NuSMV model checking: create a simple game model - logic

Question

I'm new to NuSMV and try to model this simple turn-based game. There are 10 bricks in a pile, each player can take 1-3 brick per turn, whoever take the last brick wins the game. Assume player A go first, and here is my attempt. I want to express that "eventually there is a winner",but my code doesn't work because it does not prevent player taking brick after brick=0, so eventually player a,b will both become winner.
here is my code:
MODULE main
VAR
bricks : 0..10;
i : 1..3;
j : 1..3;
turn : boolean;
winner : {none, a, b};
ASSIGN
init(winner) := none;
init(bricks) := 10;
init(turn) := TRUE;
next(turn) := case
turn : FALSE;
!turn: TRUE;
esac;
next(bricks) :=
case
bricks - j >= 0 : bricks - j;
bricks - j < 0 : 0;
TRUE:bricks;
esac;
next(winner) := case
turn=TRUE & bricks = 0: a;
turn=FALSE & bricks = 0: b;
TRUE:winner;
esac;
SPEC AF (winner = a | winner = b)
and here is my output on SPEC AF (winner = a | winner = none) to illustrate my point.
i = 1
j = 1
turn = TRUE
winner = none
State: 1.2 <-
bricks = 9
j = 3
turn = FALSE
State: 1.3 <-
bricks = 6
turn = TRUE
State: 1.4 <-
bricks = 3
turn = FALSE
State: 1.5 <-
bricks = 0
j = 1
turn = TRUE
State: 1.6 <-
turn = FALSE
winner = a
State: 1.7 <-
turn = TRUE
winner = b
as you can see, model still provide a counter example where player b win the game after playe a already won.

Answer 1

I am not sure how you provided a counter-example, since the property you specified is verified by the model:
-- specification AF (winner = a | winner = b) is true
Perhaps you simulated the program and simply observed that it behaves in an unexpected manner. The property that you seem to really want to verify is AF (AG winner = a | AG winner = b). In fact, using this property results in a counter-example similar to your own:
-- specification AF (AG winner = a | AG winner = b) is false
-- as demonstrated by the following execution sequence
Trace Description: CTL Counterexample
Trace Type: Counterexample
-> State: 1.1 <-
bricks = 10
i = 1
j = 1
turn = TRUE
winner = none
-> State: 1.2 <-
bricks = 9
turn = FALSE
-> State: 1.3 <-
bricks = 8
turn = TRUE
-> State: 1.4 <-
bricks = 7
turn = FALSE
-> State: 1.5 <-
bricks = 6
turn = TRUE
-> State: 1.6 <-
bricks = 5
turn = FALSE
-> State: 1.7 <-
bricks = 4
turn = TRUE
-> State: 1.8 <-
bricks = 3
turn = FALSE
-> State: 1.9 <-
bricks = 2
turn = TRUE
-> State: 1.10 <-
bricks = 1
turn = FALSE
-> State: 1.11 <-
bricks = 0
turn = TRUE
-- Loop starts here
-> State: 1.12 <-
turn = FALSE
winner = a
-> State: 1.13 <-
turn = TRUE
winner = b
-> State: 1.14 <-
turn = FALSE
winner = a
The problem is that you flip turns even when the game is finished, and as a result of this, the winner continuously flips among A and B too.
I re-wrote your solution in a better way:
MODULE main
VAR
bricks : 0..10;
q : 0..3;
turn : {A_TURN , B_TURN};
DEFINE
game_won := next(bricks) = 0;
a_won := game_won & turn = A_TURN;
b_won := game_won & turn = B_TURN;
ASSIGN
init(bricks) := 10;
init(turn) := A_TURN;
next(bricks) := case
bricks - q >= 0 : bricks - q;
TRUE : 0;
esac;
next(turn) := case
turn = A_TURN & !game_won: B_TURN;
turn = B_TURN & !game_won: A_TURN;
TRUE : turn;
esac;
-- forbid q values from being both larger than the remaining number of
-- bricks, and equal to zero when there are still bricks to take.
INVAR (q <= bricks)
INVAR (bricks > 0) -> (q > 0)
INVAR (bricks <= 0) -> (q = 0)
-- Sooner or later the number of bricks will always be
-- zero for every possible state in every possible path,
-- that is, someone won the game
CTLSPEC
AF AG (bricks = 0)
I think the code is quite self-explanatory.
You can run it with both NuSMV and nuXmv using the following commands:
> read_model -i game.smv
> go
> check_property
-- specification AF (AG bricks = 0) is true
If instead you want to find a possible solution, just flip the property:
> check_ctlspec -p "AF AG (bricks != 0)"
-- specification AF (AG bricks != 0) is false
-- as demonstrated by the following execution sequence
Trace Description: CTL Counterexample
Trace Type: Counterexample
-> State: 1.1 <-
bricks = 10
q = 1
turn = A_TURN
game_won = FALSE
b_won = FALSE
a_won = FALSE
-> State: 1.2 <-
bricks = 9
turn = B_TURN
-> State: 1.3 <-
bricks = 8
turn = A_TURN
-> State: 1.4 <-
bricks = 7
turn = B_TURN
-> State: 1.5 <-
bricks = 6
turn = A_TURN
-> State: 1.6 <-
bricks = 5
turn = B_TURN
-> State: 1.7 <-
bricks = 4
turn = A_TURN
-> State: 1.8 <-
bricks = 3
turn = B_TURN
-> State: 1.9 <-
bricks = 2
turn = A_TURN
-> State: 1.10 <-
bricks = 1
turn = B_TURN
game_won = TRUE
b_won = TRUE
-- Loop starts here
-> State: 1.11 <-
bricks = 0
q = 0
-> State: 1.12 <-
I hope you'll find this answer helpful.