Haskell Programming: Attractive Types

Haskell has become a laboratory and playground for advanced type hackery
Simon Peyton Jones: Wearing the hair shirt: a retrospective on Haskell. POPL 2003, invited talk.


Many ways to the fix-point combinator besides the value recursion

Value-level recursion in Haskell is built-in since definitions are implicitly recursive. The explicit polymorphic fix-point combinator can therefore be defined simply as

     fix :: (a -> a) -> a
     fix f = f (fix f)
Had the value-level recursion been unavailable, we still could have safely defined the polymorphic fix-point combinator in Haskell. In fact, in five different ways, without ever resorting to unsafe operations. Iso-recursive data types are a well-known way to fix. Less known is using type classes or families. The lazy ST approach is most puzzling: the reading of a reference cell appears to occur in pure code.

Uncharitably speaking, Haskell, taken as a logic, is inconsistent in more than two ways.

The current version is 1.0, May 2010.

Fix.hs [3K]
The complete Haskell code and the tests

All the ways to express the polymorphic fixpoint operator in OCaml


Restricted Monads

We show how to attain the gist of the restricted datatype proposal (Hughes, 1999) in Haskell, now. We need solely multi-parameter type classes; no functional dependencies, no undecidable instances, let alone more controversial extensions, are required. Restricted monads thus should be implementable in Haskell'.

By definition, monadic operations such as return :: a -> m a and bind must be fully polymorphic in the type of the value a associated with the monadic action m a. Indeed, return must be a natural transformation. A recurring discussion on Haskell mailing lists points out the occasional need to restrict that polymorphism. For example, one of the common implementations of MonadPlus collects the choices, the results of non-deterministic computations, in a list. One may wish for a more efficient data structure, such as Data.Map or a Set. That however requires the Ord constraint on the values, therefore, neither Map nor Set may be instances of Monad, let alone MonadPlus. More examples of restricted monads are discussed in the article below.

We propose a fully backward-compatible extension to the monadic interface. All monads are members of the extended monads, and all existing monadic code should compile as it is. In addition, restricted monads become expressible. The article defines the extended interface with the functions

     ret2  :: MN2 m a   => a -> m a
     fail2 :: MN2 m a   => String -> m a
     bind2 :: MN3 m a b => m a -> (a -> m b) -> m b
which have exactly the same type as the ordinary monadic operations -- only with more general constraints. Because new operations have exactly the same type, one may use them in the regular monadic code (given -fno-implicit-prelude flag) and with the do-notation (cf. `rebindable syntax' feature). Perhaps one day this more general interface becomes the default one.

The gist of our proposal is the splitting of the Monad class into two separate classes, MN2 for return and fail and MN3 for bind. The latter class implies the former. The new classes explicitly mention the type of the monadic action value in their interface. That makes it possible to attach constraints to those types. The article shows the attaching of the Ord constraint, so to make Set an instance of Monad and MonadPlus.

The current version is 1.1, Feb 8, 2006.

CPS-transforming a restricted monad to the ordinary one

RestrictedMonad.lhs [4K]
The literate Haskell source code and a few tests
The code was originally posted as Restricted Data Types Now on the Haskell' mailing list on Wed, 8 Feb 2006 00:06:23 -0800 (PST)

DoRestrictedM.hs [3K]
The illustration of the do-notation for restricted monads, which works already for GHC 6.6 and later. We demonstrate that the do-notation works uniformly for ordinary monads and restricted monads. We show the conventional-looking monadic code which nevertheless uses Data.Set as the implementation of MonadPlus -- a frequently requested feature.

John Hughes: Restricted datatypes in Haskell.
Haskell 1999 Workshop, ed. Erik Meijer. Technical Report UU-CS-1999-28, Department of Computer Science, Utrecht University.


Polymorphic stanamically balanced AVL trees

We describe a datatype of polymorphic balanced binary trees: AVL trees. The trees are polymorphic: the values in different nodes may have different type. The trees are balanced: for each non-leaf node, the heights of its two children can differ at most by one. Here, by definition the height of a node is 1 + max of the heights of its children. A leaf node has the height of 0.

The main feature of the present approach is a blended static and dynamic enforcement of the balancing constraint. The function make_node verifies the balancing constraint at compile time -- if it can. If the static check is not possible, the function delays the check till the run-time.

A detailed follow-up message by Chris Okasaki describes a very interesting representation of AVL trees with the balancing constraint ensured statically by a type-checker.

The current version is 1.5, Apr 26, 2003.

stanamically-balanced-trees.lhs [12K]
The literate Haskell source code and a few tests
The code was originally posted as Polymorphic stanamically balanced binary trees on Haskell mailing list on Sun, 20 Apr 2003 15:25:12 -0700 (PDT)

Christopher Okasaki. A follow-up message posted on Haskell mailing list on Mon, 28 Apr 2003 08:34:37 -0400


Partial signatures

The regular (full) signature of a function specifies the type of the function and -- if the type includes constrained type variables -- enumerates all of the typeclass constraints. The list of the constraints may be quite large. Partial signatures help when:

Contrary to a popular belief, both of the above are easily possible, in Haskell98.

The current version is 1.1, Aug 6, 2004.
partial-signatures.lhs [3K]
The literate Haskell code with explanations and the examples
The code was originally posted as Partial signatures on Haskell-Cafe mailing list on Fri, 6 Aug 2004 20:29:40 -0700 (PDT)

How to make a function strict without changing its body
Another application of the trick of adding a clause with an always failing guard


Reversing Haskell typechecker: converting from undefined to defined

This message shows how to make the Haskell typechecker work in reverse: to infer a term of a given type:

     rtest4 f g = rr (undefined::(b -> c) -> (a -> b) -> a -> c) HNil f g
     *HC> rtest4 (:[]) Just 'x'
     [Just 'x']
     *HC> rtest4 Just Right True
     Just (Right True)

We ask the Haskell typechecker to derive us a function of the specified type. We get the real function, which we can then apply to various arguments. The return result does behave like a `composition' -- which is what the type specifies. Informally, we converted from undefined to defined.

It must be emphasized that no modifications to the Haskell compiler are needed, and no external programs are relied upon. In particular, however surprising it may seem, we get by without eval -- because Haskell has reflexive facilities already built-in.

Our system solves type habitation for a class of functions with polymorphic types. From another point of view, the system is a prover in the implication fragment of intuitionistic logic. Essentially we turn a type into a logical program -- a set of Horn clauses -- which we then solve by SLD resolution. It is gratifying to see that Haskell typeclasses are up to that task.

The message below presents two different converters from a type to a term. Both derive a program, a term, from its specification, a type -- for a class of fully polymorphic functions. The first converter has just been demonstrated. It is quite limited in that the derived function must be used `polymorphically' -- distinct type variables must be instantiated to different types (or, the user should first instantiate their types and then derive the term). The second converter is far more useful: it can let us `visualize' what a function with a particular type may be doing. For example, it might not be immediately clear what is the function of the type

     (((a -> b -> c) -> (a -> b) -> a -> c) -> (t3 -> t1 -> t2 -> t3) -> t) -> t)
Our reifier says,
     test9 = reify (undefined::(((a -> b -> c) -> (a -> b) -> a -> c) -> 
                                 (t3 -> t1 -> t2 -> t3) -> t) -> t) gamma0
     *HC> test9
     \y -> y (\d h p -> d p (h p)) (\d h p -> d)
that is, the function in question is one of the X combinators. It is an improper combinator. Similarly the reifier can turn a point-free function into the pointful form to help really understand the former. For example, it might take time to comprehend the following expression:
     pz = (((. head . uncurry zip . splitAt 1 . repeat) . uncurry) .) . (.) . flip
Our system says
     test_pz = reify (undefined `asTypeOf` pz) gamma0
     *HC> test_pz
     \h p y -> h y (p y)
So, pz is just the S combinator.

An attempt to derive a term for the type a->b expectedly fails. The type error message essentially says that a |- b is underivable.

The examples above exhibit fully polymorphic types -- those with uninstantiated, implicitly universally quantified type variables. That is, our typeclasses can reify not only types but also type schemas. The ability to operate on and compare unground types with uninstantiated type variables is often sought but rarely attained. The contribution of this message is the set of primitives for nominal equality comparison and deconstruction of unground types.

The current version is 1.1, Mar 1, 2005.

de-typechecker.lhs [20K]
The literate Haskell code with extensive explanations and many examples, including the code and explanation for the Equality predicate on type schemas.
The code was originally posted as De-typechecker: converting from a type to a term on the Haskell mailing list on Tue, 1 Mar 2005 00:13:08 -0800 (PST)

Lennart Augustsson: Announcing Djinn, version 2004-12-11, a coding wizard
A Message posted on the Haskell mailing list on Sun Dec 11 17:32:07 EST 2005.
The user types a Haskell type at DJinn's prompt, and DJinn gives back a term of that type if one exists. The produced term is in DJinn's term language. The printed term can be cut and pasted into the Haskell code.

pointless-translation.lhs [6K]
The literate Haskell98 code for translating proper linear combinators into point-free style.
The code was originally posted as Automatic pointless translation on the Haskell-Cafe mailing list on Mon, 14 Feb 2005 22:56:04 -0800 (PST)


Impredicativity and explicit type applications

On a simple example we demonstrate that the type system of Haskell with the common rank-2 extension (not counting the extensions in GHC 6.6) is already impredicative, and it permits explicit type, i.e., big-lambda and type applications. This note is based on a message by Shin-Cheng Mu from Feb 2005, and comments by Simon Peyton-Jones and Greg Morrisett. We add the observation that big lambda and type applications are in fact present in Haskell and can be explicitly used by programmers.

Polymorphic types in Haskell can only be instantiated with monomorphic types. In other words, a type variable ranges over ground types, which do not (overtly -- see below) contain quantified type variables. In particular, in the following polymorphic type definition (of Church numerals)

     type M = forall a . (a -> a) -> a -> a
the type variable a cannot be instantiated with the type M itself. This so-called predicativity prevents defining a type implicitly in terms of itself. This property significantly simplifies type inference; otherwise, unification, typically used to solve type equations, becomes higher-order, which is in general undecidable. The restriction that polytypes can only be instantiated with monotypes is responsible for the rejection of intuitively correct programs and seemingly makes Haskell unable to faithfully reproduce second-order lambda calculus. Shin-Cheng Mu showed the simple example of that, arithmetic of Church numerals:
     zero :: M;       zero = \f a ->  a
     succ :: M -> M;  succ n = \f a -> f (n f a)
     add, mul :: M -> M -> M
     add m n = \f a -> m f (n f a)
     mul m n = \f a -> m (n f) a
     exp, exp2 :: M -> M -> M
     exp m n = n (mul m) one
     exp2 m n = n m
This program typechecks -- with the sole exception of exp. This may seem surprising as the equivalent exp2 is accepted by the typechecker. Shin-Cheng Mu pointed out that if we write exp and exp2 with the explicit big lambda (denoted ?x -> term) and type application (to be denoted term[type])
     exp  (m::M) (n::M) = n [M] ((mul m)::M->M) (one::M)
     exp2 (m::M) (n::M) = ?b -> \(f::(b->b)) -> n[b->b] (m[b]) f
then we observe that exp instantiates the polymorphic term n with the polymorphic type M -- which is prohibited in Haskell. Hence the typechecker complains, with a rather uninformative message ``Inferred type is less polymorphic than expected. Quantified type variable `a' escapes.'' The term exp2 is accepted since the argument b of the type-lambda is assumed monotype.

The above notation for explicit type-level abstractions and applications is not Haskell. Or is it? It turns out, the introduction and elimination of big lambda is already part of Haskell. We can use them to guide the typechecker when instantiating polytypes with polytypes -- which is too effectively possible. Our guidance makes the inference decidable. As Greg Morrisett pointed out on the discussion thread, Haskell is impredicative: ``You can instantiate a type variable with a newtype that contains a polymorphic type... GHC enforces a sub-kind constraint on variables that precludes them from ranging over types whose top-most constructor is a forall (and has a few more structural constraints.) The distinction is subtle, but important. A predicative version of Haskell would have a much, much simpler denotational semantics, but also prevent a number of things that are useful and interesting.'' Indeed, we can write exp after all:

     newtype N = N{un:: forall a . (a -> a) -> a -> a}
     zero :: N;   zero = N ( \f a ->  a )
     succ :: N -> N;  succ n = N ( \f a -> f (un n f a) )
     exp, exp2 :: N -> N -> N
     exp m n = un n (mul m) one
     exp2 m n = N (\f a -> un n (un m) f a)
We should compare exp and exp2 code here with the explicit type lambda code above. Where we had ?t we have N, and where we had term[t] before we have un term now. Thus N and un act as -- mark the places of -- big lambda introduction and elimination. The notation this time is Haskell. Wrapping polymorphic types in newtype such as N also permits easy, nominal rather than structural, equality of polymorphic types.

In this code, one may not replace N (...) with N $ (...). This is yet another case where ($) is not the same as application with a lower precedence.

The current version is 1.1, Feb 1, 2005.

Shin-Cheng Mu. Re: Polymorphic types without type constructors? A message that started the discussion, posted on the Haskell mailing list on Tue Feb 1 22:36:00 EST 2005

numerals-second-order.hs [2K]
The complete code of the example, which compiles with Hugs,GHC 6.6. and pre-GHC 6.6.
The code was originally posted on the above discussion thread on Tue Feb 1 22:36:00 EST 2005

Simon Peyton Jones, Dimitrios Vytiniotis, Stephanie Weirich, and Mark Shields: Practical type inference for arbitrary-rank types. To appear in the Journal of Functional Programming.

Didier Le Botlan, Didier Remy: Raising ML to the Power of System F. ICFP 2003.


Translucent applicative functors

ML is known for its sophisticated, higher-order module system, one of the most interesting examples of which is a translucent applicative functor such as SET parameterized by the element-comparison function. If we make two instances of the SET with the same (>) comparison on integers, we can take an element from one set and put in in the other: the element types are `transparent' and the compiler can clearly see they are both integers. We can also take a union of the two sets. The type of the set itself is opaque -- set values can only be manipulated by the operations of SET. Now the compiler cannot see the concrete representation of the set types and verify they are the same. The compiler knows however that instantiations of SET with the identical element comparisons are type-compatible.

It turns out translucent functors can be implemented in Haskell idiomatically, taking the full use of type classes. We also show that type sharing constraints can be expressed in a scalable manner, so that the whole translation is practically usable. Thus we demonstrate that Haskell already has a higher-order module language. No new extensions are required; furthermore, we avoid even undecidable let alone overlapping instances.

In the process of translating OCaml module expressions into Haskell, we have noted several guidelines:

In our experience these guidelines are also useful when translating from Haskell to OCaml.

TranslucentAppFunctor.lhs [18K]
The article with the juxtaposed literate Haskell and OCaml code
It was posted as Applicative translucent functors in Haskell on the Haskell mailing list on Fri, 27 Aug 2004 16:51:43 -0700 (PDT).

Chung-chieh Shan. Higher-order modules in System F-omega and Haskell

Stefan Wehr and Manuel M. T. Chakravarty. ML Modules and Haskell Type Classes: A Constructive Comparison
Proc. Sixth Asian Symposium on Programming Languages and Systems. LNCS 5356, Springer, 2008.


ML-style modules with type sharing by name

This second article on higher-order module programming in Haskell deals with type-equality, or sharing constraints. Recall a module is a collection of type and value definitions. Here is an example, in OCaml notation.

      module TIF = struct
         type a = int
         type b = float
         let  app x = float_of_int x
The module interface can be described by the following signature (module type)
     module type FN = sig
       type a
       type b
       val app  : a -> b

A higher-order module, a functor, is a module parameterized by other modules, such as the following module that composes two instances of FN:

     module Compose(L: FN)(R: (FN with type a = L.b)) = struct
       type a = L.a
       type t = L.b
       type b = R.b
       let  app x = R.app (L.app x)
The composition of the R.app function with L.app is well-typed only when the result type of L.app is the same as the argument type of the R.app. Since these types (namely, L.b and R.a) are abstract, the compiler cannot know if they are the same. We have to specify the type equality L.b = R.a as a sharing constraint: (R: (FN with type a = L.b)).

The naive implementation of sharing constraints -- sharing by position -- leads to the exponential explosion of type parameters, as was shown by Harper and Pierce in 2003. One often hears suggestions of type-level records, to fix the problem.

In the joint article with Chung-chieh Shan, we translate Harper and Pierce's example into Haskell, using only the most common Haskell extensions to give type-equality constraints by name and avoid the exponential blowup. We can indeed refer to type parameters `by name' without any type-level records, taking advantage of the ability of a Haskell compiler to unify type expressions and bind type variables. We hope this message helps clarify the difference between the two sharing styles, and relate the ML and Haskell orthodoxies.

Fibration.lhs [16K]
The article with the juxtaposed literate Haskell and OCaml code
It was posted by Chung-chieh Shan on the discussion thread Applicative translucent functors in Haskell on the Haskell mailing list on Mon Sep 13 15:20:33 EDT 2004.

Typeable makes Haskell98 unsound

The class Typeable provides run-time representation of types and a type-safe cast operation. According to the documentation, ``To this end, an unsafe cast is guarded by a test for type (representation) equivalence.'' Alas, that test is trivial to fake, which gives us the total function of the inferred type a->b. This unsound cast can indeed lead to the Segmentation fault.

     module C where
     import Data.Typeable
     import Data.Maybe
     newtype W a = W{unW :: a}
     instance Typeable (W a) where typeOf _ = typeOf ()
     bad_cast x = unW . fromJust . cast $ W x
     -- inferred type: bad_cast :: a -> b
     test1 = bad_cast True ++ ""
When we load the above Haskell98 code in GHCi and try to evaluate test1 (which casts a boolean to a string), we see:
     $ ghci /tmp/c.hs 
     Loading package base ... linking ... done.
     [1 of 1] Compiling C                ( /tmp/c.hs, interpreted )
     Ok, modules loaded: C.
     *C> test1
     segmentation fault: 11
The current version is 1.1, Jul 31, 2007.

State Monad as a term algebra

We show the implementation of the State monad as a term algebra. We represent monadic computation by a term built from the following constructors:

     data Bind t f   = Bind t f
     data Return v   = Return v
     data Get        = Get
     data Put v      = Put v
For example, the term Get `Bind` (\v -> Put (not v) `Bind` (\() -> Return v)) denotes the action of negating the current state and returning the original state.
     runst :: RunState t s a => t -> s -> (s, a)
The function runst is the observer of our terms, or the interpreter of monadic actions. Given the term t and the initial state of the type s, the function interprets Get, Bind, etc. actions and returns the final state and the resulting value. The type of the result, a, is uniquely determined by the term and the state. The only non-trivial part is the interpretation of Bind, due to the polymorphism of the monadic bind operation. We use an auxiliary class RunBind for that purpose.

For completeness, we show that our term representation of state monadic actions is an instance of MonadState. We can then use the familiar notation to write our sample term above: do{v <- get; put (not v); return v}

Our implementation statically guarantees that only well-formed and well-typed terms can be evaluated.

The current version is 1.1, Jan 4, 2005.
StateAlgebra.hs [4K]
The commented Haskell source code and a few tests
The code was originally posted as Initial (term) algebra for a state monad on the Haskell-Cafe mailing list on Tue, 4 Jan 2005 00:08:04 -0800 (PST)

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