This question is in one of the Haskell textbook exercises, not so smart question.

Question: What are the types of the following values?

[tail, init, reverse]

Solution: [[a] -> [a]]

But why?

CodePudding user response：

tail, init and reverse are functions defined on a generic list, [a]. You read [a] as "a list of any type abstracted as a", in particular, if a = Int, you get [Int] - list of integers.

Now, function that takes a list and outputs a list of the same type has signature [c] -> [c] or, if c = a, [a] -> [a]. The letter picked up does not matter as long as you have the same latter on both sides, because [a] -> [b] would mean that you take somehow a list of as and map it to a list of bs - which - in general - a different type. You could say that a = Int and b = String and therefore you'd have to map Int to String anyhow; also you could say that b = a an thus reduce to the [a] -> [a] case.

The last one: [tail, init, reverse] is obviously yet another list, where a = [b] -> [b]. So you got [ [b] -> [b] ] type or, equivalently, [ [a] -> [a] ] or [ [k] -> [k] ]. Again: particular letter does not matter as long as you stick to the chosen one.

CodePudding user response：

The type of the following value, not "values". singular. List of three elements is a value. Or did you mean "each of the values in the given list"? then they all must have the same type. since a list has a type [t] for some t. which implies all its elements have that same type t, whatever that is. then since tail :: [a] -> [a], so must be the other two that appear in the same list.

On their own they could each refer to a different letter i.e. "type variable". but inside the same list, they all must be in agreement. thus the type of the list as a whole is [ [a] -> [a] ].