NuSMV 模型检查：创建简单的游戏模型

Question

我是 NuSMV 的新手，我尝试为这款简单的回合制游戏建模。 10块积木，每个玩家每回合可以拿1-3块积木，谁拿走最后一块积木获胜。假设玩家 A 先走，这是我的尝试。我想表达“最终会有赢家”，但我的代码不起作用，因为它不会阻止玩家在brick = 0之后拿砖，所以最终玩家a，b都会成为赢家。

这是我的代码：

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)

这是我在 SPEC AF 上的输出（获胜者 = a | 获胜者 = 无）来说明我的观点。

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

如您所见，模型仍然提供了一个反例，即玩家 b 在玩 a 已经获胜后赢得游戏。

Answer 1

我不确定您是如何提供反例的，因为您指定的属性已由模型验证：

-- specification AF (winner = a | winner = b)  is true

也许您模拟了程序并只是观察到它的行为方式出乎意料。您似乎真正想要验证的属性是AF (AG winner = a | AG winner = b)。事实上，使用这个属性会产生一个类似于你自己的反例：

-- 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

问题是，即使游戏结束，你也会翻转回合，因此，获胜者也会不断地在 A 和 B 之间翻转。

我以更好的方式重写您的解决方案：

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)

我认为代码是不言自明的。

您可以使用以下命令同时使用 NuSMV 和 nuXmv 运行它：

> read_model -i game.smv
> go
> check_property
-- specification AF (AG bricks = 0)  is true

如果您想找到可能的解决方案，只需翻转属性：

> 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 <-

希望这个答案对您有所帮助。