Parametric polymorphism is the parameterization of a term by a type,
permitting the term to be used in differently-typed contexts. Types
of polymorphic terms contain universal quantification. For the sake
of inference and ease of type checking, mainstream functional
programming languages restrict where in the type the quantifiers may
appear and what they quantify over. The Hindley-Milner system, or core
ML, limits polymorphism to rank-1 (prenex): all quantifiers in a type
must be at the front and quantify over monomorphic types. OCaml and
Haskell relax the rank restriction, permitting quantifiers inside
certain types such as records. Haskell also permits parameterization
by arbitrary type constructors: we can define a tree data type
abstracting not only over the type of leaf values but also over the
type of the collection to hold node's children.

From time to time we run into the limits of polymorphism in Haskell, let alone in ML. We may need to parameterize by arbitrary type functions rather than mere constructors; we may need to quantify over not only types but also kinds. Or we think we need. Some of these demanding problems turn out solvable even in ML. On this page we collect such examples. Most of the code is simple OCaml, demonstrating that the plain Hindley-Milner system can be surprisingly expressive.

- Warm-up: polyvariadic functions
- Polymorphic selection
- Higher-order polymorphic selection
- Parametric polymorphism over a type class
- Getting around polymorphism restrictions using Category Theory and pretense
- Leibniz equality can be made injective
- Haskell Programming: Attractive Types

Many more examples of seemingly unattainable polymorphism - Typeclass overloading and bounded polymorphism in ML
- Representing existential data types with isomorphic simple types

- As a warm-up, we show the trivial OCaml code for the problem that
may at first blush appear unsolvable within the Hindley-Milner system.
The problem is to write a function that takes the number
`n`

followed by`n`

other arguments, returning them as the`n`

-element list. It may seem that the function should have the type`n -> ('a ->)^n -> 'a list`

, which is of course not possible in OCaml.The solution takes three lines; the first two define the building blocks for our number representation: zero and the successor. The third line defines the desired function

`p`

. An indented line after each definition shows the response of the OCaml top-level, with the inferred type for each function.let n0 = fun k -> k [];; val n0 : ('a list -> 'b) -> 'b = <fun> let s n k x = n (fun v -> k (x::v));; val s : (('a list -> 'b) -> 'c) -> ('a list -> 'b) -> 'a -> 'c = <fun> let p sel = sel (fun x -> x);; val p : (('a -> 'a) -> 'b) -> 'b = <fun>

We use the function`p`

as follows. (Again, an indented line underneath each statement is the response of the OCaml top-level.)p n0;; - : 'a list = [] p (s n0) 1;; - : int list = [1] p (s (s n0)) 1 2;; - : int list = [1; 2] p (s (s (s n0))) 'a' 'b' 'c';; - : char list = ['a'; 'b'; 'c']

**Version**- The current version is September 2008
**References**- Our problem is a variation of so-called `functional unparsing':

Olivier Danvy: Functional Unparsing. J. Functional Programming, 1998, v8, N6, pp. 621-625Polyvariadic

`zipWith`

and the tautology checker of boolean functions of arbitrary many arguments are described in the Functional Pearl

Daniel Fridlender and Mia Indrika: Do We Need Dependent Types? J. Functional Programming, 2000, v10, N4, pp. 409-415

The concluding section of the paper thoroughly discusses the custom numeral approach.Polyvariadic functions and keyword arguments

The introduction section of that page describes several other simple encodings of polyvariadic functions, including a simple Forth-like interpreter.double_generic.ml [6K]

A simple generalization: a generic `function' (or, recipe) that can be instantiated to different data types (sums, products, lists) and to the different number of arguments. In effect, we unify the generic`map`

with the polyvariadic`zipWith`

.

- Selection from heterogeneous structures such as nested tuples seems to
require parameterization of a term by a general type function (not a
mere type constructor). We ran into such selection problems on
several occasions; last time, when designing typed printf and scanf
sharing the same formatting specification. The problem turns out
solvable the with mere Hindley-Milner polymorphism, after we exploit the
deep connection between polymorphism and inlining.
The problem is to select a component from two tuples. If the tuples are homogeneous, this is easy: we define the function

`f1`

that takes a selector as the argument. To select a component, we apply`f1`

either to`fst`

or to`snd`

. (An indented line below each definition or statement shows the response of the OCaml top-level.)let f1 sel = (sel (1,2), sel (3,4));; val f1 : (int * int -> 'a) -> 'a * 'a = <fun> f1 fst;; - : int * int = (1, 3) f1 snd;; - : int * int = (2, 4)

Heterogeneous tuples cause serious trouble, however. For example:let f2 sel = (sel (1,'b'), sel (true,"four"));; Error: This expression has type bool * string but an expression was expected of type int * char

The first problem is indicated in the error message. In a Hindley-Milner system, function arguments must be monomorphic. Therefore, it is not possible to apply`sel`

(received as the argument to`f2`

) to the pairs of different types. There is another problem with`f2`

. Let us try to define this function in the less restrictive System F:let f2 (sel: forall a b. (a,b) -> a) = (sel @ int @ char (1,'b'), sel @ bool @ string (true,"four"))

where`@`

is the type application. We can apply`f2`

to`fst`

without trouble:`f2 (Fun ta tb -> fun (x:ta,y:tb) -> x)`

assuming the notation`Fun`

for the type abstraction. Alas, we cannot apply`f2`

to`snd`

, which has a different polymorphic type`forall a b. (a,b) -> b`

. Even System F is not expressive enough to write`f2`

. We need an abstraction not only over types but over arbitrary type functions of the kind`* -> * -> *`

.It turns out the selection from heterogeneous tuples is, after all, expressible in a Hindley-Milner system. Looking back at

`f2`

we notice that there are only two choices for`sel`

. Therefore, we can compute the applications`f2 fst`

and`f2 snd`

separately, and later choose between the two results. We use thunks to delay the computations until chosen.let f3 = (* f2 applied to fst *) let f2_fst () = (fst (1,'b'), fst (true,"four")) in (* f2 applied to snd *) let f2_snd () = (snd (1,'b'), snd (true,"four")) in fun sel -> sel (f2_fst, f2_snd) () val f3 : ((unit -> int * bool) * (unit -> char * string) -> unit -> 'a) -> 'a = <fun> f3 fst;; - : int * bool = (1, true) f3 snd;; - : char * string = ('b', "four")

The trick is a higher-order version of the technique commonly used in partial evaluation. It is related to `narrowing' in functional-logic programming. It is essentially the eta-rule for sums:fun x -> C[match x with V1 -> e1 | V2 -> e2] ==> fun x -> match x with V1 -> C[e1] | V2 -> C[e2]

where the context`C[]`

does not bind`x`

. **Version**- The current version is July 18, 2008
**References**- Interpreting
types as abstract values

The second part of these lecture notes expounds the deep relation between parametric polymorphism and `inlining'.Generating Code with Polymorphic let: A Ballad of Value Restriction, Copying and Sharing

Yet another illustration of the connection, in Sec 2.1 of the paper

- This article shows an example of how looking at a problem in a
different way turns it from impossible (at least, in System F) to
elementary, expressible in the Hindley-Milner system. The problem is
a higher-order version of the selection from heterogeneous
structures, described above. Recall, we wanted to write the
function
`g`

that takes a selector such as the tuple selectorfst :: forall a b. (a,b) -> a snd :: forall a b. (a,b) -> b

as an argument and applies it to several*heterogeneous*tuples. For example:g sel = (sel (1,'b'), sel (true,"four"))

(We have switched to Haskell, because the higher-order version of the problem was posed on Haskell-Cafe.)It is already a challenge to give a type to such a function in System F, let alone in the Hindley-Milner system. But we want more: a function that takes functions like

`g`

as an argument. The original problem was posed by Vladimir Reshetnikov on the Haskell-Cafe mailing list on June 6, 2009. We describe here an elaborated version: to type check the following definition of the function`fs`

along with several examples of its use.fs g = (g snd, (), g fst) t1 = fs id t2 = fs (:[]) t3 = fs (\sel -> sel (True,False)) t4 = fs (\sel -> sel (True,"False")) t5 = fs (\sel -> head $ sel ([True],"False")) t6 = fs (\sel -> (sel (1,'b'), sel (true,"four")))

The examples`t1`

through`t3`

type check in Haskell98; the others are flagged as ill-typed. Indeed, the inferred type of`fs`

is`(((a, a) -> a) -> t) -> (t, (), t)`

, betraying the selection from tuples whose components have the same type. In the example`t4`

the selector is applied to the patently heterogeneous tuple. We saw earlier that the type of a function taking a tuple selector as an argument must be parameterized by a type function of the kind`* -> * -> *`

. Such polymorphism is already not representable in System F. Now we wish to pass such a function as an argument to`fs`

.First we show a brute-force solution, emulating the necessary higher-rank polymorphism. Then we change the point of view, and the problem becomes trivial. The complex solution introduces an extra level of interpretation:

{-# LANGUAGE MultiParamTypeClasses, FlexibleInstances #-} {-# LANGUAGE FunctionalDependencies, UndecidableInstances #-} data Fst = Fst data Snd = Snd fs g = (apply g Snd, (), apply g Fst)

replacing the functions like`fst`

and`snd`

with their `encodings'`fstE`

and`sndE`

, to be interpreted by the extensible interpreter`Apply`

. The base cases of the interpreter are:class Apply f x y | f x -> y where apply :: f -> x -> y instance Apply (x->y) x y where apply = ($) instance Apply Fst (x,y) x where apply _ = fst instance Apply Snd (x,y) y where apply _ = snd

To write the simple tests`t1`

through`t3`

we add the encoding for functions that can be passed as arguments of`fs`

. Representing functions by their encoding lets us get around the too complex polymorphism:newtype Sel a w = Sel (((a,a) -> a) -> w) t1 = fs (Sel id) t2 = fs (Sel (:[])) t3 = fs (Sel (\sel -> sel (True,False))) -- (False,(),True) instance Apply (Sel a w) Fst w where apply (Sel f) _ = f fst instance Apply (Sel a w) Snd w where apply (Sel f) _ = f snd

That encoding is too simple for`t4`

. Hence we introduce a more complex encoding, and extend our interpreter:newtype PSel obj = PSel obj instance Apply Fst obj w => Apply (PSel obj) Fst w where apply (PSel obj) _ = apply Fst obj instance Apply Snd obj w => Apply (PSel obj) Snd w where apply (PSel obj) _ = apply Snd obj t13 = fs (PSel (True,False)) t4 = fs (PSel (True,"False")) -- ("False",(),True)

Alas, to express`t5`

and`t6`

we have to complicate our encoding further still.Let us pause however. The number of selectors is finite and small; for example, there are only two selectors from tuples,

`fst`

and`snd`

. This gives us the idea to represent the function`g`

as a `table', `indexed' by the selector argument. This change of representation is quite like applying the eta-rule for sums. The function`fs`

will receive the table: a pair of the results`(g fst, g snd)`

.g' = (g_fst, g_snd) where g_fst = (fst (1,'b'), fst (True,"four")) g_snd = (snd (1,'b'), snd (True,"four")) fs g' = (snd g', (), fst g') t1 = fs (id,id) t2 = fs ((:[]),(:[])) t3 = fs (True,False) t4 = fs (True,"False") -- ("False",(),True) t5 = fs (head [True], head "False") -- ('F',(),True) t6 = fs g' -- (('b',"four"),(),(1,True))

Thanks to the non-strict evaluation of Haskell, the results are not actually computed unless needed. This code type checks in Haskell98, or any Hindley-Milner system. Changing the representation made the problem trivial. **Version**- The current version is 1.2, October 2009
**References**- Type-class overloaded functions:
second-order typeclass programming with backtracking

A bigger example of using the interpreter`Apply`

Haskell with only one typeclass

Another extensive example of`Apply`

(called`C`

in that code)

- Leibniz equality (reminded below) can be used to prove that if two
types
`a`

and`b`

are equal, then so are`c a`

and`c b`

where`c`

is an arbitrary type constructor. The proof is constructive, giving us the substitution function that converts any term of the type`c a`

to the corresponding term of the type`c b`

. The inverse direction is called injectivity: from the equality of`c a`

and`c b`

obtain the equality of`a`

and`b`

. For example, injectivity tells us that if two list types`[a]`

and`[b]`

are equal then the types of their elements are equal as well. Obtaining the constructive proof of injectivity using Leibniz equality, without any compiler magic, has been considered impossible. We now demonstrate how to do that. The key is the realization that a type-constructor polymorphism extends to the polymorphism over arbitrary type functions. Leibniz injectivity is also the example of type functions seemingly more expressive than type-class functional dependencies.``Typing Dynamic Typing'' (Baars and Swierstra, ICFP 2002) demonstrated the first implementation and application of Leibniz equality witnesses in Haskell:

newtype EQU a b = EQU{subst:: forall c. c a -> c b}

The term`eq :: EQU a b`

witnesses the equality of the types`a`

and`b`

: in any context`c`

, the type`a`

can be substituted with the type`b`

. The context is represented by the type constructor`c`

. The witness gives the constructive proof of substitutability: given a term of the type`c a`

, we can always obtain a term of the type`c b`

. Since we do not know anything about the context`c`

, such a substitution may only happen if the two types are indeed equal. Hence the sole non-bottom witness isrefl :: EQU a a refl = EQU id

testifying that equality is reflexive. Leibniz equality became quite popular since it lets us represent GADTs to a large extent. The power of the Leibniz equality comes from the freedom to choose the context`c`

. The following example illustrates that power, by concisely proving that the equality is transitive. Here is the 1-line witness of the proof:tran :: EQU a u -> EQU u b -> EQU a b tran au ub = subst ub au

We treat the type`EQU a u`

as`(EQU a) u`

, that is, as an application of the `type constructor'`(EQU a)`

to the type`u`

. We then apply the witness`EQU u b`

to replace the type`u`

with the type`b`

in the context`(EQU a)`

, giving us the desired`EQU a b`

.Implementing type checkers and type inferencers in Haskell along the lines of typing dynamic typing requires the proof that two arrow types are equal if and only if their components are equal. In one direction,

eq_arr :: EQU a1 a2 -> EQU b1 b2 -> EQU (a1->b1) (a2->b2)

witnesses that if the argument and the result types of two arrow types are equal, the arrow types themselves are equal. Fortunately, for this, commonly used direction, the proof is easy, after we define suitable type-level combinators`F1`

and`F2`

to place the types being equated in the right context:newtype F1 t b a = F1{unF1:: EQU t (a->b)} newtype F2 t a b = F2{unF2:: EQU t (a->b)} eq_arr a1a2 b1b2 = unF2 . subst b1b2 . F2 . unF1 . subst a1a2 . F1 $ refl

In the type`EQU (a1->b1) (a2->b2)`

, the type `constructor'`(F1 (a1->b1) b2)`

represents the context of the type`a2`

, and the type constructor`(F2 (a1->b1) a2)`

is the context of the type`b2`

. Leibniz equality only applies if the context of a type is represented as a type constructor; therefore, we sometimes have to define auxiliary newtypes to put the contexts in the required form.In the reverse direction, we have to prove that if two arrow types are equal, their components (e.g., the argument types) are equal as well. The proof will witness the injectivity of the arrow type constructor. It was thought that injectivity cannot be proven with Leibniz equality at all. For example, the paper ``Implementing Cut Elimination: A Case Study of Simulating Dependent Types in Haskell'' by Chen, Zhu and Xi, PADL'04, needed the reverse, elimination direction and found it impossible to obtain with Leibniz equality witnesses. We demonstrate that type families help.

Type families let us define ``subtractive contexts'', so that we can view a type

`a`

as the type`(a->b)`

placed into the context that removes`(->b)`

. Again we use an auxiliary newtype to make the subtractive context appear as a type constructor.type family Fst a :: * type instance Fst (x->y) = x type instance Fst (x,y) = x -- etc. newtype FstA a b = FstA{unFstA :: EQU (Fst a) (Fst b)} eq_arr1 :: EQU (a1->b1) (a2->b2) -> EQU a1 a2 eq_arr1 eq = unFstA . subst eq $ ra where ra :: FstA (a1->b1) (a1->b1) ra = FstA refl

Jeremy Yallop has suggested a simple generalization that lets us write generic injectivity witnesses, for arbitrary type constructors`f`

and`f1`

:eq_f :: EQU (f a) (f1 b) -> EQU a b eq_f2 :: EQU (f a1 b1) (f1 a2 b2) -> EQU a1 a2

These witnesses testify that Leibniz equality is indeed stronger than mere bijection. **Version**- The current version is March 2011
**References**- LeibnizInjective.hs [4K]

The complete code for the article, with a simple test of the injectivity. The code implements Jeremy Yallop's suggestion. The first version of the code was posted on the Haskell-Cafe mailing list on Sun, 2 May 2010 18:28:02 -0700 (PDT).Proving False with impredicativity, injectivity, and type case analysis

- We describe a subtle and little known restriction on polymorphic
functions (class methods) in OCaml, and the general way of getting
around them. Befitting Category Theory, the approach is very general
and works in many languages, to write unrestricted polymorphic
functions such as maps over collections that are not really
polymorphic (examples below). We do not actually use any Category
Theory or draw any diagrams. We pretend -- which is the point. One
of my hard-won insights from Category Theory is that many of its
formal constructions are about pretending to do something without
doing anything. We put the pretense to real use -- and actually
solve a practical problem.
It all started with the question asked by Spiros Eliopoulos on the caml-mailing list on On 19 October 2015, describing a simple example distilled from the problem encountered in his work on js_of_ocaml. He wanted to define a polymorphic container class with a `map'-like method -- something like the following:

class ['a] container (x:'a) = object val v = x method map : 'b. ('a -> 'b) -> 'b container = fun f -> new container (f v) end

To be able to access fields and (private) methods of container objects,`map`

should be a method. It should return a new instance of the container, with a potentially different element type, depending on the type of the transformation function`f`

. Alas, OCaml rejects the definition, with a rather cryptic message that ``The universal type variable 'b cannot be generalized: it escapes its scope.'' Can such a container class be implemented?A similar class with the less-polymorphic (and less useful) map method

class ['a] containerM (x:'a) = object val v = x method mapM : ('a -> 'a) -> 'a containerM = fun f -> new containerM (f v) end

is however accepted. To understand why, it is worth writing the`['a] containerM`

object type in full, expanding all abbreviations:type 'a contM = <map : ('a->'a) -> <map : ('a->'a) -> <... > > >

This is an equi-recursive type, as object types are in OCaml, represented as the infinite tree. Crucially, it is a*regular*tree: the set of its distinct subtrees is finite (to be precise, the distinct subtrees are:`'a`

,`'a->'a`

,`map: ('a->'a) -> 'a contM`

, and`'a contM`

). In contrast, the object type of the original`['a] container`

has an infinite number of distinct subtrees: one for each instantiation of the type variable`'b`

. For the sake of type checking, object (and polymorphic variant) types in OCaml must be regular. In more detail, the regularity requirement was explained in the message by Jeremy Yallop, posted in response to Spiros Eliopoulos.Thus, strictly speaking, the desired polymorphic container class with the polymorphic map method cannot be defined. That is where Category Theory comes in -- at least, that is how it how it came to my mind and lead to the solution.

Let us consider a closely related problem: defining a polymorphic map function over a collection that is not really polymorphic, like Bigarrays in OCaml (we show one-dimensional arrays for simplicity):

open Stdlib.Bigarray type ('a,'elt) arr = ('a, 'elt, c_layout) Array1.t

Although it looks like the type variable`'a`

can be instantiated to any type, in reality big arrays may contain only integer and floating-point numbers. Thus the unrestricted polymorphic map-like function`amap: ('a -> 'b) -> ('a,'elt) arr -> ('b,'elt) arr`

is not possible. Luckily, left Kan extension (along the identity) tells us how to overcome the restriction: rather than actually performing the mapping, just collect its arguments and declare the operation done.Here is the realization of the idea:

type 'b karr = KanArr : ('a,'elt) arr * ('a -> 'b) -> 'b karr

The data type`'b karr`

collects the arguments of a mapping operation: the array`arr`

and the transformation function`f`

. The parameters`'a`

and`'elt`

are existentially quantified. Any (one-dimensional) big array`('a,'elt) arr`

can be converted to`'a karr`

, as illustrated below:let ar1 : (int,int_elt) arr = Array1.of_array Int c_layout [|1;2;3;4;5|] val ar1 : (int, int_elt) arr = <abstr> let kar1 = KanArr (ar1,fun x->x);; val kar1 : int karr = KanArr (<abstr>, <fun>)

(The indented lines below the definitions show the responses of the OCaml top-level.) The data type`'a karr`

permits the fully polymorphic map without any restrictions:let kmap : ('a -> 'b) -> 'a karr -> 'b karr = fun f (KanArr (ar,g)) -> KanArr (ar, fun x -> f (g x))

to be used aslet kar2 = kmap string_of_int kar1;; val kar2 : string karr = KanArr (<abstr>, <fun>)

This is just a pretense: although`kmap string_of_int`

creates`string karr`

it certainly does not create`(string,int) arr`

; the latter is just impossible. It may not matter however: if all we want in the end is, say, to reduce the`arr`

, we can do the left folding just as well on`karr`

instead:let kfold : ('b -> 'a -> 'b) -> 'b -> 'a karr -> 'b = fun f z (KanArr (ar,g)) -> let sum = ref z in for i=0 to Array1.dim ar -1 do sum := f !sum (g ar.{i}) done; !sum kfold (^) "" kar2;; - : string = "12345"

The lesson carries over to the original container problem. The map method does not have to actually create a new container instance. It could pretend: that is, merely collect all data needed to create the instance. Here is the whole solution:type 'a cont_proxy = P of 'a class ['a] container (x:'a) = object val v = x method map' : 'b. ('a -> 'b) -> 'b cont_proxy = fun f -> P (f v) method get : 'a = v end

We added the accessor`get`

for the sake of examples. The data type`cont_proxy`

is defined to hold the arguments to the container constructor (on our case, a single value). The method`map'`

does not actually construct anything; it merely returns the data needed for the construction. As a method,`map'`

has access to fields and (private) methods of container objects. When all data needed to build a container instance is at hand, constructing the instance is trivial, and can be done by an external function:let map : ('a -> 'b) -> ('a container -> 'b container) = fun f c -> match c#map' f with P x -> new container x

Here are the tests (the responses of the OCaml top-level are shown under each test, indented).let c = new container 3;; val c : int container = <obj> let _ = c#get;; - : int = 3 let c' = map string_of_int c;; val c' : string container = <obj> let _ = c'#get;; - : string = "3"

As Jacques Garrigue said on the mailing list: ``The externalizing solution has been known since the beginning of OCaml, but it is nice to know that it has such a cute name.'' **Version**- The current version is October 2015
**References**- Thread "map"-ing parameterized class types on the caml-list, October 2015.

- Bas van Dijk, in a message posted on Haskell-Cafe on 30 September 2008
has posed a problem of parameterizing a function by a type class
rather than by a type. The standard Prelude functions
read :: Read a => String -> a fromInteger :: Num a => Integer -> a

are the examples of parameterizing a term by a type, which is constrained to be a member of a type class, Read or Num, respectively. With rank-2 polymorphism, we can write terms that accept polymorphic functions like`read`

and`fromInteger`

as arguments, for example:foo :: (Num c, Num d) => (forall b. Num b => a -> b) -> a -> (c, d) foo f x = (f x, f x) bar :: (Read c, Read d) => (forall b. Read b => a -> b) -> a -> (c, d) bar f x = (f x, f x) testFoo = foo fromInteger 1 :: (Int, Float) -- (1,1.0) testBar = bar read "1" :: (Int, Float) -- (1,1.0)

Since higher-rank types in general cannot be inferred, we must supply signatures for`foo`

and`bar`

. Writing signatures for top-level functions is overall a good habit. If`foo`

and`bar`

were local however, defined in a`where`

clause, we must still write their full signatures -- and that is annoying.The functions

`foo`

and`bar`

illustrate parameterizing terms by polymorphic terms. Bas van Dijk has observed that`foo`

and`bar`

look quite alike. It is tempting to abstract away the difference between them, generalizing`foo`

and`bar`

to the single`baz`

to be used as:testBaz1 = baz fromInteger 1 :: (Int, Float) testBaz2 = baz read "1" :: (Int, Float)

The only difference between`foo`

and`bar`

is the type class constraint:`Num`

for`foo`

and`Read`

for`bar`

. The desired`baz`

must therefore be parameterized by a type class. This article develops the solution without resorting to the recently added Constraint kind, showing that first-class constraints have always been available in Haskell.It turns out that quantification over a type class is easy, after we add a layer of indirection. We introduce a type class that relates, or interprets, types as type classes. That type class acts as a look-up table from types to type class constraints. We can now use ordinary types to

*represent*type classes. Ordinary polymorphism -- parameterization over types -- becomes in effect parameterization over type classes.We implement the idea as follows, using these extensions:

{-# LANGUAGE MultiParamTypeClasses, FunctionalDependencies #-} {-# LANGUAGE FlexibleInstances, TypeSynonymInstances #-}

Although functional dependencies are required, overlapping or undecidable instances are not. We also get by without exotic extensions such as mutually recursive instances, required by `SYB with class'.We define the ``look-up table'' from types to type classes as the following type class:

class Apply l a b | l b -> a where apply :: l -> a -> b

The type class is the generalization of the ordinary function application. We can even give the following instance for it (although we won't be needing it here).instance Apply (a->b) a b where apply = ($)

We now add two entries into our look-up table from types to type classes. The types`LRead`

and`LFromInt`

below will act as look-up keys.data LRead = LRead instance Read b => Apply LRead String b where apply _ = read data LFromInt = LFromInt instance Num b => Apply LFromInt Integer b where apply _ = fromInteger

The type`LRead`

indeed relates to the type class constraint`Read`

while`LFromInt`

relates to`Num`

. We are done. The function`baz`

is simplybaz f x = (apply f x, apply f x)

to be used as follows:testBaz1 = baz LFromInt 1 :: (Int, Float) testBaz2 = baz LRead "1" :: (Int, Float)

Incidentally, if it were not for the polymorphic numeric literal`1`

, we could have defined`Apply`

with only two parameters and without the functional dependency.We have thus attained a seemingly high-ranking polymorphism, using only rank-1 types, which are all inferrable. We do not have to write the signature for

`baz`

. It is still interesting to see what the type checker has inferred:*Main> :t baz baz :: (Apply l a b, Apply l a b1) => l -> a -> (b, b1)

The type of`baz`

does show it to be a generalization of both`foo`

and`bar`

. The term is polymorphic over the type`l`

of keys in our look-up table of type class constraints. In effect,`baz`

is polymorphic over type classes -- as desired.Since a type class acts as a type predicate in Haskell, quantification over arbitrary type predicates gives us the unrestricted set comprehension. We should be able to write something like Russell paradox. Indeed we can. The following code understandably needs the UndecidableInstances extension.

data Delta = Delta instance Apply l l b => Apply Delta l b where apply _ x = apply x x

The definition of`apply`

is*not*recursive: the method`apply`

for the instance`Apply Delta l b`

is defined in terms of a generally different`apply`

method, for instance`Apply l l b`

. However, we may, later on, instantiate`l`

to be`Delta`

. After all, the quantification in the above instance is impredicative, over a class that includes the very instance being defined. We introduceomega () = apply Delta Delta

whose inferred type*Main> :t omega omega :: () -> b

leaves no doubts about the behavior of the term:`omega ()`

promises to produce a value of any type whatsoever. Of course the evaluation of`omega ()`

does not terminate. Since`apply`

is sort of a functional application, the term`apply x x`

above does look quite like the self-application in`\x -> x x`

, which is the `kernel' of fixed-point combinators. The self-application and hence fixed-point combinators are not expressible in the simply-typed lambda-calculus. Our calculus is certainly not simply typed. **Version**- The current version is October 1, 2008
**References**- Type-class overloaded functions:
second-order typeclass programming with backtracking

A bigger example of`Apply`

Class-parameterized classes, and the type-level logarithm An illustration of parameterizing a type class by a type class