Finite State Model Checking

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1 Finite State Model Checking

2 Finite State Model Checking Finite State Systems System Descrition A Requirement F CTL TOOL No! Debugging Information Yes, Prototyes Executable Code Test sequences Tools: visualstate, SPIN, Statemate, Verilog, Formalcheck,...

3 From Programs to Networks P1 P1 :: :: while True do do T1 T1 : wait(turn=1) C1 C1 : turn:=0 endwhile P2 P2 :: :: while True do do T2 T2 : wait(turn=0) C2 C2 : turn:=1 endwhile Mutual Exclusion Program

4 From Network Models to Krike Structures T1 I2 I1 I2 T1 T2 I1 T2 T1 C2 I1 C2 C1 I2 T1 I2 C1 T2 I1 I2 T1 T2 I1 T2

5 CTL Models = Krike Structures

6 Comutation Tree Logic, CTL Clarke & Emerson 1980 Syntax

7 Path s s 1 s 2 s 3... The set of ath starting in s

8 Formal Semantics ( )

9 ossible inevitable AF CTL, Derived Oerators EF

10 otentially always always EG CTL, Derived Oerators AG

11 Theorem A All oerators are derivable from EX EX f f EG EG f f E[ E[ f f U g ] and boolean connectives [ f U g] E[ gu( f g) ] EG g

12 Examle 1 2 4,q q 3

13 Examle EX 1 2 4,q q 3

14 Examle EX 1 2 4,q q 3

15 Examle AX 1 2 4,q q 3

16 Examle AX 1 2 4,q q 3

17 Examle EG 1 2 4,q q 3

18 Examle EG 1 2 4,q q 3

19 Examle AG 1 2 4,q q 3

20 Examle AG 1 2 4,q q 3

21 Examle A[ U q ] 1 2 4,q q 3

22 Examle A[ U q ] 1 2 4,q q 3

23 Proerties of MUTEX examle? T1 I2 I1 I2 T1 T2 AG (C AG[ EG AG C I1 T2 T 1 C AF(C [ C1] [ A[ C U ( C A[ C U C ]) ] 1 1 I1 C2 2 ) 1 1 )] 1 C1 I2 1 T1 I2 2 HOW to DECIDE IN GENERAL I1 I2 T1 T2 I1 T2 T1 C2 C1 T2

24 CTL Model Checking Algorithms

25 Fixoint Characterizations EF EXEF or let A be the set of states satisfying A EX A in fact A is the smallest such set (the least fixoint) EF then

26 Examle 1 2,q q 3 EF q 4 A q EX A

27 Fixed oints of monotonic functions Let τ be a function 2 S 2 S Say τ is monotonic when Fixed oint of τ is y such that If τ monotonic, then it has x imlies least fixed oint μy. τ(y) greatest fixed oint νy. τ(y) y τ ( y ) = y τ ( x) τ ( y)

28 Iteratively comuting fixed oints Suose S is finite The least fixed oint μy. τ(y) is the limit of false τ (false) τ ( τ (false)) L The greatest fixed oint νy. τ(y) is the limit of true τ (true) τ ( τ (true)) L Note, since S is finite, convergence is finite

29 Examle: EF EF is characterized by EF = μy. ( EX y) Thus, it is the limit of the increasing series EX EX( EX )

30 Examle: EG EG is characterized by EG = ν y. ( EX y) Thus, it is the limit of the decreasing series EX( EX ) EX

31 Examle, continued EF q 1 2,q q 3 EF q = μy. ( q EX y) A A A A = Ø 4 = {2,3} = {1,2,3} = {1,2,3}

32 Remaining oerators )) (.( ) ( )) (.( ) ( ).( ).( y AX q y U q A y EX q y U q E y AX y AG y AX y AF = = = = μ μ ν μ

33 Proerties of MUTEX examle? T1 I2 I1 I2 T1 T2 I1 T2 AG[ T1 AF(C AF(C )] I1 C2 1 1 C1 I2 )] T1 I2 I1 I2 T1 T2 I1 T2 T1 C2 C1 T2

36 ({ s s '.( s, s ') R s ' Q } Sat ( φ ))

37 More Efficient Check EG SCC SCC SCC

38 Examle EG q,q q

39 Examle,q EG Reduced Model

40 Examle EG Non trivial Strongly Connected Comonent

41 Proerties of MUTEX examle? T1 I2 I1 I2 T1 T2 I1 T2 [ ] EG C 1 I1 C2 C1 I2 T1 I2 I1 I2 T1 T2 I1 T2 T1 C2 C1 T2

42 Proerties of MUTEX examle? T1 I2 I1 I2 T1 T2 I1 T2 [ ] EG C 1 I1 C2 T1 I2 I1 I2 T1 T2 I1 T2 T1 C2 Reduced Model which are the non-trivial SCC s?

43 Comlexity However SS sys may sys be beexponential in in number of ofarallel comonents! FIXPOINT COMPUTATIONS may be becarried out out using ROBDD s (Reduced Ordered Binary Decision Diagrams) Bryant, 86 86

Computation Tree Logic

Computation Tree Logic Comutation Tree Logic Finite State Model Checking of Branching Time Logic Kim Guldstrand Larsen BRICS@Aalborg 1 Tool Suort Finite State Systems System Descrition A Reuirement F CTL TOOL Course Objectives: