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CSCI.4430/6969 Programming Languages
Lecture Notes
August 26, 2003
The mathematical notation for defining a function is with a statement such as f (x) = x^2 , f : Z → Z,
where Z is the set of all integers. The first Z is called the domain of the function, or the set of values x can take. The second Z is called the range of the function, or the set containing all possible values of f (x). Suppose f (x) = x^2 and g(x) = x + 1. Traditional function composition is defined as f ◦ g = f (g(x)).
With our functions f and g,
f ◦ g = f (g(x)) = f (x + 1) = x^2 + 2x + 1.
Similarly g ◦ f = g (f (x)) = g(x^2 ) = x^2 + 1. Therefore, function composition is not commutative. In lambda (λ) calculus, we can use a different notation to define these same concepts. To define a function f (x) = x^2 , we instead write
λx.x^2.
Similarly for g(x) = x + 1 we instead write
λx.x + 1.
To describe a function application such as f (2) = 4, we write
(λx.x^2 2) ⇒ 22 ⇒ 4.
The syntax for lambda calculus expressions is
e ::= v – variable | λv.e – lambda expression | (e e) – procedure call
The semantics of the lambda calculus, or the way of evaluating or simplifying expressions, is defined by the rule
(λx.E M ) ⇒ E{M/x}.
The new expression E{M/x} can be read as “replace ‘fresh’ x’s in E with M ”. Informally, a “fresh” x is an x that is not nested inside another lambda expres- sion. We will cover free and bound variable occurrences in detail in upcoming lectures. For example, in the expression
(λx.x^2 2),
E = x^2 and M = 2. To evaluate the expression, we replace x’s in E with M , to obtain (λx.x^2 2) ⇒ 22 ⇒ 4. In lambda calculus, all functions may only have one variable. Functions with more than one variable may be expressed as a function of one variable through currying. Suppose we have a function of two variables expressed in the normal way h(x, y) = x + y, h : (Z × Z) → Z.
With currying, we can input one variable at a time into separate functions. The first function will take the first argument, x, and return a function that will take the second variable, y, and will in turn provide the desired output. To create the same function with currying, let
f : Z → (Z → Z)
and g : Z → Z.
That is, f maps integers to a function, and g maps integers to integers. The function f (x) returns a function gx that provides the appropriate result when supplied with y. For example,
f (2) = g 2 , where g 2 (y) = 2 + y.
So f (2)(3) = g 2 (3) = 2 + 3 = 5.
In lambda calculus this function would be described with currying by
λx.λy.x + y.
For function application, we nest two application expressions
((λx.λy.x + y 2) 3).
defines a function f (x) = x^2. To perform a procedure call, use the code
(f x [y z ...])
where y, z, etc. are additional parameters that f may require. The code
(f 2)
evaluates f (2) = 4.
