how to create a structure of kripke in NuSMV? - model-checking

Question

i must to create a structure of Kripke in NuSMV and i must to check some properties.
Anybody help me? The structure and the properties(LTL, CTL and CTL*) are in the pictures.
Here there is a structure and properties:
http://cl.ly/image/1x0b1v3E0P0D/Screen%20Shot%202014-10-16%20at%2016.52.34.png

Answer 1

I found a simpler and seemingly more reliable NuSMV code for your Kripke Structure. Thanks to dejvuth for his answer to my question. The code is as follows
MODULE main
VAR
state : {s0,s1,s2,s3,s4};
ASSIGN
init(state) := s0;
next(state):=
case
state = s0 : {s1,s2};
state = s1 : {s1,s2};
state = s2 : {s1,s2,s3};
state = s3 : {s1,s4};
state = s4 : {s4};
esac;
DEFINE
p := state = s1 | state = s2 | state = s3 | state = s4;
q := state = s1 | state = s2;
r := state = s3;
SPEC
EG p;
SPEC
AG p;
SPEC
EF (AG p);

Answer 2

As far as I know NuSMV only handles LTL and CTL formulas (see NuSMV in Wikipedia). The formulas in problem 1-3 are CTL formulas, hence it can be model-checked by NuSMV. However the formulas in problem 4 & 5 are CTL* formulas, and thus we cannot straightforwardly use them as an input to NuSMV. You also need to understand that the set of all CTL* formulas is the proper superset of the union of all LTL and CTL formulas. This conditions implies that some CTL* formulas do not have their equivalent LTL or CTL formulas (see CTL* in Wikipedia). Your Kripke structure can be defined in NuSMV by following code:
MODULE main
VAR
p : boolean;
q : boolean;
r : boolean;
state : {s0,s1,s2,s3,s4};
ASSIGN
init (state) := s0;
next (state) :=
case
state = s0 : {s1, s2};
state = s1 : {s1, s2};
state = s2 : {s1, s2, s3};
state = s3 : {s1, s4};
state = s4 : {s4};
TRUE : state;
esac;
init (p) := FALSE;
init (q) := FALSE;
init (r) := FALSE;
next(p) :=
case
state = s1 | state = s2 | state = s3 | state = s4 : TRUE;
TRUE : p;
esac;
next(q) :=
case
state = s1 | state = s2 : TRUE;
state = s3 | state = s4 : FALSE;
TRUE : q;
esac;
next(r) :=
case
state = s3 : TRUE;
state = s1 | state = s2 | state = s4 : FALSE;
TRUE : r;
esac;
SPEC
EG p;
SPEC
AG p;
SPEC
EF (AG p);
Of course, there is another way to define your Kripke structure in NuSMV, but I think this is one of the easiest. (Anyway, thanks for helping me with my problem).
As for the formulas in problem 4 & 5, here is my answer.
The formula AF [p U EG ( p -> q)] is of the form AF [\phi], where \phi is an LTL formula p U EG (p->q). Since the LTL formula \phi is satisfied in a Kripke model if for every path starting at s0 we have the satisfaction of \phi, then we translate AF [p U EG ( p -> q)] into AF A[p U EG ( p -> q)].
By similar argument, we translate EG[(( p & q ) | r) U ( r U AG p)] into EG[A(( p & q ) | r) U A( r U AG p)].