分享一道伯克利 CS 61A 关于高阶函数的一道Python作业题（1）

Church numerals

题目大意是，用函数来表示自然数。具体见下文， 这里我就不翻译了。

The logician Alonzo Church invented a system of representing non-negative integers entirely using functions. The purpose was to show that functions are sufficient to describe all of number theory: if we have functions, we do not need to assume that numbers exist, but instead we can invent them.

Your goal in this problem is to rediscover this representation known as Church numerals. Here are the definitions of zero, as well as a function that returns one more than its argument:

def zero(f):
    return lambda x: x

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

First, define functions one and two such that they have the same behavior as successor(zero) and successsor(successor(zero)) respectively, but do not call successor in your implementation.

Next, implement a function church_to_int that converts a church numeral argument to a regular Python integer.

Finally, implement functions add_church, mul_church, and pow_church that perform addition, multiplication, and exponentiation on church numerals.

def one(f):
    """Church numeral 1: same as successor(zero)"""
    "*** YOUR CODE HERE ***"

def two(f):
    """Church numeral 2: same as successor(successor(zero))"""
    "*** YOUR CODE HERE ***"

three = successor(two)

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 >>> church_to_int(three) 3 """
    "*** YOUR CODE HERE ***"

def add_church(m, n):
    """Return the Church numeral for m + n, for Church numerals m and n. >>> church_to_int(add_church(two, three)) 5 """
    "*** YOUR CODE HERE ***"

def mul_church(m, n):
    """Return the Church numeral for m * n, for Church numerals m and n. >>> four = successor(three) >>> church_to_int(mul_church(two, three)) 6 >>> church_to_int(mul_church(three, four)) 12 """
    "*** YOUR CODE HERE ***"

def pow_church(m, n):
    """Return the Church numeral m ** n, for Church numerals m and n. >>> church_to_int(pow_church(two, three)) 8 >>> church_to_int(pow_church(three, two)) 9 """
    "*** YOUR CODE HERE ***"

此外，在同一组作业中，还提供了几个可能会用到的函数

def square(x):
    return x * x

def triple(x):
    return 3 * x

def identity(x):
    return x

def increment(x):
    return x + 1

为了让那些有兴趣的读者有自己尝试的机会，这里先不给出个人的解答，个人解答见下一篇。