Re: [Haskell-cafe] Generating arbitrary function in QuickCheck

2009/4/8 Yusaku Hashimoto :

Thanks for all replies! these gave me many enlightenments.

I have been looking in about free theorem for a while, and I want to try to describe what I learned, and please collect me where I am wrong or misunderstood.

Generating random functions for sortBy is meaningless, because, for testing sortBy, casual comparison function is enough to it. The reason of this comes from free theorem. By the free theorem, sortBy should satisfy this theorem (copied from ftshell).

   forall t1,t2 in TYPES, f :: t1 -> t2.     forall p :: t1 -> t1 -> Ordering.      forall q :: t2 -> t2 -> Ordering.       (forall x :: t1. p x = q (f x) . f)       ==> (map f . sortBy p = sortBy q . map f)

Any linear orderings can be defined by mapping value to Integer. And above theorem says comparing functions in the theorem must be homomorphic to each other. Hence comparing function is meaningful only when it is homomorhpic to Integer's that. (I have not tried to Generate such function yet, and it may be unnecessary, but) taking such function is easy.

Kind of like that. The theorem says that for any function of sortBy's type, it will not change permutation of the list if we replace the comparison function by another one that is monotonic with respect to it. For any correct comparison function F, there exists a mapping to integers M and 'compare' as the comparison function agreeing with F. Therefore, if you check yourSortBy on integers with 'compare', you will also be sure of all other types with correct comparison functions, because you'll be sure that it permutes the list in a correct way.

Furthermore, it should be satisfied by all functions that have same type signitures to sortBy, because it is implied from only its own type.

Reference: - [Theorem for free!](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.38.9875) - [ftshell](http://hackage.haskell.org/cgi-bin/hackage-scripts/package/ftshell-0.3)

Thanks, Yusaku Hashimoto

On 2009/04/07, at 15:19, Janis Voigtlaender wrote:

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Jason Dagit wrote:

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On Mon, Apr 6, 2009 at 10:09 PM, Eugene Kirpichov wrote:

...

Since the argument to sortBy must impose a linear ordering on its arguments, and any linear ordering may as well be generated by assigning an integer to each element of type 'a', and your sorting function is polymorphic, from the free theorem for the sorting function we may deduce that it suffices to test your function on integer lists with a casual comparison function (Data.Ord.compare), and there is no need to generate a random comparison function.

Interesting.  How is this free theorem stated for the sorting function?  Intuitively I understand that if the type is polymorphic, then it seems reasonable to just pick one type and go with it.

You can try free theorems here:

http://linux.tcs.inf.tu-dresden.de/~voigt/ft/

For example, for

sort :: Ord a => [a] -> [a]

it generates the following:

forall t1,t2 in TYPES(Ord), f :: t1 -> t2, f respects Ord.  forall x :: [t1]. map f (sort x) = sort (map f x)

where:

f respects Ord if f respects Eq and  forall x :: t1.   forall y :: t1. compare x y = compare (f x) (f y)  forall x :: t1. forall y :: t1. (<) x y = (<) (f x) (f y)  forall x :: t1. forall y :: t1. (<=) x y = (<=) (f x) (f y)  forall x :: t1. forall y :: t1. (>) x y = (>) (f x) (f y)  forall x :: t1. forall y :: t1. (>=) x y = (>=) (f x) (f y)

f respects Eq if  forall x :: t1. forall y :: t1. (==) x y = (==) (f x) (f y)  forall x :: t1. forall y :: t1. (/=) x y = (/=) (f x) (f y)

Assuming that all the comparison functions relate to each other in the mathematically sensible way, the latter can be reduced to:

f respects Ord if  forall x :: t1. forall y :: t1. (x <= y) = (f x <= f y)

For sortBy you would get a similar free theorem.

To see how the free theorem allows you to switch from an arbitrary type to just integers, set t2=Int and simply use f to build a order-preserving bijection between elements in the list x and a prefix of [1,2,3,4,...]

Ciao, Janis.

-- Dr. Janis Voigtlaender http://wwwtcs.inf.tu-dresden.de/~voigt/ mailto:voigt@tcs.inf.tu-dresden.de

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