Documentation.SBV.Examples.WeakestPreconditions.GCD

Proof of correctness of an imperative GCD (greatest-common divisor) algorithm, using weakest preconditions. The termination measure here illustrates the use of lexicographic ordering. Also, since symbolic version of GCD is not symbolically terminating, this is another example of using uninterpreted functions and axioms as one writes specifications for WP proofs.

Program state

data GCDS a

The state for the sum program, parameterized over a base type a.

Constructors

GCDS

Fields

• x :: a

  First value

• y :: a

  Second value

• i :: a

  Copy of x to be modified

• j :: a

  Copy of y to be modified

Instances

Functor GCDS

Defined in Documentation.SBV.Examples.WeakestPreconditions.GCD

Methods

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

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

Foldable GCDS

Defined in Documentation.SBV.Examples.WeakestPreconditions.GCD

Methods

fold :: Monoid m => GCDS m -> m

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

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

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

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

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

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

foldr1 :: (a -> a -> a) -> GCDS a -> a

foldl1 :: (a -> a -> a) -> GCDS a -> a

toList :: GCDS a -> [a]

null :: GCDS a -> Bool

length :: GCDS a -> Int

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

maximum :: Ord a => GCDS a -> a

minimum :: Ord a => GCDS a -> a

sum :: Num a => GCDS a -> a

product :: Num a => GCDS a -> a

Traversable GCDS

Defined in Documentation.SBV.Examples.WeakestPreconditions.GCD

Methods

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

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

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

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

SymVal a => Fresh IO (GCDS (SBV a))

Fresh instance for the program state

Defined in Documentation.SBV.Examples.WeakestPreconditions.GCD

Methods

fresh :: QueryT IO (GCDS (SBV a))

Show a => Show (GCDS a)

Defined in Documentation.SBV.Examples.WeakestPreconditions.GCD

Methods

showsPrec :: Int -> GCDS a -> ShowS

show :: GCDS a -> String

showList :: [GCDS a] -> ShowS

(SymVal a, Show a) => Show (GCDS (SBV a))

Show instance for GCDS. The above deriving clause would work just as well, but we want it to be a little prettier here, and hence the OVERLAPS directive.

Defined in Documentation.SBV.Examples.WeakestPreconditions.GCD

Methods

showsPrec :: Int -> GCDS (SBV a) -> ShowS

show :: GCDS (SBV a) -> String

showList :: [GCDS (SBV a)] -> ShowS

Generic (GCDS a)

Defined in Documentation.SBV.Examples.WeakestPreconditions.GCD

Associated Types

type Rep (GCDS a) :: Type -> Type

Methods

from :: GCDS a -> Rep (GCDS a) x

to :: Rep (GCDS a) x -> GCDS a

Mergeable a => Mergeable (GCDS a)

Defined in Documentation.SBV.Examples.WeakestPreconditions.GCD

Methods

symbolicMerge :: Bool -> SBool -> GCDS a -> GCDS a -> GCDS a

select :: (Ord b, SymVal b, Num b) => [GCDS a] -> GCDS a -> SBV b -> GCDS a

type Rep (GCDS a)

Defined in Documentation.SBV.Examples.WeakestPreconditions.GCD

type Rep (GCDS a) = D1 ('MetaData "GCDS" "Documentation.SBV.Examples.WeakestPreconditions.GCD" "sbv-8.10-9ZT5qtGqUieJfHfo5fo8OY" 'False) (C1 ('MetaCons "GCDS" 'PrefixI 'True) ((S1 ('MetaSel ('Just "x") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Just "y") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a)) :*: (S1 ('MetaSel ('Just "i") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Just "j") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a))))

type G = GCDS SInteger

Helper type synonym

The algorithm

algorithm :: Stmt G

The imperative GCD algorithm, assuming strictly positive x and y:

   i = x
   j = y
   while i != j      -- While not equal
     if i > j
        i = i - j    -- i is greater; reduce it by j
     else
        j = j - i    -- j is greater; reduce it by i

When the loop terminates, i equals j and contains GCD(x, y).

gcd :: SInteger -> SInteger -> SInteger

Symbolic GCD as our specification. Note that we cannot really implement the GCD function since it is not symbolically terminating. So, we instead uninterpret and axiomatize it below.

NB. The concrete part of the definition is only used in calls to traceExecution and is not needed for the proof. If you don't need to call traceExecution, you can simply ignore that part and directly uninterpret. In that case, we simply use Prelude's version.

axiomatizeGCD :: Symbolic ()

Constraints and axioms we need to state explicitly to tell the SMT solver about our specification for GCD.

pre :: G -> SBool

Precondition for our program: x and y must be strictly positive

post :: G -> SBool

Postcondition for our program: i == j and i = gcd x y

noChange :: Stable G

Stability condition: Program must leave x and y unchanged.

imperativeGCD :: Program G

A program is the algorithm, together with its pre- and post-conditions.

Correctness

correctness :: IO (ProofResult (GCDS Integer))

With the axioms in place, it is trivial to establish correctness:

>>> correctness
Total correctness is established.
Q.E.D.

Note that I found this proof to be quite fragile: If you do not get the algorithm right or the axioms aren't in place, z3 simply goes to an infinite loop, instead of providing counter-examples. Of course, this is to be expected with the quantifiers present.

Concrete execution

Example concrete run. As we mentioned in the definition for gcd, the concrete-execution function cannot deal with uninterpreted functions and axioms for obvious reasons. In those cases we revert to the concrete definition. Here's an example run:

>>> traceExecution imperativeGCD $ GCDS {x = 14, y = 4, i = 0, j = 0}
*** Precondition holds, starting execution:
  {x = 14, y = 4, i = 0, j = 0}
===> [1.1] Conditional, taking the "then" branch
  {x = 14, y = 4, i = 0, j = 0}
===> [1.1.1] Skip
  {x = 14, y = 4, i = 0, j = 0}
===> [1.2] Assign
  {x = 14, y = 4, i = 14, j = 4}
===> [1.3] Loop "i != j": condition holds, executing the body
  {x = 14, y = 4, i = 14, j = 4}
===> [1.3.{1}] Conditional, taking the "then" branch
  {x = 14, y = 4, i = 14, j = 4}
===> [1.3.{1}.1] Assign
  {x = 14, y = 4, i = 10, j = 4}
===> [1.3] Loop "i != j": condition holds, executing the body
  {x = 14, y = 4, i = 10, j = 4}
===> [1.3.{2}] Conditional, taking the "then" branch
  {x = 14, y = 4, i = 10, j = 4}
===> [1.3.{2}.1] Assign
  {x = 14, y = 4, i = 6, j = 4}
===> [1.3] Loop "i != j": condition holds, executing the body
  {x = 14, y = 4, i = 6, j = 4}
===> [1.3.{3}] Conditional, taking the "then" branch
  {x = 14, y = 4, i = 6, j = 4}
===> [1.3.{3}.1] Assign
  {x = 14, y = 4, i = 2, j = 4}
===> [1.3] Loop "i != j": condition holds, executing the body
  {x = 14, y = 4, i = 2, j = 4}
===> [1.3.{4}] Conditional, taking the "else" branch
  {x = 14, y = 4, i = 2, j = 4}
===> [1.3.{4}.2] Assign
  {x = 14, y = 4, i = 2, j = 2}
===> [1.3] Loop "i != j": condition fails, terminating
  {x = 14, y = 4, i = 2, j = 2}
*** Program successfully terminated, post condition holds of the final state:
  {x = 14, y = 4, i = 2, j = 2}
Program terminated successfully. Final state:
  {x = 14, y = 4, i = 2, j = 2}

As expected, gcd 14 4 is 2.