, etc. Communication Mechanism binary (rendez-vous) broadcast Neighbour global

11 Simplest Case: Set + Finite-state + Rendez-vous W C W C W C Example: Parameterized mutual exclusion R=0? R:=1 R:=0R=0? R:=1 R:=0 R=0? R:=1 R:=0 Counter abstraction = Petri net

12 Petri Net Model W C R=0? R:=1 R:=0 W C R=0 Initial marking No token in C, 1 token in (R=0) Bad markings At least 2 tokens in C

13 Parameterized System of Finite-Sate Processes (Geman & Sistla)   Finite-state process Synchronize:  Parameterized System Petri Net Representation

14 Parameterized System of Timed Processes – (Timed Networks)   timed process Synchronize:  Parameterized System Timed Petri Net Representation x:=0 x<5 [0:0] [0:5]

15 Array of Finite-State Processes  in general: undecidable  use Regular Model Checking [Kesten et al 97]

16 Example: Szymanski’s Algorithm Pseudocode for process i 1: await  j : j  i ::  s j 2: w i, s i := true,true 3: if  j : j  i :: (pc j  1 /\  w j ) then s i := false; goto 4 else w i := false; goto 5 4: await  j : j  i :: (s j /\  w j ) then w i, s i := false,true 5: await  j : j  i ::  w j 6: await  j : j  i ::  s j 7: s i := false; goto 1

17 Linear Process Networks: Token Passing T NNNN 

18 N TNNN 

19 N NTNN 

20  Alphabet : S = {N, T }  Configurations : words over S  Initial Configurations : T N* (regular lang.)  Transition Relation : transducer : N/N T/NN/T N/N Token Passing: Model

21 N/N T/NN/T N/N T N N NInitial configuration (T N*) A Run of the Transducer : R

22 N/N T/NN/T N/N T N N N N T N N Initial configuration (T N*) A Run of the Transducer : R R

23 N/N T/NN/T N/N T N N N N T N N N N T N Initial configuration (T N*) A Run of the Transducer : R R R

24 N/N T/NN/T N/N T N N N N T N N N N T N N N N T Initial configuration (T N*) A Run of the Transducer : R R R R

25 N/N T/NN/T N/N T N* Initial configurations Symbolic Run of the Transducer : R

26 N/N T/NN/T N/N T N* N T N* Initial configurations Symbolic Run of the Transducer : R R

27 N/N T/NN/T N/N T N* N T N* N N T N* Initial configurations Symbolic Run of the Transducer : R R R

28 N/N T/NN/T N/N T N* N T N* N N T N* N N N T N* Initial configurations Symbolic Run of the Transducer : R R R R  Termination ?  Ideally: compute: R* (T N*) = N* T N*

29 N/N T/NN/T N/N T N N N N Column Transducer R q 0 q 1 q 2

30 N/N T/NN/T N/N T N N N N Column Transducer R q 0 q 1 q 2 q 2 q 0 q 1 q 2 q 2 q 2 N T N N N

31 N/N T/NN/T N/N T N N N N Column Transducer R q 0 q 1 q 2 q 2 q 0 q 1 q 2 q 2 q 2 N T N N N q 1 q 0 q 0 q 2 q 2 q 2 N N T N N

32 N/N T/NN/T N/N T N N N N Column Transducer R q 0 q 1 q 2 q 2 q 0 q 1 q 2 q 2 q 2 N T N N N q 1 q 0 q 0 q 2 q 2 q 2 N N T N N q 0 q 0 q 0 q 1 q 2 q 2 N N N T N

33 N/N T/NN/T N/N T N N N N Column Transducer R q 0 q 1 q 2 q 2 q 0 q 1 q 2 q 2 q 2 N T N N N q 1 q 0 q 0 q 2 q 2 q 2 N N T N N q 0 q 0 q 0 q 1 q 2 q 2 N N N T N q 0 q 0 q 0 q 0 q 1 q 2 N N N N T

34 N/N T/NN/T N/N T N N N N Column Transducer R q 0 q 1 q 2 q 2 q 0 q 1 q 2 q 2 q 2 N T N N N q 1 q 0 q 0 q 2 q 2 q 2 N N T N N q 0 q 0 q 0 q 1 q 2 q 2 N N N T N q 0 q 0 q 0 q 0 q 1 q 2 N N N N T

35 Column Transducer  Configurations: columns – members of S  Transitions :  Initial configurations : columns of initial states  Final configurations : columns of final states a q 0 r 0 b q 1 r 1 q 2 r 2 q 3 r 3 c d e x yx a e + y

36 N/N T/NN/T N/N Example : Token passing R q 0 q 1 q 2 q 0 q 0 q 0 q 0 q 0 q 0 initial columns : q 0 q 0 q 0 q 0 q 2 q 2 q 2 q 2 q 2 q 2 final columns : q 2 q 2 q 2 q 2 q 2 q 1 q 0 q 0 q 2 q 2 q 1 q 0 N N q 2 q 1 q 0 q 0 q 2 q 2 q 1 q 0 N N N T N and therefore transitions : e.g.

37 N/N T/NN/T N/N Example : Token passing R q 0 q 1 q 2 q 0 q 0 q 0 q 0 q 0 q 0 initial columns : q 0 q 0 q 0 q 0 q 2 q 2 q 2 q 2 q 2 q 2 final columns : q 2 q 2 q 2 q 2  Transducer language = transitive closure  Problem : number of columns infinite !!  Solution: abstraction !! =

38 Computing Abstract Transducer  Start with original transducer  repeat  Define equivalence on columns  until construction stabilizes

39 Computing Abstract Transducer  Start with initial configurations (columns)  repeat then add  Define equivalence on columns xz a b y w b c if and XyXy zwzw a c  until construction stabilizes

40 Computing Abstract Transducer  Start with initial configurations (columns)  repeat then add  Define equivalence on columns if x y then merge x and y xz a b y w b c if and XyXy zwzw a c  until construction stabilizes

41 Defining Left-copying states Right-copying states Non-copying states N T N T T T N N T T

42 Defining Left-copying states Right-copying states Non-copying states N T N T T T N N T T x y if x = y modulo deletion of identical left- or right-copying neighbours

43 N/N T/NN/T N/N Example : Token passing R q 0 q 1 q 2 Left-copying state : Right-copying state : q 02 q q 0 q 0 q 1 q 2 q 2 q 0 q 1 q 2 q 2

44 N/N N/T N/N Example : Token passing q 2 T/N q 1 q 0

45 N/N T/N N/T N/N Example : Token passing q 2 q 0 q 0 q 1 q 0 T/N q 1 q 0

46 N/N T/N N/T N/N Example : Token passing q 2 q 0 q 0 q 1 q 0 T/N q 1 q 0

47 N/T N/N Example : Token passing q 2 q 1 q 0 T/N q 1 q 0 N/N

48 T/N N/T N/N Example : Token passing q 2 q 1 q 0 T/N q 1 q 0 q 2 q 1 N/N

49 T/N N/T N/N Example : Token passing q 2 q 1 q 0 T/N q 1 q 0 q 2 q 1 q 2 q 2 N/T N/N

50 T/N N/T N/N Example : Token passing q 2 q 1 q 0 T/N q 1 q 0 q 2 q 1 q 2 q 2 N/T N/N

51 T/N N/T N/N Example : Token passing q 2 q 1 q 0 T/N q 1 q 0 q 2 q 1 N/T N/N

52 T/N N/T N/N Example : Token passing q 2 q 1 q 0 T/N q 1 q 0 q 2 q 1 N/T q 0 q 2 q 1 q 0 q 1 q 0 N/N

53 T/N N/T N/N Example : Token passing q 2 q 1 q 0 T/N q 1 q 0 q 2 q 1 N/T q 0 q 2 q 1 q 0 q 1 q 0 N/N

54 T/N N/T N/N Example : Token passing q 2 T/N q 1 q 0 q 2 q 1 N/T q 2 q 1 q 0 q 1 q 0 N/N

55 T/N N/T N/N Example : Token passing q 2 T/N q 1 q 0 q 2 q 1 N/T q 2 q 1 q 0 q 1 q 0 N/N q 2 q 2 q 1

56 T/N N/T N/N Example : Token passing q 2 T/N q 1 q 0 q 2 q 1 N/T q 2 q 1 q 0 q 1 q 0 N/N q 2 q 2 q 1

57 T/N N/T N/N Example : Token passing q 2 T/N q 1 q 0 q 2 q 1 N/T q 2 q 1 q 0 q 1 q 0 N/N

58 T/N N/T N/N Example : Token passing q 2 T/N q 1 q 0 q 2 q 1 N/T q 2 q 1 q 0 q 1 q 0 N/N q 2 q 1 q 0 q 0

59 T/N N/T N/N Example : Token passing q 2 T/N q 1 q 0 q 2 q 1 N/T q 2 q 1 q 0 q 1 q 0 N/N q 2 q 1 q 0 q 0

60 T/N N/T N/N Example : Token passing q 2 T/N q 1 q 0 q 2 q 1 N/T q 2 q 1 q 0 q 1 q 0 N/N

61 initial states equivalence class final states x y Exactness of

62 initial states equivalence class final states x y Exactness of z

63 initial states equivalence class final states x y Exactness of z How to define ?

64 Forward Simulation F x1x1 x2x2 y1y1 F

65 F x1x1 x2x2 y1y1 F  y2y2 F

66 F x1x1 x2x2 y1y1 F  y2y2 F Backward Simulation B x1x1 y1y1 y2y2 B

67 Forward Simulation F x1x1 x2x2 y1y1 F  y2y2 F Backward Simulation B x1x1 y1y1 y2y2 B  y1y1 B

68 x y  z  w y x F F B B iff Equivalence FB, independent: y  w z F B F B x

69 Example B xyx = y modulo deletion of identical left-copying neighbours

70 Example B xyx = y modulo deletion of identical left-copying neighbours q 0 q 0 q 1 q 2 q 0 q 1 q 2 B

71 Example B xyx = y modulo deletion of identical left-copying neighbours q 0 q 0 q 1 q 2 q 0 q 1 q 2 B F xy q 0 q 1 q 2 F x = y modulo deletion of identical right-copying neighbours q 0 q 1 q 2 q 2

72 q 0 q 0 q 1 q 2 q 0 q 0 q 0 q 1 q 2 q 0 q 0 q 0 q 1 q 2 q 0 q 2 Independence F B

73 q 0 q 0 q 1 q 2 q 0 q 0 q 0 q 1 q 2 q 0 q 0 q 0 q 1 q 2 q 0 q 2 F B F B

74 q 0 q 0 q 1 q 2 q 0 q 0 q 0 q 1 q 2 q 0 q 0 q 0 q 1 q 2 q 0 q 2 q 0 F B F B

75 q 0 q 0 q 1 q 2 q 0 q 0 q 0 q 1 q 2 q 0 q 0 q 0 q 1 q 2 q 0 q 2 q 0 q 1 F B F B

76 q 0 q 0 q 1 q 2 q 0 q 0 q 0 q 1 q 2 q 0 q 0 q 0 q 1 q 2 q 0 q 2 q 0 q 1 q 2 q 2 F B F B

77 q 0 q 0 q 1 q 2 q 0 q 0 q 0 q 1 q 2 q 0 q 0 q 0 q 1 q 2 q 0 q 2 q 0 q 0 q 1 q 2 q 0 q 2 F B F B

78 Example B xyx = y modulo deletion of identical left-copying neighbours F xy x = y modulo deletion of identical right-copying neighbours xx = y modulo deletion of identical left- or right-copying neighbours y Induced equivalence :

79 Consequence w F x y

80 w F x y z  B F

81 [x 0 ][x 1 ][x 2 ][x 3 ] y1y1 y2y2 y3y3

82 [x 0 ][x 1 ][x 2 ][x 3 ] y1y1 w0w0 F y2y2 y3y3 x 0 =

83 [x 0 ][x 1 ][x 2 ][x 3 ] y1y1 w0w0 v1v1 FF y2y2 y3y3 x 0 =

84 [x 0 ][x 1 ][x 2 ][x 3 ] y1y1 w0w0 v1v1 FF B y2y2 w1w1 F y3y3 x 0 =

85 [x 0 ][x 1 ][x 2 ][x 3 ] y1y1 w0w0 v1v1 FF B y2y2 w1w1 v2v2 FF B y3y3 F w2w2 x 0 =

86 [x 0 ][x 1 ][x 2 ][x 3 ] y1y1 w0w0 v1v1 FF B y2y2 w1w1 v2v2 FF B y3y3 w3w3 v3v3 FF B w2w2 F x 0 =

87 [x 0 ][x 1 ][x 2 ][x 3 ] y1y1 w0w0 v1v1 FF B y2y2 w1w1 v2v2 FF B y3y3 w3w3 v3v3 FF B w2w2 F z3z3 B x 0 = w3w3

88 [x 0 ][x 1 ][x 2 ][x 3 ] y1y1 w0w0 v1v1 FF B y2y2 w1w1 v2v2 FF B y3y3 w3w3 v3v3 FF B w2w2 F z3z3 B z2z2 B x 0 = w3w3

89 [x 0 ][x 1 ][x 2 ][x 3 ] y1y1 w0w0 v1v1 FF B y2y2 w1w1 v2v2 FF B y3y3 w3w3 v3v3 FF B w2w2 F z3z3 B z2z2 B z1z1 B x 0 = w3w3

90 [x 0 ][x 1 ][x 2 ][x 3 ] y1y1 w0w0 v1v1 FF B y2y2 w1w1 v2v2 FF B y3y3 w3w3 v3v3 FF B w2w2 F z3z3 B z2z2 B z1z1 B z0z0 B x 0 = w3w3

91 Other Examples: Szymanski’s Algorithm (idealized) Pseudocode for process i 1: await  j : j  i ::  s j 2: w i, s i := true,true 3: if  j : j  i :: (pc j  1 /\  w j ) then s i := false; goto 4 else w i := false; goto 5 4: await  j : j  i :: (s j /\  w j ) then w i, s i := false,true 5: await  j : j  i ::  w j 6: await  j : j  i ::  s j 7: s i := false; goto 1

92 Built states in transitive closures

93 All implementation available Implementation of automata with symbolic edges (BDDs) Source available under GPL

94 Future Work Tree-like Topologies Liveness properties Non-structure-preserving Other kinds of systems: stacks, queues, timed, etc


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