Adding church numerals using lambda functions in python

Question

I'm trying to learn python and CS on my own using a course online that is based off SICP. I understand the basics of church numerals, but I am having trouble on adding church numerals using lambda functions in python.

This is my code below for context:

def zero(f):
    return lambda x: x


def successor(n):
    return lambda f: lambda x: f(n(f)(x))


def one(f):
    """Church numeral 1."""
    return lambda x: f(x)


def two(f):
    """Church numeral 2."""
    return lambda x: f(f(x))


def church_to_int(n):
    """Convert the Church numeral n to a Python integer.

    >>> church_to_int(zero)
    0    
    >>> church_to_int(one)
    1
    >>> church_to_int(two)
    2
    """
    return n(lambda x: x + 1)(0)


def mul_church(m, n):
    """Return the Church numeral for m * n, for Church numerals m and n.

    >>> three = successor(two)
    >>> four = successor(three)
    >>> church_to_int(mul_church(two, three))
    6
    >>> church_to_int(mul_church(three, four))
    12
    """
    return lambda x: m(n(x))

This is add_church function that I am having trouble with:

def add_church(m, n):
    """Return the Church numeral for m + n, for Church numerals m and n.
    >>> three = successor(two)
    >>> church_to_int(add_church(two, three))
    5
    """
    return lambda f: lambda x: m(f(x))(n(x))

I have concluded that a way to add church numerals would be to somehow have one of the functions in add_church(m, n) to be the input or "x" in the other's lambda function. However, I keep getting errors that imply that I am not using the right arguments in my function call.

For example, when I call:

church_to_int(add_church(one, two))

I get an "int object not callable" error amongst others and have also tried other different methods with no success.

I think there is something that I am not seeing about lambda functions which is causing me to have trouble with implementing add_church. I've been spending a while on figuring this out, so any help that will guide me to the answer will be greatly appreciated.

ecatmur · Accepted Answer · 2023-07-17T16:40:31.373

Recall that Church encoding can be understood as repeated application of a function to an argument. So to add m + n, we need to apply a function f to an argument x m + n times, or equivalently apply it n times and then apply it m times:

def add_church(m, n):
    def m_plus_n(f):
        def f_repeated_m_plus_n_times(x)                # f ** (m + n)
            intermediate_result = (n(f))(x)             # (f ** n) (x)
            final_result = (m(f))(intermediate_result)  # (f ** m) ((f ** n) (x))
            return final_result
        return f_repeated_m_plus_n_times
    return m_plus_n

In lambda form, removing redundant parentheses:

def add_church(m, n):
    """Return the Church numeral for m + n, for Church numerals m and n.
    >>> three = successor(two)
    >>> church_to_int(add_church(two, three))
    5
    """
    return lambda f: lambda x: m(f)(n(f)(x))

Your example using functions was really insightful and cleared up a lot of confusion for understanding lambda functions, and I found it very clever. I am actually going to play around with translating functions into lambdas to really understand it. Thanks! — Andrew, Jul 23 '14 at 09:04

Adding church numerals using lambda functions in python

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