Lab 3: Midterm Review

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Lab3 部分question

- Q5：It's Always a Good Prime
- Q6：Church numerals

Q5：It’s Always a Good Prime

Implement div_by_primes_under, which takes in an integer nand returns an n-divisibility checker. An n-divisibility-checker is a function that takes in an integer k and returns whether kis divisible by any integers between 2 and n, inclusive. Equivalently, it returns whether k is divisible by any primes less than or equal to n.

Review the Disc 01 is_primeproblem for a reminder about prime numbers.

You can also choose to do the no lambda version, which is the same problem, just with defining functions with def instead of lambda.

def div_by_primes_under(n):
	"""
	>>> div_by_primes_under(10)(11)
	False
	>>> div_by_primes_under(10)(121)
	False
	>>> div_by_primes_under(10)(12)
	True
	>>> div_by_primes_under(5)(1)
	False
	"""
	checker = lambda x: False
	i = _____________________________
	while _______________________________:
		if not checker(i):
			checker = (lambda f, i: lambda x: ___________)(checker, i)
		i = _____________________________
	return __________________________

这里是 Disc 1 的 is_prime函数：

def is_prime(n):
	"""
	>>> is_prime(10)
	False
	>>> is_prime(7)
	True
	>>> is_prime(1) #one is not a prime number !!
	False
	"""
	if n == 1:
		return False
	k = 2
	while k < n:
		if n % k == 0:
			return False
		k += 1
	return True

根据 s_prime函数，我们可以推出 i 就类似与is_prime中 k，checker是分离出 n 的质数。最后通过checker判断 x 是否能够整除这些质数，如果能，则返回 True。

根据题目给出的框架，我们可以进一步实现：

def div_by_primes_under(n):
	checker = lambda x: False
	i = 2
	while i <= n:
		if not checker(i):
			checker = (lambda f, i: lambda x: _____________)(checker, i)
		i += 1
	return checker

checker = (lambda f, i: lambda x: _____________)(checker, i)

我们分析这个赋值语句，从左到右，第一个 lambda 函数有两个参数 f，i；第二个 lambda 函数有一个参数 x。
如果 checker函数返回 True（只要不是 n 的质数），这个语句就不会执行。很自然，空白处肯定有
x % i == 0。

checker = (lambda f, i: lambda x: x % i == 0)(checker, i)

上式是会一直得到 True，由于 i % i 会一直是 0。我们需要能够“存储”小于n的质数。f 就起作用了。

checker = (lambda f, i : lambda x: x % i == 0 or f(x))(checker, i)

上述是div_by_primes_under(5)(2)的例子：
通过反复调用 checker来保存小于 n 的质数，最后当 x = 2 时，通过不断回退函数，从 i = 5 、i = 3 最后到 i = 2，一步步判断，得到结果。

最终答案：

def div_by_primes_under(n):
    checker = lambda x: False
    i = 2
    while i <= n:
        if not checker(i):
             checker = (lambda f, i: lambda x: (x % i == 0) or f(x))(checker, i)
        i = i + 1
    return checker

Q6：Church numerals

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. Church numerals are a way to represent non-negative integers via repeated function application. Specifically, church numerals (such as zero, one, and two below) are functions that take in a function f and return a new function which, when called, repeats f a number of times on some argument x. Here are the definitions of zero, as well as a successor function, which takes in a church numeral n as an argument and returns a function that represents the church numeral one higher than n:

def zero(f):
    return lambda x: x

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

First, define functions oneand 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.

要求是实现丘奇数，其中zero、one、two分别是丘奇数中的 0、1、2。

最终实现如下：

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: same as successor(zero)"""
    "*** YOUR CODE HERE ***"
    return lambda x: f(x)


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

#three = successor(two)
def three(f):
    return lambda x: f(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
    >>> church_to_int(three)
    3
    """
    "*** YOUR CODE HERE ***"
    return n(increment)(0)



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 ***"
    return lambda f: lambda x: m(f)(n(f)(x))


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 ***"
    return lambda f : m(n(f))


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 ***"
    return n(m)