ℂDuce: Documentation: Tutorial: Polymorphism in ℂDuce (development only)

*What follows is code that can be executed in the current development version but it is not included in the stable version yet. The syntax is not finalized and may change before inclusion in a stable release.*## Deep flatten

In ℂDuce there is a**operator that transforms a sequence of sequences into a sequence. To show the power of the polymorphic typing of ℂDuce let us define the deep variant of flatten, that we call**

`flatten`**, which recursively flattens all nested sequences into a single flat one. Nested sequences are a convenient representation of trees, whose leaves are values that are not sequences. So let us first define the type of α-trees, that is of trees whose leaves are of type α:**

`flatten_d`type Tree('a) = ('a\[Any*]) | [ (Tree('a))* ]

**is a type variable (read "alpha") and**

`'a`**is the type of trees whose leaves are of type**

`Tree('a)`**. These are either a leaf (i.e., any**

`'a`**value that is not a sequence:**

`'a`**) or a (possibly empty) sequence of**

`'a\[Any*]`**. The deep flatten function can then be defined as**

`Tree('a)`let flatten_d ( (Tree('a)) -> ['a*] ) | [] -> [] | [h ; t] -> (flatten_d h)@(flatten_d t) | x -> [x]

ℂDuce is the only language that can type this function truly polymorphically, in the sense that the function can be applied to any well typed expression. Indeed, notice that it is possible to unify the type

**with**

`Tree('a)`*every*type (in particular

**is a subtype of**

`Any`**). Therefore**

`Tree(Any)`**can be applied to every well-typed expression. So for instance we have:**

`flatten_d`# flatten_d [ 3 'r' [ 4 [ `true 5 ]] [ "quo" [[ `false ] "stop" ] ] ];; - : [ (Bool | 3--5 | 'o'--'u')* ] = [ 3 'r' 4 true 5 'quo' false 'stop' ]

**was unified with the union of all the types of the leaves of the tree, that is**

`'a`**. Without polymorphism the best type we can give to**

`[ (Bool | 3--5 | 'o'--'u')* ]`**would be**

`flatten_d`**, which is not very useful (it just states that the function returns a sequence that does not contain other sequences).**

`Any->[(Any/[Any*])*]`If we want to define a version of deep flatten that can be applied only to sequences, then this can be done by the following further definition

let flatten_tree (x: [(Tree('a))*]) : ['a*] = flatten_d x;;

## Partial applications

ℂDuce type inference is pretty sofisticated. While in ML-like languages unification suffices to type the application of a function to an argument, to type the same application in ℂDuce requires to combine unification with subtyping and expansion. As an example consider the following definitionlet even (Int -> Bool; ('a \ Int) -> ('a \ Int)) | x&Int -> x mod 2 = 0 | x -> x;;

**is a function that applied to an integer it returns a Boolean (i.e., whether the integer is even or not) and applied to an argument of any other type, returns a result of the same type (i.e., the argument itself). Now take the classic functional map function which can be defined as follows (alternatively in ℂDuce you can call**

`even`**function of the OCaml library):**

`Stdlib.List.map`let fmap (f : 'a -> 'b) (l : ['a*]) : ['b*] = match l with | [] -> [] | (a,b) -> (f a, fmap f b)

**. For the partial application**

`('a -> 'b) -> ['a*] -> ['b*]`**ℂDuce returns the following type:**

`fmap even`# fmap even;; - : ([ Int* ] -> [ Bool* ]) & ([ ('a \ Int)* ] -> [ ('a \ Int)* ]) & ([ ] -> [ ]) & ([ (Int | ('b \ Int))* ] -> [ (Bool | ('b \ Int))* ]) = <fun>

**is a function that (in the order): applied to a list of integers it returns a list of Booleans; applied to a list that does not contain integers it returns a list of the same type (actually, the same list); applied to the empty list returns the empty list; applied to a list that**

`(fmap even)`*may*contain some integers and other elements that are not integers (of type

**) it returns a list that contains some Booleans (that replaced all the integers of the list) and the other elements that are not integers. To obtain this high degree of precision the type system of ℂDuce has instantiated type type of**

`'a \Int`**in four different ways (corresponding to all the possibilities of instantiation of the type**

`fmap`**with the type of**

`'a -> 'b`**), computed the results of the applications, and taken their intersection.**

`even`While the definition of

**is somewhat artificial, we show next that its type is not strange, since it follows a common pattern that, for instance, is used to type the**

`even`**function we present next.**

`balance`## Red-Black trees

As a second example we show that the use of polymorphic set-theoretic types yields a better definition of Okasaki's implementation of red-black trees that can be found in his excellent monograph [1].A red and black tree is a colored binary search tree in which all nodes are colored either black or red and that satisfies 4 invariants:

- the root of the tree is black;
- the leaves of the tree are black;
- no red node has a red child;
- very path from the root to a leaf contains the same number of black nodes.

*by typing*that the operations on red-black trees (notably, the insertion) satisfy the first three invariants, as well as, that the

**function, local to insertion, never returns empty trees (yet another important property that, in ML/Haskell Okasaki's version, types cannot ensure). We use α and β to pretty print the**

`ins_aux`*type variables*

**and**

`'a`**respectively. Also we have slightly simplified the code. For the executable code see this file red-black.cd.**

`'b`type RBtree(α) = Btree(α) | Rtree(α) (* Black rooted RB tree: *) type Btree(α) = [] | <black elem=α>[ RBtree(α) RBtree(α) ] (* Red rooted RB tree: *) type Rtree(α) = <red elem=α>[ Btree(α) Btree(α) ] type Wrongtree(α) = <red elem=α>( [ Rtree(α) Btree(α) ] | [ Btree(α) Rtree(α) ]) type Unbalanced(α) = <black elem=α>( [ Wrongtree(α) RBtree(α) ] | [ RBtree(α) Wrongtree(α) ]) let balance ( Unbalanced(α) -> Rtree(α) ;β\Unbalanced(α) ->β\Unbalanced(α) ) | <black (z)>[ <red (y)>[ <red (x)>[ a b ] c ] d ] | <black (z)>[ <red (x)>[ a <red (y)>[ b c ] ] d ] | <black (x)>[ a <red (z)>[ <red (y)>[ b c ] d ] ] | <black (x)>[ a <red (y)>[ b <red (z)>[ c d ] ] ] -> <red (y)>[ <black (x)>[ a b ] <black (z)>[ c d ] ] | x -> x let insert (x :α) (t : Btree(α)) : Btree(α) = let ins_aux ( [] -> Rtree(α); Btree(α)\[] -> RBtree(α)\[]; Rtree(α) -> Rtree(α)|Wrongtree(α) ) | [] -> <red elem=x>[ [] [] ] | (<(color) elem=y>[ a b ]) & z -> if x << y then balance <(color) elem=y>[ (ins_aux a) b ] else if x >> y then balance <(color) elem=y>[ a (ins_aux b) ] else z in match ins_aux t with | <_ (y)>[ a b ] -> <black (y)>[ a b ]

**s) are black-rooted**

`Btree`**s (invariant 1), which are themselves black-rooted trees or red-rooted trees. The difference between the last two is that the latter cannot be leaves (invariant 2) and their children can only be black-rooted trees (invariant 3).**

`RBtree`The

**function takes an element**

`insert`**of type**

`x`*α*, and returns a function that maps red-black trees into red-black trees.

insert ::α-> Btree(α) -> Btree(α)

**is empty a red-rooted tree is returned, otherwise the element is inserted in the appropriate subtree and the whole tree is then balanced by the function**

`tree`**. This has the following type (which follows the same typing pattern as the function even defined in the introduction):**

`balance`balance :: ( Unbalanced(α) -> Rtree(α) ) & (β\Unbalanced(α) ->β\Unbalanced(α) )

**transforms an unbalanced tree into a (balanced) red-rooted tree and leaves all other trees (in particular the balanced ones) unchanged. The core of our definition is the type of**

`balance`**:**

`ins_aux`ins_aux :: ( [] -> Rtree(α) ) & ( Btree(α)\[] -> RBtree(α)\[] ) & ( Rtree(α) -> Rtree(α)|Wrongtree(α) )

**(which is the union of the three domains of the arrows forming its intersection type) is exactly**

`ins_aux`**. The intersection type describes the behaviour of**

`RBtree`**for each form of an**

`ins_aux`**—**

`RBtree`*ie*, empty, black-rooted, and red-rooted—. The type system needs the full precision of this type to infer whether the calls to

**in the body of**

`balance`**are applied to a balanced or an unbalanced tree: even a slight approximation of this type, such as**

`ins_aux`ins_aux :: ( Btree(α)\[] -> RBtree(α)\[] ) & ( Rtree(α)|[] -> Rtree(α)|Wrongtree(α) )

**it is easy to see that**

`ins_aux`**always returns either a (balanced) black-rooted tree or a tree with a red root in which one of the children may be a**

`ins_aux`**. In case of a tree with a red root, a balanced red-black tree is then obtained by changing the color of the root to black, as it is done in the last line of**

`Rtree`**.**

`insert`The implementation above must be compared with the corresponding version in monomorphic ℂDuce:

type RBtree = Btree | Rtree;; type Btree = [] | <black elem=Int>[ RBtree RBtree ];; type Rtree = <red elem=Int>[ Btree Btree ];; type Wrongtree = Wrongleft | Wrongright;; type Wrongleft = <red elem=Int>[ Rtree Btree ];; type Wrongright = <red elem=Int>[ Btree Rtree ];; type Unbalanced = <black elem=Int>([ Wrongtree RBtree ] | [ RBtree Wrongtree ]) let balance ( Unbalanced -> Rtree ; Rtree -> Rtree ; Btree\[] -> Btree\[] ; [] -> [] ; Wrongleft -> Wrongleft ; Wrongright -> Wrongright) | <black (z)>[ <red (y)>[ <red (x)>[ a b ] c ] d ] | <black (z)>[ <red (x)>[ a <red (y)>[ b c ] ] d ] | <black (x)>[ a <red (z)>[ <red (y)>[ b c ] d ] ] | <black (x)>[ a <red (y)>[ b <red (z)>[ c d ] ] ] -> <red (y)>[ <black (x)>[ a b ] <black (z)>[ c d ] ] | x -> x let insert (x : Int) (t : Btree) : Btree = let ins_aux ( [] -> Rtree ; Btree\[] -> RBtree\[]; Rtree -> Rtree|Wrongtree) | [] -> <red elem=x>[ [] [] ] | (<(color) elem=y>[ a b ]) & z -> if x << y then balance <(color) elem=y>[ (ins_aux a) b ] else if x >> y then balance <(color) elem=y>[ a (ins_aux b) ] else z in match ins_aux t with | <_ (y)>[ a b ] -> <black (y)>[ a b ]

**and**

`Wrongleft`**) in order to describe the polymorphic behavior of**

`Wrongright`**—whose type results, thus, much more obscure—. As far as we know ours types are the only system that can statically ensure the invariants above simply by decorating (with types) the original Okasaki's code without any further modification.**

`balance`ℂDuce: Documentation: Tutorial: Polymorphism in ℂDuce (development only)