Documentation.SBV.Examples.ProofTools.BMC

Author : Levent Erkok License : BSD3 Maintainer: [e-mail] Stability : experimental

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

Queriable IO (S SInteger) (S Integer)

Queriable instance for our state

Defined in Documentation.SBV.Examples.ProofTools.BMC

Methods

fresh :: QueryT IO (S SInteger)

extract :: S SInteger -> QueryT IO (S Integer)

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

distinct :: [S a] -> SBool

allEqual :: [S a] -> SBool

sElem :: 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),(0,2),(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.