# Introduction to Software Verification

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1 Introduction to Software Verification Orna Grumberg Lectures Material winter

2 Lecture

3 Model Checking Automated formal verification: A different approach to formal verification

4 Model Checking [CE81,QS82] An efficient procedure that receives: A finite-state model describing a system A temporal logic formula describing a property It returns yes, if the system has the property no + Counterexample, otherwise 4

5 Building a model from a program 5

6 Back to the Mutual Exclusion Example Two processes with a joint Boolean signal sem Each process P i has a variable v i describing its state: v i := N Non critical v i := T Trying v i := C Critical

7 Mutual Exclusion Example Each process runs the following program: Atomic action P i :: while (true) { } if (v i == N) v i = T; else if (v i == T && sem) { v i = C; sem = 0; } else if (v i == C) {v i = N; sem = 1; } The full program is: P 1 P 2 Initial state: (v 1 =N, v 2 =N, sem) The execution is interleaving

8 Mutual Exclusion Example v 1 =N, v 2 =N, sem v 1 =T, v 2 =N, sem v 1 =N, v 2 =T, sem v 1 =C, v 2 =N, sem v 1 =T, v 2 =T, sem v 1 =N, v 2 =C, sem v 1 =C, v 2 =T, sem v 1 =T, v 2 =C, sem

9 v 1 =N, v 2 =N, sem T 1 v 1 =T, v 2 =N, sem v 1 =N, v 2 =T, sem C 1 v 1 =C, v 2 =N, sem v 1 =T, v 2 =T, sem v 1 =N, v 2 =C, sem C 1,T 2 v 1 =C, v 2 =T, sem v 1 =T, v 2 =C, sem We define atomic propositions: AP={C 1,C 2,T 1,T 2 ) A state is marked with T i if v i =T A state is marked with C i if v i =C

10 T 1 T 2 C 1 T 1,T 2 C 2 C 1,T 2 T 1,C 2 Property 1: AG (C 1 C 2 ) Property 2: AG((T 1 FC 1 ) (T 2 FC 2 ))

11 Temporal Logic

12 Kripke Structure M=(S,R,L,S 0 ) Given AP finite set of atomic proposition S (finite) set of states R S S total transition relation For every s S there exists s S such that (s,s ) R. Totality means that every path is infinite L:S 2 AP - labeling function that associates every state with the atomic propositions true in that state S 0 S set of initial states (optional)

13 = s 0,s 1, is a path in M from a state s if s = s 0 and R(s i,s i+1 ) for every I i the suffix of starting at s i 13

14 Temporal Logics Express properties of event orderings in time Linear Time Every moment has a unique successor Infinite sequences (words) Linear Time Temporal Logic (LTL) Branching Time Every moment has several successors Infinite tree Computation Tree Logic (CTL) 14

15 Propositional Temporal Logic AP a set of atomic propositions, p AP Temporal operators: Xp Gp Fp puq Path quantifiers: A for all path E there exists a path 15

16 State formulas: p AP CTL* g 1, g 1 g 2, g 1 g 2 where g 1,g 2 are state formulas Ef, Af where f is a path formula Path formulas: Every state formula g is a path formula f 1, f 1 f 2, f 1 f 2, Xf 1, Gf 1, Ff 1, f 1 Uf 2 where f 1,f 2 are path formulas CTL* - set of all state formulas 16

17 Semantics of CTL* The truth value of CTL* formula f is determined with respect to A Kripke structure M = (S, R, L), and A state s S, if f is a state formula A path, if f is a path formula 17

18 Semantics of CTL* 18

19 Semantics of CTL* 19

20 Semantics of CTL* 20

21 Semantics of CTL* 21

22 Semantics of CTL* 22

23 Semantics of CTL* 23

24 Semantics of CTL* 24

25 Semantics of Path Formulas If p,q are state formulas then: Xp Gp Fp puq But in the general case, they can be path formulas 25

26 Semantics of CTL* 26

27 LTL/CTL/CTL* LTL state formulas of the form A - path formula, contains no path quantifiers interpreted over infinite computation paths CTL state formulas where path quantifiers and temporal operators appear in pairs: AG, AU, AF, AX, EG, EU, EF, EX interpreted over infinite computation trees CTL* - Allows any combination of temporal operators and path quantifiers. Includes both LTL and CTL 27

28 State formulas: LTL Af where f is a path formula Path formulas: p AP f 1, f 1 f 2, f 1 f 2, Xf 1, Gf 1, Ff 1, f 1 Uf 2 where f 1,f 2 are path formulas LTL - set of all state formulas 28

29 CTL CTL - set of all state formulas p AP g 1, g 1 g 2, g 1 g 2 AG g 1, A g 1 U g 2, AF g 1, AX g 1, EG g 1, E g 1 U g 2, EF g 1, EX g 1 where g 1,g 2 are state formulas 29

30 Semantics of CTL Recall: path =s 0,s 1, M,s p p L(s) for p AP M,s 1 2 M,s 1 or M,s 2 M,s EX there is s s.t. R(s,s ) and M,s M,s EG there is a path from s, s.t. for every i 0, M,s i 30

31 Semantics of CTL M,s E[ 1 U 2 ] there is a path from s and there is k 0 s.t. M,s k 2 and for every k>i 0, M,s i 1 M,s AG for every path from s and for every i 0, M,s i M,s AF for every path from s there exists i 0 s.t. M,s i 31

32 Examples (LTL) AG (start ready) AG (req F ack) A GF enbled A FG deadlock A (GF enbled GF running) Cannot express existential properties: from any state the system can 32

33 Examples (CTL) EF (start ready) AG (req AF ack) AG (AF enbled) AF (AG deadlock) AG (EF restart) AG (non_critical EX tryring) AG (try A[try U succeed]) 33

34 Illustration of CTL Semantics EFp : AFp : exists reachable state s.t. EGp : AGp : all reachable states. 34

35 Property types Universal Existential Safety AGp EGp Liveness AFp EFp 35

36 Examples 36

37 Equivalence 37

38 Expressiveness,, X, U, E suffice to express all CTL*: Ff true U f Gf F ( f) Af E ( f) EG f E F ( f) E [true U ( f)] In CTL: EX, EG, EU are sufficient A [puq] ( EG q) E[ q U ( p q)] 38

39 LTL vs. CTL A (FG p) has no equivalent in CTL in all paths, p globally holds from some point on Failed attempts: AFAGP : in every path there is a point from which all reachable states satisfy p. s 0 p s 1 s 2 p - s 0,s 0,s 0, - s 0,s 0, s 0,s 1,s 2,s 2,s 2 All paths satisfy FGp But first one does not sat FAGp 39

40 LTL vs. CTL A (FG p) has no equivalent in CTL in all paths, p globally holds from some point on What about AFEGP? in every path there is a point from which there is a path where p globally holds s 0 s All paths satisfy FEGp 1 p - since s 1 sat EGp But s 0,s 1,s 0,s 1,s 0,s 1 does not sat FGp 40

41 LTL vs. CTL AG (EFp) has no equivalent in LTL all reachable states can reach p Failed attempt: AGFp : in all paths, p holds infinitely many times. s 0 s 1 All reachable states (s 0, s 1 ) satisfy EFp p But s 0,s 0,s 0, does not satisfy GFp 41

42 LTL and CTL vs. CTL* E (GFp) has no equivalent in LTL or CTL 42

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