了解 Haskell 中的类型和术语

Question

Which of the types below are the same type:

i. a -> b -> c -> d

ii. (a -> b) -> c -> d

iii. a -> (b -> c) -> d

iv. a -> b -> (c -> d)

v. (a -> b) -> (c -> d)

vi. ((a -> b) -> c) -> d

vii. a -> (b -> (c -> d))

(b) In the following terms what are the types of the functions f given that x,y,z:: Integer:

i. (fx) (y, z)

ii. fxyz

iii. f (x,y,z)

iv. f (x,(y,z))

(c) Give the types of the following terms (if indeed they type) and indicate which are equal:

i. "abcd"

ii. [('a','b'),('c','d')]

iii. ('a':['b']):('c':['d'])

iv. 'a':('b':'c':'d':[])

v. ["ab","cd"]

I'm not looking for a solution but I need help understanding the usage of () and its meaning. thank you.

Answer 1

In types, -> associates to the right, ie a -> b -> c actually means a -> (b -> c) . This is a function which takes an argument of type a and returns a function of type b -> c .

By comparison, (a -> b) -> c is a function which takes as argument a function of type a -> b , and returns a value of type c .

Here are a few examples

foo :: Int -> Bool -> String
-- the same as foo :: Int -> (Bool -> String)
-- takes Int, returns function
foo n = \b -> if b && n> 5 then "some result" else "some other result"

bar :: (Int -> Bool) -> String
-- takes function, returns string
bar f = if f 43 then "hello" else "good morning"

-- bar can be called in this way
test :: String
test = bar (\n -> n > 34)    -- evaluates to "hello"

When calling a function, as in fxyz , application associates to the left, as in (((fx) y) z) . For example, these are equivalent:

foo 5 True
(foo 5) true

By contrast, (,,,,) with commas inside is the way to form tuples, and is unrelated to application. Hence, [('a','b'),('c','d')] is a list of pairs. Instead, in your example ('a':['b']):('c':['d']) there are no commas, so parentheses are only for grouping, and the expression has the same meaning of

x : y
where x = 'a':['b']
      y = 'c':['d']

Try thinking about what types should x and y have, and then think about the type of x: y , if any.