Chapter 1: Introduction


This tutorial by Adam Chlipala is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License.

This is a tutorial for the Ur/Web programming language.

Briefly, Ur is a programming language in the tradition of ML and Haskell, but featuring a significantly richer type system. Ur is functional, pure, statically-typed, and strict. Ur supports a powerful kind of metaprogramming based on row types.

Ur/Web is Ur plus a special standard library and associated rules for parsing and optimization. Ur/Web supports construction of dynamic web applications backed by SQL databases, with mixed server-side and client-side applications generated from source code in one language.

Ur inherits its foundation from ML and Haskell, then going further to add fancier stuff. This first chapter of the tutorial reviews the key ML and Haskell features, giving their syntax in Ur. I do assume reading familiarity with ML and Haskell and won't dwell too much on explaining the imported features.

For information on compiling applications (and for some full example applications), see the intro page of the online demo, with further detail available in the reference manual.

Basics


Let's start with features shared with both ML and Haskell. First, we have the basic numeric, string, and Boolean stuff. (In the following examples, == is used to indicate the result of evaluating an expression. It's not valid Ur syntax!)

1 + 1
== 2

1.2 + 3.4
== 4.6

"Hello " ^ "world!"
== "Hello world!"

1 + 1 < 6
== True

0.0 < -3.2
== False

"Hello" = "Goodbye" || (1 * 2 <> 8 && True <> False)
== True

We also have function definitions with type inference for parameter and return types.

fun double n = 2 * n

double 8
== 16

fun fact n = if n = 0 then 1 else n * fact (n - 1)

fact 5
== 120

fun isEven n = n = 0 || isOdd (n - 1)
and isOdd n = n = 1 || isEven (n - 1)

isEven 32
== True


Of course we have anonymous functions, too.

val inc = fn x => x + 1

inc 3
== 4

Then there's parametric polymorphism. Unlike in ML and Haskell, polymorphic functions in Ur/Web often require full type annotations. That is because more advanced features (which we'll get to in the next chapter) make Ur type inference undecidable.

fun id [a] (x : a) : a = x

id "hi"
== "hi"

fun compose [a] [b] [c] (f : b -> c) (g : a -> b) (x : a) : c = f (g x)

compose inc inc 3
== 5

The option type family is like ML's option or Haskell's Maybe. We also have a case expression form lifted directly from ML. Note that, while Ur follows most syntactic conventions of ML, one key difference is that type family names appear before their arguments, as in Haskell.

fun predecessor (n : int) : option int = if n >= 1 then Some (n - 1) else None

predecessor 6
== Some(5)

predecessor 0
== None

Naturally, there are lists, too!

val numbers : list int = 1 :: 2 :: 3 :: []
val strings : list string = "a" :: "bc" :: []

fun length [a] (ls : list a) : int =
    case ls of
        [] => 0
      | _ :: ls' => 1 + length ls'

length numbers
== 3

length strings
== 2

And lists make a good setting for demonstrating higher-order functions and local functions. (This example also introduces one idiosyncrasy of Ur, which is that map is a keyword, so we name our "map" function mp.)

fun mp [a] [b] (f : a -> b) : list a -> list b =
    let
        fun loop (ls : list a) =
            case ls of
                [] => []
              | x :: ls' => f x :: loop ls'
    in
        loop
    end

mp inc numbers
== 2 :: 3 :: 4 :: []

mp (fn s => s ^ "!") strings
== "a!" :: "bc!" :: []

We can define our own polymorphic datatypes and write higher-order functions over them.

datatype tree a = Leaf of a | Node of tree a * tree a

fun size [a] (t : tree a) : int =
    case t of
        Leaf _ => 1
      | Node (t1, t2) => size t1 + size t2

size (Node (Leaf 0, Leaf 1))
== 2

size (Node (Leaf 1.2, Node (Leaf 3.4, Leaf 4.5)))
== 3

fun tmap [a] [b] (f : a -> b) : tree a -> tree b =
    let
        fun loop (t : tree a) : tree b =
            case t of
                Leaf x => Leaf (f x)
              | Node (t1, t2) => Node (loop t1, loop t2)
    in
        loop
    end

tmap inc (Node (Leaf 0, Leaf 1))
== Node(Leaf(1), Leaf(2))

We also have anonymous record types, as in Standard ML. The next chapter will show that there is quite a lot more going on here with records than in SML or OCaml, but we'll stick to the basics in this chapter. We will add one tantalizing hint of what's to come by demonstrating the record concatention operator ++ and the record field removal operator --.

val x = { A = 0, B = 1.2, C = "hi", D = True }

x.A
== 0

x.C
== "hi"

type myRecord = { A : int, B : float, C : string, D : bool }

fun getA (r : myRecord) = r.A

getA x
== 0

getA (x -- #A ++ {A = 4})
== 4

val y = { A = "uhoh", B = 2.3, C = "bye", D = False }

getA (y -- #A ++ {A = 5})
== 5


Borrowed from ML


Ur includes an ML-style module system. The most basic use case involves packaging abstract types with their "methods."

signature COUNTER = sig
    type t
    val zero : t
    val increment : t -> t
    val toInt : t -> int
end

structure Counter : COUNTER = struct
    type t = int
    val zero = 0
    val increment = plus 1
    fun toInt x = x
end

Counter.toInt (Counter.increment Counter.zero)
== 1

We may package not just abstract types, but also abstract type families. Here we see our first use of the con keyword, which stands for constructor. Constructors are a generalization of types to include other "compile-time things"; for instance, basic type families, which are assigned the kind Type -> Type. Kinds are to constructors as types are to normal values. We also see how to write the type of a polymorphic function, using the ::: syntax for type variable binding. This ::: differs from the :: used with the con keyword because it marks a type parameter as implicit, so that it need not be supplied explicitly at call sites. Such an option is the only one available in ML and Haskell, but, in the next chapter, we'll meet cases where it is appropriate to use explicit constructor parameters.

signature STACK = sig
    con t :: Type -> Type
    val empty : a ::: Type -> t a
    val push : a ::: Type -> t a -> a -> t a
    val peek : a ::: Type -> t a -> option a
    val pop : a ::: Type -> t a -> option (t a)
end

structure Stack : STACK = struct
    con t = list
    val empty [a] = []
    fun push [a] (t : t a) (x : a) = x :: t
    fun peek [a] (t : t a) = case t of
                                 [] => None
                               | x :: _ => Some x
    fun pop [a] (t : t a) = case t of
                                 [] => None
                               | _ :: t' => Some t'
end

Stack.peek (Stack.push (Stack.push Stack.empty "A") "B")
== Some("B")

Ur also inherits the ML concept of functors, which are functions from modules to modules.

datatype order = Less | Equal | Greater

signature COMPARABLE = sig
    type t
    val compare : t -> t -> order
end

signature DICTIONARY = sig
    type key
    con t :: Type -> Type
    val empty : a ::: Type -> t a
    val insert : a ::: Type -> t a -> key -> a -> t a
    val lookup : a ::: Type -> t a -> key -> option a
end

functor BinarySearchTree(M : COMPARABLE) : DICTIONARY where type key = M.t = struct
    type key = M.t
    datatype t a = Leaf | Node of t a * key * a * t a

    val empty [a] = Leaf

    fun insert [a] (t : t a) (k : key) (v : a) : t a =
        case t of
            Leaf => Node (Leaf, k, v, Leaf)
          | Node (left, k', v', right) =>
            case M.compare k k' of
                Equal => Node (left, k, v, right)
              | Less => Node (insert left k v, k', v', right)
              | Greater => Node (left, k', v', insert right k v)

    fun lookup [a] (t : t a) (k : key) : option a =
        case t of
            Leaf => None
          | Node (left, k', v, right) =>
            case M.compare k k' of
                Equal => Some v
              | Less => lookup left k
              | Greater => lookup right k
end

structure IntTree = BinarySearchTree(struct
                                         type t = int
                                         fun compare n1 n2 =
                                             if n1 = n2 then
                                                 Equal
                                             else if n1 < n2 then
                                                 Less
                                             else
                                                 Greater
                                     end)

IntTree.lookup (IntTree.insert (IntTree.insert IntTree.empty 0 "A") 1 "B") 1
== Some("B")

It is sometimes handy to rebind modules to shorter names.

structure IT = IntTree

IT.lookup (IT.insert (IT.insert IT.empty 0 "A") 1 "B") 0
== Some("A")

One can even use the open command to import a module's namespace wholesale, though this can make it harder for someone reading code to tell which identifiers come from which modules.

open IT

lookup (insert (insert empty 0 "A") 1 "B") 2
== None

Ur adopts OCaml's approach to splitting projects across source files. When a project contains files foo.ur and foo.urs, these are taken as defining a module named Foo whose signature is drawn from foo.urs and whose implementation is drawn from foo.ur. If foo.ur exists without foo.urs, then module Foo is defined without an explicit signature, so that it is assigned its principal signature, which exposes all typing details without abstraction.


Borrowed from Haskell


Ur includes a take on type classes. For instance, here is a generic "max" function that relies on a type class ord. Notice that the type class membership witness is treated like an ordinary function parameter, though we don't assign it a name here, because type inference figures out where it should be used. The more advanced examples of the next chapter will include cases where we manipulate type class witnesses explicitly.

fun max [a] (_ : ord a) (x : a) (y : a) : a =
    if x < y then
        y
    else
        x

max 1 2
== 2

max "ABC" "ABA"
== "ABC"

The idiomatic way to define a new type class is to stash it inside a module, like in this example:

signature DOUBLE = sig
    class double
    val double : a ::: Type -> double a -> a -> a
    val mkDouble : a ::: Type -> (a -> a) -> double a

    val double_int : double int
    val double_string : double string
end

structure Double : DOUBLE = struct
    con double a = a -> a

    fun double [a] (f : double a) (x : a) : a = f x
    fun mkDouble [a] (f : a -> a) : double a = f

    val double_int = mkDouble (times 2)
    val double_string = mkDouble (fn s => s ^ s)
end

open Double

double 13
== 26

double "ho"
== "hoho"

val double_float = mkDouble (times 2.0)

double 2.3
== 4.6

That example had a mix of instances defined with a class and instances defined outside its module. Its possible to create closed type classes simply by omitting from the module an instance creation function like mkDouble. This way, only the instances you decide on may be allowed, which enables you to enforce program-wide invariants over instances.

signature OK_TYPE = sig
    class ok
    val importantOperation : a ::: Type -> ok a -> a -> string
    val ok_int : ok int
    val ok_float : ok float
end

structure OkType : OK_TYPE = struct
    con ok a = unit
    fun importantOperation [a] (_ : ok a) (_ : a) = "You found an OK value!"
    val ok_int = ()
    val ok_float = ()
end

open OkType

importantOperation 13
== "You found an OK value!"

Like Haskell, Ur supports the more general notion of constructor classes, whose instances may be parameterized over constructors with kinds beside Type. Also like in Haskell, the flagship constructor class is monad. Ur/Web's counterpart of Haskell's IO monad is transaction, which indicates the tight coupling with transactional execution in server-side code. Just as in Haskell, transaction must be used to create side-effecting actions, since Ur is purely functional (but has eager evaluation). Here is a quick example transaction, showcasing Ur's variation on Haskell do notation.

val readBack : transaction int =
   src <- source 0;
   set src 1;
   n <- get src;
   return (n + 1)

We get ahead of ourselves a bit here, as this example uses functions associated with client-side code to create and manipulate a mutable data source.