1. Chapter 4: Model Checking of Finite State Systems
Albert M. K. Cheng

2. Model Checking Is the finite-state graph a model of the temporal
logic formula?
Specification
represented as
a labeled
finite-state
Graph (Kripke
structure)
Safety assertion
written as temporal
logic formula

3. 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

4. Example: Solution to Mutual Exclusion Problem
N1,N2
T1,N2
N1,T2
C1,N2
T1,T2
T1,T2
N1,C2
C1,T2
T1,C2

5. 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)

6. 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);
case ax:
axf(fi,s,b);
case ex:
exf(fi,s,b);

7. Explicit-State Model Checking
case au:
for (i=0; i <= numstates; i++)
marked[i] = false;
if (!marked[i])
auf(fi,s,b);
break;
case eu:
euf(fi,s,b); }

8. 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

9. 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