NuSMV model checking: create a simple game model

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.

Patrick Trentin Patrick Trentin · Accepted Answer · 2016-04-01T13:24:13

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.

NuSMV model checking: create a simple game model

1 Answers