Higher-order QuickCheck

A QuickCheck extension for properties of higher-order values.

Examples

Higher-order properties are properties which may:

1. quantify over functions;
2. state equalities between functions.

Some examples:

fmap_dot :: forall a b c. (b -> c) -> (a -> b) -> Equation (Maybe a -> Maybe c)
fmap_dot g f x = (fmap g . fmap f) :=: fmap (f . g)

callCC_bind :: forall r a. Cont r a -> Equation (Cont r a)
callCC_bind m = callCC ((>>=) m) :=: m

quickcheck-higherorder makes it easy to define and test such properties.

main :: IO ()
main = do
  quickCheck' (fmap_dot @Int @Int @Int)
  quickCheck' (callCC_bind @Int @Int)

(Additional setup is required for the callCC example.)

Summary

QuickCheck has a cute trick to implicitly convert functions Thing -> Bool or Thing -> Gen Bool to testable properties, provided Thing is an instance of Arbitrary and Show, i.e., there is a random generator, a shrinker, and a printer for Thing. Sadly, those constraints limit the range of types that Thing can be. In particular, they rule out functions and other values of “infinite size”.

This library, quickcheck-higherorder, lifts that limitation, generalizing that technique to arbitrary types, and provides other related quality-of-life improvements for property-based test suites.

The key idea is to separate the Thing manipulated by the application under test, of arbitrary structure, from its representation, which is manipulated by QuickCheck and needs to be “concrete” enough to be possible to generate, shrink, and show.

Constructible types

The Constructible type class relates types a to representations Repr a from which their values can be constructed. Constraints for Arbitrary and Show are thus attached to those representations instead of the raw type that will be used in properties.

class (Arbitrary (Repr a), Show (Repr a)) => Constructible a where
  type Repr a :: Type
  fromRepr :: Repr a -> a

To illustrate what it enables, here’s an example of higher-order property:

prop_bool :: (Bool -> Bool) -> Bool -> Property
prop_bool f x =
  f (f (f x)) === f x

In vanilla QuickCheck, it needs a little wrapping to actually run it:

main :: IO ()
main = quickCheck (\(Fn f) x -> prop_bool f x)

The simpler expression quickCheck prop_bool would not typecheck because Bool -> Bool is not an instance of Show.

With “higher-order” QuickCheck, that wrapping performed by Fn is instead taken care of by the Constructible class, so we can write simply:

main :: IO ()
main = quickCheck' prop_bool

This is especially convenient when the function type is not directly exposed in the type of the property (as in prop_bool), but may be hidden inside various data types or newtypes.

Testable equality

In a similar vein, the Eq class is limited to types with decidable equality, which typically requires them to have values of “finite size”. Most notably, the type of functions a -> b cannot be an instance of Eq in general.

But testing can still be effective with a weaker constraint, dubbed testable equality. To compare two functions, we can generate some random arguments and compare their images. That is useful even if we can’t cover the whole domain:

• if we find two inputs with distinct outputs, then the two functions are definitely not equal, and we now have a very concrete counterexample to contemplate;

• if we don’t find any difference, then we can’t conclude for sure, but:

  1. we can always try harder (more inputs, or rerun the whole property from scratch);
  2. in some situations, such as implementations of algebraic structures, bugs cause extremely obvious inequalities. If only we would look at them. The point of this new feature is to lower the bar for testing equations between higher-order values in the first place.

This package introduces a new type class TestEq, for testable equality.

class TestEq a where
  (=?) :: a -> a -> Property

The codomain being Property offers some notable capabilities:

1. as explained earlier, we can use randomness to choose finite subsets of infinite values (such as functions) to compare;

2. we can also provide detailed context in the case of failure, by reporting the observations which lead to unequal outcomes.

For example, we can rewrite prop_bool as an algebraic property of functions using TestEq:

prop_bool :: (Bool -> Bool) -> Property
prop_bool f = (f . f . f) =? f

More types of properties

Many common properties are quite simple, like prop_bool. However, QuickCheck’s way of declaring properties as functions with result type Property introduces some unexpected complexity in the types.

For example, try generalizing the property prop_bool above to arbitrary types instead of Bool. Since we use testable equality of functions a -> a, we incur constraints that the domain must be Constructible, and the codomain itself must have testable equality.

prop_fun :: (Constructible a, TestEq a) => (a -> a) -> Property
prop_fun f = (f . f . f) =? f

This type tells us both too much and too little. Too much, because the constraints leak details about the very specific way in which the comparison is performed. Too little, because a Property can do a lot of things besides testing the equality of two values; in fact that is one cause for the previous concern.

A more precise formulation is the following:

prop_fun :: (a -> a) -> Equation (a -> a)
prop_fun f = (f . f . f) :=: f

This does not actually do the comparison, but exposes just the necessary amount of information to do it in whatever way one deems appropriate. Indeed, Equation is simply a type of pairs:

data Equation a = a :=: a

It is equipped with a Testable instance that will require a TestEq constraint indirectly at call sites only.

Full example

import Test.QuickCheck (quickCheck)
import Test.QuickCheck.HigherOrder (property', Equation((:=:)), CoArbitrary)

import Control.Monad.Cont (Cont, ContT(..), callCC)


-- Example property
callCC_bind :: forall r a. Cont r a -> Equation (Cont r a)
callCC_bind m = callCC ((>>=) m) :=: m


main :: IO ()
main = quickCheck' (callCC_bind @Int @Int)


-- Newtype boilerplate

import Test.QuickCheck (Gen)
import Test.QuickCheck.HigherOrder (CoArbitrary, TestEq(..), Constructible(..))

-- Constructible instances
instance (CoArbitrary Gen (m r), Constructible a, Constructible (m r)) => Constructible (ContT r m a) where
  type Repr (ContT r m a) = Repr ((a -> m r) -> m r)
  fromRepr = ContT . fromRepr

instance (TestEq ((a -> m r) -> m r)) => TestEq (ContT r m a) where
  ContT f =? ContT g = f =? g