Function Types and Partial Application

A function type A→B says allowed inputs come from domain A and results belong to co-domain B. The actual range may use only part of B. A function taking a pair of integers can be modelled as integer×integer→integer, where the Cartesian product supplies the pair.

Function application supplies actual input to obtain a result. Referring to a function without applying it can instead produce the function value. In Python, square names the function while square(3) calls it; confusing these leads to wrong higher-order arguments or premature evaluation.

A first-class function can appear in expressions, be assigned to a name, be passed as an argument and be returned as a result. Other common values, such as integers, are first class too; the term describes permitted uses rather than computational speed.

Returning a function lets a program produce specialised behaviour. A closure can retain access to values from its defining environment. These capabilities can support functional styles in multi-paradigm languages, without claiming that every function written in Python is pure or free of side effects.

A curried type integer→integer→integer is read as integer→(integer→integer). Supplying the first integer returns a function waiting for the second. Partial application fixes some arguments of a multi-argument function, creating a new function that needs the remaining ones.

Argument order matters. Fixing the first argument of subtract(a,b)=a-b to 10 gives a function producing 10-b, not b-10. Python functools.partial can demonstrate partial application even though an ordinary two-argument Python function is not automatically a curried function.

from functools import partial
def subtract(a, b):
    return a - b

from_ten = partial(subtract, 10)
print(from_ten(3))

def add(a):
    return lambda b: a + b

print(add(4)(6))

from functools import partial

def volume(length, width, height):
    return length * width * height

fixed_base = partial(volume, 2, 3)
print(fixed_base(4))

Attempt the task before opening its worked response. Record each change of state, explain the rule used, and check the result against the original requirements. Code examples labelled Python 3 use that language’s conventions; pseudocode states its own assumptions.