Introduction to Computational Linguistics

8. Combinators 22
You can of course now define
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# let p x y = k i x y;;
val p = ’a −> ’b −> ’b =
and this defines the function pxy = y. Notice that you cannot ask OCaML to
evaluate a function abstractly. This is because it does not know that when you ask
k i x y the letters x and y are variables. There are no free variables!
So, you can apply combinators to concrete values. You cannot calculate abstractly
using the inbuilt functions of OCaML. One thing you can do, however, is
check whether an expression is typable. For example, the combinator SII has no
type. This is because no matter how we assign the primitive types, the function is
not well–defined. OCaML has this to say:
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# s i i;;
This expression has type (’a −> ’b’) −> ’a −> ’b but
is used here with type (’a -> ’b) -> ’a
How does it get there? It matches the definition of s and its first argument i.
Since its first argument must be of type ’a -> ’b -> ’c, this can only be the
case if ’a = ’b -> ’c. Applying the function yields an expression of type (’a
-> ’b) -> ’a -> ’c. Hence, the next argument must have type (’b -> ’c)
-> ’b. This is what OCaML expects to find. It communicates this using ’a
and ’b in place of ’b and ’c, since these are variables and can be renamed at
will. (Yes, OCaML does use variables, but these are variables over types, not over
actual objects.) However, if this is the case, and the argument is actually i, then
the argument actually has type (’b -> ’c) -> ’b -> ’c. Once again, since
OCaML did some renaming, it informs us that the argument has type (’a ->
’b) -> ’a -> ’b. The types do not match, so OCaML rejects the input as not
well formed.
Now, even though OCaML does not have free variables, it does have variables.
We have met them already. Every let–clause defines some function, but this
definition involves saying what it does on its arguments. We have defined i, k,
and s in this way. The identifiers x, y and z that we have used there are unbound
outside of the let–clause. Now notice that upon the definition of k, OCaML
issued its type as ’a -> ’b -> ’a. This means that it is a function from objects