The type signature of a combinator does not match the type signature of its equivalent Lambda function

Question

Consider this combinator:

S (S K)

Apply it to the arguments X Y:

S (S K) X Y

It contracts to:

X Y

I converted S (S K) to the corresponding Lambda terms and got this result:

(\x y -> x y)

I used the Haskell WinGHCi tool to get the type signature of (\x y -> x y) and it returned:

(t1 -> t) -> t1 -> t

That makes sense to me.

Next, I used WinGHCi to get the type signature of s (s k) and it returned:

((t -> t1) -> t) -> (t -> t1) -> t

That doesn't make sense to me. Why are the type signatures different?

Note: I defined s, k, and i as:

s = (\f g x -> f x (g x))
k = (\a x -> a)
i = (\f -> f)

Answer 1

Thanks to all who responded to my question. I have studied your responses. To be sure that I understand what I've learned I have written my own answer to my question. If my answer is not correct, please let me know.

We start with the types of k and s:

   k  :: t1 -> t2 -> t1 
   k  =  (\a x -> a) 

   s  :: (t3 -> t4 -> t5) -> (t3 -> t4) -> t3 -> t5 
   s  =  (\f g x -> f x (g x)) 

Let's first work on determing the type signature of (s k).

Recall s's definition:

s = (\f g x -> f x (g x))

Substituting k for f results in (\f g x -> f x (g x)) contracting to:

(\g x -> k x (g x))

Important The type of g and x must be consistent with k's type signature.

Recall that k has this type signature:

   k :: t1 -> t2 -> t1

So, for this function definition k x (g x) we can infer:

• The type of x is the type of k's first argument, which is the type t1.

• The type of k's second argument is t2, so the result of (g x) must be t2.

• g has x as its argument which we've already determined has type t1. So the type signature of (g x) is (t1 -> t2).

• The type of k's result is t1, so the result of (s k) is t1.

So, the type signature of (\g x -> k x (g x)) is this:

   (t1 -> t2) -> t1 -> t1

Next, k is defined to always return its first argument. So this:

k x (g x)

contracts to this:

x

And this:

(\g x -> k x (g x))

contracts to this:

(\g x -> x)

Okay, now we have figured out (s k):

   s k  :: (t1 -> t2) -> t1 -> t1 
   s k  =  (\g x -> x)

Now let's determine the type signature of s (s k).

We proceed in the same manner.

Recall s's definition:

s = (\f g x -> f x (g x))

Substituting (s k) for f results in (\f g x -> f x (g x)) contracting to:

(\g x -> (s k) x (g x))

Important The type of g and x must be consistent with (s k)'s type signature.

Recall that (s k) has this type signature:

   s k :: (t1 -> t2) -> t1 -> t1

So, for this function definition (s k) x (g x) we can infer:

• The type of x is the type of (s k)'s first argument, which is the type (t1 -> t2).

• The type of (s k)'s second argument is t1, so the result of (g x) must be t1.

• g has x as its argument, which we've already determined has type (t1 -> t2). So the type signature of (g x) is ((t1 -> t2) -> t1).

• The type of (s k)'s result is t1, so the result of s (s k) is t1.

So, the type signature of (\g x -> (s k) x (g x)) is this:

   ((t1 -> t2) -> t1) -> (t1 -> t2) -> t1

Earlier we determined that s k has this definition:

   (\g x -> x)

That is, it is a function that takes two arguments and returns the second.

Therefore, this:

   (s k) x (g x)

Contracts to this:

   (g x)

And this:

   (\g x -> (s k) x (g x))

contracts to this:

   (\g x -> g x)

Okay, now we have figured out s (s k).

   s (s k)  :: ((t1 -> t2) -> t1) -> (t1 -> t2) -> t1 
   s (s k)  =  (\g x -> g x)

Lastly, compare the type signature of s (s k) with the type signature of this function:

   p = (\g x -> g x)

The type of p is:

   p :: (t1 -> t) -> t1 -> t

p and s (s k) have the same definition (\g x -> g x) so why do they have different type signatures?

The reason that s (s k) has a different type signature than p is that there are no constraints on p. We saw that the s in (s k) is constrained to be consistent with the type signature of k, and the first s in s (s k) is constrained to be consistent with the type signature of (s k). So, the type signature of s (s k) is constrained due to its argument. Even though p and s (s k) have the same definition the constraints on g and x differ.

Answer 2

We start with the types of k and s

k :: t1 -> t2 -> t1
k = (\a x -> a)

s :: (t3 -> t4 -> t5) -> (t3 -> t4) -> t3 -> t5
s = (\f g x -> f x (g x))

So passing k as the first argument of s, we unify k's type with the type of s's first argument, and use s at the type

s :: (t1 -> t2 -> t1) -> (t1 -> t2) -> t1 -> t1

hence we obtain

s k :: (t1 -> t2) -> t1 -> t1
s k = (\g x -> k x (g x)) = (\g x -> x)

Then in s (s k), the outer s is used at the type (t3 = t1 -> t2, t4 = t5 = t1)

s :: ((t1 -> t2) -> t1 -> t1) -> ((t1 -> t2) -> t1) -> (t1 -> t2) -> t1

applying that to s k drops the type of the first argument and leaves us with

s (s k) :: ((t1 -> t2) -> t1) -> (t1 -> t2) -> t1

As a summary: In Haskell, the type of s (s k) is derived from the types of its constituent subexpressions, not from its effect on its argument(s). Therefore it has a less general type than the lambda expression that denotes the effect of s (s k).

Answer 3

The type system you are using is basically the same as simply-typed lambda calculus (you are not using any recursive functions or recursive types). Simply-typed lambda calculus is not fully general; it is not Turing-complete, and it cannot be used to write general recursion. SKI combinator calculus is Turing-complete, and can be used to write fixed-point combinators and general recursion; therefore, the full power of SKI combinator calculus cannot be expressed in simply-typed lambda calculus (although it can be in untyped lambda calculus).