CSE 428: Lecture 26 (additional)


The term "currying" comes from Haskell Curry, a famous logician who worked in State College (Dept of Mathematics) and developed the Combinatory Logic, which, together with the Lambda Calculus (Alonzo Church) constitutes the foundation of Higher Order Functional Programming.

Essentially, the idea is the following: consider a function f: 'a * 'b -> 'c. This function takes a pair of arguments, the first of type 'a, and the second of type 'b, and gives back a result in 'c. There is only one way f can be applied: by passing the two arguments together. Given x:'a, and y:'b, we write f(x,y) for the application of f to the pair(x,y).

We could imagine now a variant of f that, instead of taking the arguments together, takes them "one at the time", and gives the same result as f when it is supplied with both arguments. This variant is called curried version of f. Let us denote it by fc. The type of this function is fc: 'a -> 'b -> 'c, and, by definition, we have that for every x:'a, and y:'b, f(x,y) = fc x y holds. The main difference between f and fc is that the latter can be applied also to the first argument only: The expression fc x (for x:'a) is perfectly legal and denotes a function of type 'b -> 'c (the function which, when provided with an input y:'b, will give as result f(x,y)).

In order to provide support for curried functions, a language needs to be Higher Order. The currying adds flexibility and expressivity to the language, and it is part of the general principle of abstraction. In a sense, currying is intrinsic to the philosophy and design of an higher-order language. Below we will see some examples of how this feature adds expressivity.

Let us see how the currying possibility can be used in ML.

Defining curried functions

Let us consider a function with two arguments like the append of two lists. A possible definition is:
   - fun append([],k) = k
       | append(x::l,k) = x::append(l,k);
   val append = fn : 'a list * 'a list -> 'a list
The curried version of append can be defined in the following way:
   - fun append_c [] k = k
       | append_c (x::l) k = x::(append_c l k);
   val append_c = fn : 'a list -> 'a list -> 'a list
or, equivalently:
   - fun append_c [] = (fn k => k)
       | append_c (x::l) = fn k => x::(append_c l k);
   val append_c = fn : 'a list -> 'a list -> 'a list
We can now write expressions like append_c [1], for instance in a declaration:
   - val append_one = append_c [1];
   val append_one = fn : int list -> int list
Note that the system does not evaluate the expression append_c [1], because it is a function. It only computes its type. The evaluation will be performed only when we provide also the second argument. For instance:
   - append_one [5,6];
   val it = [1,5,6] : int list

Using currying for abstraction

We illustrate here how Higher Order and currying provide powerful and elegant mechanisms for abstraction.

Consider the functions sum_all : int list -> int and product_all : int list -> int (respectively sum and product of all the elements in a list of integers). They can be defined as follows:

   - fun sum_all [] = 0
       | sum_all (x::l) = x + sum_all l;
   val sum_all = fn : int list -> int

   - fun product_all [] = 1
       | product_all (x::l) = x * product_all l;
   val product_all = fn : int list -> int
Note that these two function work according to the same scheme: they scan the list (recursively) element by element, and perform a certain operation on every element (and on the result of the recursive call), and give a certain initial result when the list is empty.

This scheme is common to several other functions. We could then think of defining an abstract function (abstract wrt the operation and the initial element), which represent the general scheme. The particular functions (like sum_all and product_all can then be defined by providing the particular operation and initial value. This general function is commonly called reduce, and it is "more natural" to define it by using currying, as follows:

   fun reduce f v [] = v
     | reduce f v (x::l) = f(x, reduce f v l);
   val reduce = fn : ('a * 'b -> 'b) -> 'b -> 'a list -> 'b
Here, f: 'a * 'b -> 'b represents the operation, and v:'b represents the initial value.

The definitions of sum_all and product_all can now be given as follows:

   - val sum_all = reduce (op +) 0;
   val sum_all = fn : int list -> int

   -  val product_all = reduce (op * ) 1;
   val product_all = fn : int list -> int
We need to use (op +) and (op * ) instead of + and * because the latter are infix, while in the definition of reduce the parameter f is prefix. The operator op changes a function from infix to prefix.

We can now apply these functions to lists of integers, as illustrated in the following examples:

   - sum_all [];         
   val it = 0 : int
   - sum_all [1,2,3,4];
   val it = 10 : int

   - product_all [];  
   val it = 1 : int
   - product_all [1,2,3,4];
   val it = 24 : int
Another example of function that we can define by using reduce is the function forall, which checks whether a certain property p: 'a -> bool holds for all elements of a list (of type 'a list). We can define it as follows:
   fun forall p = let fun f(x,b) = p x andalso b
                   in reduce f true
   val forall = fn : ('a -> bool) -> 'a list -> bool
Examples of uses (note that (fn x => x>0) represents the property of being positive; (fn x => x mod 2 = 0) represents the property of being even).
   - forall (fn x => x>0) [1,2,3];
   val it = true : bool
   - forall (fn x => x>0) [~1,2,3];
   val it = false : bool
   - forall (fn x => x mod 2 = 0) [2,0,4];
   val it = true : bool
   - forall (fn x => x mod 2 = 0) [2,1];
   val it = false : bool
Other examples can be found in Assignment 8 (CSE 428, Spring 99). Note that in the assignment the type of reduce is restricted. The reason is due to a certain limitation of the type system of the present implementation of SML. We won't go into that because it's a bit complicated, and not so important.

The concept of currying extends naturally to arbitrary tuples of arguments.

Type theory and programming languages

Types are very useful in programming languages. The advantages of using a typed language include:
  1. (substantial) help in detecting errors
  2. increasing program readability
  3. increasing efficiency in implementation (because of better allocation of resources)

Some terminology