Documentation.SBV.Examples.ProofTools.BMC

A BMC example, showing how traditional state-transition reachability problems can be coded using SBV, using bounded model checking.

We imagine a system with two integer variables, x and y. At each iteration, we can either increment x by 2, or decrement y by 4.

Can we reach a state where x and y are the same starting from x=0 and y=10?

What if y starts at 11?

System state

data S a

System state, containing the two integers.

Constructors

S

Fields

• x :: a
• y :: a

Instances

Functor S

Defined in Documentation.SBV.Examples.ProofTools.BMC

Methods

fmap :: (a -> b) -> S a -> S b

(<$) :: a -> S b -> S a

Foldable S

Defined in Documentation.SBV.Examples.ProofTools.BMC

Methods

fold :: Monoid m => S m -> m

foldMap :: Monoid m => (a -> m) -> S a -> m

foldMap' :: Monoid m => (a -> m) -> S a -> m

foldr :: (a -> b -> b) -> b -> S a -> b

foldr' :: (a -> b -> b) -> b -> S a -> b

foldl :: (b -> a -> b) -> b -> S a -> b

foldl' :: (b -> a -> b) -> b -> S a -> b

foldr1 :: (a -> a -> a) -> S a -> a

foldl1 :: (a -> a -> a) -> S a -> a

toList :: S a -> [a]

null :: S a -> Bool

length :: S a -> Int

elem :: Eq a => a -> S a -> Bool

maximum :: Ord a => S a -> a

minimum :: Ord a => S a -> a

sum :: Num a => S a -> a

product :: Num a => S a -> a

Traversable S

Defined in Documentation.SBV.Examples.ProofTools.BMC

Methods

traverse :: Applicative f => (a -> f b) -> S a -> f (S b)

sequenceA :: Applicative f => S (f a) -> f (S a)

mapM :: Monad m => (a -> m b) -> S a -> m (S b)

sequence :: Monad m => S (m a) -> m (S a)

Fresh IO (S SInteger)

Fresh instance for our state

Defined in Documentation.SBV.Examples.ProofTools.BMC

Methods

fresh :: QueryT IO (S SInteger)

Show a => Show (S a)

Show the state as a pair

Defined in Documentation.SBV.Examples.ProofTools.BMC

Methods

showsPrec :: Int -> S a -> ShowS

show :: S a -> String

showList :: [S a] -> ShowS

EqSymbolic a => EqSymbolic (S a)

Symbolic equality for S.

Defined in Documentation.SBV.Examples.ProofTools.BMC

Methods

(.==) :: S a -> S a -> SBool

(./=) :: S a -> S a -> SBool

(.===) :: S a -> S a -> SBool

(./==) :: S a -> S a -> SBool

distinct :: [S a] -> SBool

distinctExcept :: [S a] -> [S a] -> SBool

allEqual :: [S a] -> SBool

sElem :: S a -> [S a] -> SBool

sNotElem :: S a -> [S a] -> SBool

Encoding the problem

problem :: Int -> (S SInteger -> SBool) -> IO (Either String (Int, [S Integer]))

We parameterize over the initial state for different variations.

Examples

ex1 :: IO (Either String (Int, [S Integer]))

Example 1: We start from x=0, y=10, and search up to depth 10. We have:

>>> ex1
BMC: Iteration: 0
BMC: Iteration: 1
BMC: Iteration: 2
BMC: Iteration: 3
BMC: Solution found at iteration 3
Right (3,[(0,10),(0,6),(2,6),(2,2)])

As expected, there's a solution in this case. Furthermore, since the BMC engine found a solution at depth 3, we also know that there is no solution at depths 0, 1, or 2; i.e., this is "a" shortest solution. (That is, it may not be unique, but there isn't a shorter sequence to get us to our goal.)

ex2 :: IO (Either String (Int, [S Integer]))

Example 2: We start from x=0, y=11, and search up to depth 10. We have:

>>> ex2
BMC: Iteration: 0
BMC: Iteration: 1
BMC: Iteration: 2
BMC: Iteration: 3
BMC: Iteration: 4
BMC: Iteration: 5
BMC: Iteration: 6
BMC: Iteration: 7
BMC: Iteration: 8
BMC: Iteration: 9
Left "BMC limit of 10 reached"

As expected, there's no solution in this case. While SBV (and BMC) cannot establish that there is no solution at a larger depth, you can see that this will never be the case: In each step we do not change the parity of either variable. That is, x will remain even, and y will remain odd. So, there will never be a solution at any depth. This isn't the only way to see this result of course, but the point remains that BMC is just not capable of establishing inductive facts.