Chapter 4: Model Checking of Finite State Systems

Albert M. K. Cheng

Model Checking

Specificationrepresented asa labeledfinite-stateGraph (Kripkestructure)

Safety assertionwritten as temporallogic formula

Is the finite-state grapha model of the temporallogic formula?

Computation Tree Logic CTL Propositional,

branching-time temporal logic

Next-time operator X, Until operator U

A(E)X f : f holds in every (some) immediate successor of current state

A(E)[f1 U f2] : for every (some) computation path, there exists an initial prefix of the path such that f2 holds at the last state of the prefix and f1 holds at all other states along the prefix

Example; Solution to Mutual Exclusion Problem

N1,N2

T1,N2

C1,T2

N1,T2

C1,N2 T1,T2 T1,T2 N1,C2

T1,C2

CTL abbreviations AF(f) = A[True U f]: f holds in the future

along every path from the initial state s0, so f is inevitable

EG(f) = NOT AF(NOT f)

EF(f) = E[True U f]: there is some path from the initial state s0 that leads to a state at which f holds, so f potentially holds

AG(f) = NOT EF(NOT f)

Explicit-State Model Checking

for (fi=flength; fi >= 1; fi--) labelgraph(fi,s,&correct);labelgraph (fi,s,b)short fi, s;Boolean *b;{ short i; switch(nf[fi-1][0].opcode) { case atomic: atf(fi,s,b); break;

case nt: ntf(fi,s,b); break; case ad: adf(fi,s,b); break; case ax: axf(fi,s,b); break; case ex: exf(fi,s,b); break;

case au:

for (i=0; i <= numstates; i++)

marked[i] = false;

for (i=0; i <= numstates; i++)

if (!marked[i])

auf(fi,s,b);

break;

case eu:

euf(fi,s,b);

break;

}

}

Explicit-State Model Checking

Symbolic Model Checking Transition relation between the values of

the variables in the current and the next states can be stated as a Boolean formula

Use Binary Decision Diagrams (BDDs) to present this Boolean formula

Apply model checker to finite-state graph represented as BBDs

Real-Time CTL Existentially Bounded Until operator: E[f_1 U[x,y] f_2] at state s_0 means there

exists a path beginning at s_0 and some i such that x <= i <= y and f_2 holds at state s_i and forall j < i, f_1 holds at state s_j

Min/max delays Min/max number of condition occurrences