Introduction to Computational Linguistics

3. Welcome To The Typed Universe 7
Each type is interpreted by particular elements. Typically, the elements of different
types are different from each other. (This is when no type conversion occurs.
But see below.) If x is of type α and y is of type β α then x is distinct from
y. Here is how it is done. We interpret each type for b ∈ B by a set M b in
such a way that M a ∩ M b = ∅ if a b. Notice, for example, that OCaML
has a basic type char and a basic type string, which get interpreted by the
set of ASCII–characters, and strings of ASCII–characters, respectively. We may
construe strings over characters as function from numbers to characters. Then,
even though a single character is thought of in the same way as the string of
length 1 with that character, they are now physically distinct. The string k, for
example, is the function f : {0} → M char such that f (0) = k. OCaML makes
sure you respect the difference by displaying the string as "k" and the character
as ’k’. The quotes are not part of the object, they tell you what its type is.
Given two types α and β we can form the type of functions from α–objects to
β–objects. These functions have type α → β. Hence, we say that
(14) M α→β = { f : f is a function from M α to M β }
For example, we can define a function f by saying that f (x) = 2x + 3. The way to
do this in OCaML is by issuing
(15)
# let f x = (2 * x) + 3;;
val f :
int −> int =
Now we understand better what OCaML is telling us. It says that the value of the
symbol f is of type int -> int because it maps integers to integers, and that
it is a function. Notice that OCaML inferred the type from the definition of the
function. We can take a look how it does that.
First, if x is an element of type α → β and y is an element of type α, then x is
a function that takes y as its argument. In this situation the string for x followed
by the string for y (but separated by a blank) is well–formed for OCaML, and it is
of type β. OCaML allows you to enclose the string in brackets, and sometimes it
is even necessary to do so. Hence, f 43 is well–formed, as is (f 43) or even (f
(43)). Type that in and OCaML answers:
(16)
# f 43;;
- : int = 89