Copyright	(c) Levent Erkok
License	BSD3
Maintainer	erkokl@gmail.com
Stability	experimental
Safe Haskell	None
Language	Haskell2010

Documentation.SBV.Examples.WeakestPreconditions.GCD

Contents

Program state
The algorithm
Correctness
Concrete execution

Description

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.

Synopsis

data GCDS a = GCDS {
- x :: a
- y :: a
- i :: a
- j :: a
}
type G = GCDS SInteger
algorithm :: Stmt G
gcd :: SInteger -> SInteger -> SInteger
axiomatizeGCD :: Symbolic ()
pre :: G -> SBool
post :: G -> SBool
noChange :: Stable G
imperativeGCD :: Program G
correctness :: IO (ProofResult (GCDS Integer))

Program state

data GCDS a Source #

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

Instances details

Functor GCDS Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.GCD Methods fmap :: (a -> b) -> GCDS a -> GCDS b # (<$) :: a -> GCDS b -> GCDS a #
Foldable GCDS Source #
Instance details 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 Source #
Instance details 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)) Source #	`Fresh` instance for the program state
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.GCD Methods fresh :: QueryT IO (GCDS (SBV a)) Source #
Show a => Show (GCDS a) Source #
Instance details 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)) Source #	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.
Instance details 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) Source #
Instance details 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) Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.GCD Methods symbolicMerge :: Bool -> SBool -> GCDS a -> GCDS a -> GCDS a Source # select :: (Ord b, SymVal b, Num b) => [GCDS a] -> GCDS a -> SBV b -> GCDS a Source #
type Rep (GCDS a) Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.GCD type Rep (GCDS a) = D1 ('MetaData "GCDS" "Documentation.SBV.Examples.WeakestPreconditions.GCD" "sbv-8.9-ATrzvBVsNws6CzdQ7iCv1r" '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 Source #

Helper type synonym

The algorithm

algorithm :: Stmt G Source #

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 Source #

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 () Source #

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

pre :: G -> SBool Source #

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

post :: G -> SBool Source #

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

noChange :: Stable G Source #

Stability condition: Program must leave x and y unchanged.

imperativeGCD :: Program G Source #

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

Correctness

correctness :: IO (ProofResult (GCDS Integer)) Source #

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.