NuSMV: difference between IVAR and VAR - models

Question

I know the difference between IVAR (input variables) and VAR (state variables) in NuSMV. However, I am able to understand the counterexample when VAR is used, but I am not in the other case.
Let me show it with an example.
MODULE main
VAR
v1: 0..20;
v2: 0..20;
v3: 0..100;
INIT
v3 = 0;
TRANS
((v2+v1 = 0) -> (next(v3) = 10)) &
(!(v2+v1 = 0) -> (next(v3) = v1 + v2))
LTLSPEC
G(v3 = 10);
The counterexample (clear enough) given by NuSMV is:
Trace Type: Counterexample
-> State: 1.1 <-
v1 = 0
v2 = 0
v3 = 0
-- Loop starts here
-> State: 1.2 <-
v3 = 10
-> State: 1.3 <-
v1 = 7
v2 = 6
-> State: 1.4 <-
v1 = 0
v2 = 0
v3 = 13
-> State: 1.5 <-
v3 = 10
Now, change v1 and v2 to be IVAR.
MODULE main
IVAR
v1: 0..20;
v2: 0..20;
VAR
v3: 0..100;
INIT
v3 = 0;
TRANS
((v2+v1 = 0) -> (next(v3) = 10)) &
(!(v2+v1 = 0) -> (next(v3) = v1 + v2))
LTLSPEC
G(v3 = 10);
The counterexample is:
Trace Type: Counterexample
-> State: 1.1 <-
v3 = 0
-> Input: 1.2 <-
v1 = 7
v2 = 3
-- Loop starts here
-> State: 1.2 <-
v3 = 10
-> Input: 1.3 <-
-- Loop starts here
-> State: 1.3 <-
-> Input: 1.4 <-
-- Loop starts here
-> State: 1.4 <-
-> Input: 1.5 <-
-- Loop starts here
-> State: 1.5 <-
-> Input: 1.6 <-
-- Loop starts here
-> State: 1.6 <-
-> Input: 1.7 <-
-> State: 1.7 <-
Could someone explain why this counterexample is so strange? It has several nested loops. What does the output mean?

Answer 1

Both counter-examples falsify the property in the very first state of the execution trace, so what happens next is somewhat not much relevant.
From the docs of nuXmv:
4.7 Traces
A trace consists of an initial state, optionally followed by a
sequence of states-inputs pairs corresponding to a possible execution
of the model. Apart, from the initial state, each pair contains the
inputs that caused the transition to the new state, and the new state
itself. The initial state has no such input values defined as it does
not depend on the values of any of the inputs. [...]
So a trace generally has the following structure:
S_0 | I_0 -> S_1 | I_1 -> S_2 | ... | I_{N-1} -> S_N
The idea is that a state S_{k+1} is obtained by applying inputs I_k over state S_k.
One can try using the commands goto_state and print_current_state, to navigate the counter-example and print each state's content. Alternatively, one may recall that NuSMV and nuXmv only print changing variables from one state to the next, so the execution trace should look like this:
-> State: 1.1 <-
v3 = 0
-> Input: 1.2 <-
v1 = 7
v2 = 3
-- Loop starts here
-> State: 1.2 <-
v3 = 10
-> Input: 1.3 <-
v1 = 7
v2 = 3
-- Loop starts here
-> State: 1.3 <-
v3 = 10
-> Input: 1.4 <-
v1 = 7
v2 = 3
-- Loop starts here
-> State: 1.4 <-
v3 = 10
-> Input: 1.5 <-
v1 = 7
v2 = 3
-> State: 1.5 <-
v3 = 10
So basically after the first transition we end up in the same state forever with never-changing inputs and outputs.
You may want to contact NuSMV or nuXmv mailing list and mention this issue with the output routine.