10. Higher-order Polymorphism

Higher-order Polymorphism

Higher-order polymorphism extends polymorphism to higher-order functions, allowing functions to be polymorphic not only in their arguments but also in their return types.

This enables more expressive programming constructs where functions can operate on other functions in a polymorphic manner.

In higher-order polymorphism, type variables can be universally quantified over both types and functions.

For example, in a higher-order polymorphic language, we can define a function apply that takes a function and applies it to an argument:

apply : ∀α. (∀β. β → α) → α

This type signature indicates that apply is polymorphic in both its argument type α and the return type α.

Lab Activity: Implementing Higher-order Polymorphism

Let's implement a basic example of higher-order polymorphism in Python.

def apply(f):
    return f(f)

def identity(x):
    return x

def main():
    result = apply(identity)
    print("Result of apply(identity):", result)

if __name__ == "__main__":
    main()
  

In the lab activity, we define a function `apply` that takes a function `f` and applies it to itself.

We also define a simple identity function `identity`.

We then demonstrate higher-order polymorphism by applying the `apply` function to the `identity` function.