61A Homework 2

Due by 11:59pm on Tuesday, 9/17.

Instructions. We have provided a hw2.py starter file for the questions below. See the online submission instructions.

Readings. Section 1.6 of the online text.

Several doctests refer to the square function, which we define as:

def square(x):
    """Return x squared."""
    return x * x

Q1. The summation function from lecture is only the simplest of a vast number of similar abstractions that can be captured as higher-order functions. Write a similar product function that returns the product of the values of a function for n natural number arguments. Show how to define the factorial function in terms of product:

def product(n, term):
    """Return the product of the first n terms in a sequence.

    term -- a function that takes one argument

    >>> product(4, square)
    "*** YOUR CODE HERE ***"

def factorial(n):
    """Return n factorial for n >= 0 by calling product.

    >>> factorial(4)
    "*** YOUR CODE HERE ***"

Q2. Show that both summation and product are instances of a more general function, called accumulate, with the following signature:

def accumulate(combiner, start, n, term):
    """Return the result of combining the first n terms in a sequence."""
    "*** YOUR CODE HERE ***"

Accumulate takes as arguments the same arguments term and n as summation and product, together with a combiner function (of two arguments) that specifies how the current term is to be combined with the accumulation of the preceding terms and a start value that specifies what base value to use to start the accumulation. Implement accumulate and show how summation and product can both be defined as simple calls to accumulate:

def summation_using_accumulate(n, term):
    """An implementation of summation using accumulate.

    >>> summation_using_accumulate(4, square)
    "*** YOUR CODE HERE ***"

def product_using_accumulate(n, term):
    """An implementation of product using accumulate.

    >>> product_using_accumulate(4, square)
    "*** YOUR CODE HERE ***"

Q3. Define a function double that takes a function of one argument as an argument and returns a function that applies the original function twice. For example, if successor is a function that returns 1 more than its argument, then double(inc) should be a function that returns two more:

def double(f):
    """Return a function that applies f twice.

    f -- a function that takes one argument

    >>> double(square)(2)
    "*** YOUR CODE HERE ***"

Q4. If f is a numerical function and n is a positive integer, then we can form the nth repeated application of f, which is defined to be the function whose value at x is f(f(...(f(x))...)). For example, if f adds 1 to its argument, then the nth repeated application of f adds n. Write a function that takes as inputs a function f and a positive integer n and returns the function that computes the nth repeated application of f:

def repeated(f, n):
    """Return the function that computes the nth application of f.

    f -- a function that takes one argument
    n -- a positive integer

    >>> repeated(square, 2)(5)
    >>> repeated(square, 4)(5)
    "*** YOUR CODE HERE ***"

Hint: You may find it convenient to use compose1 from the lecture notes:

def compose1(f, g):
    """Return a function h, such that h(x) = f(g(x))."""
    def h(x):
        return f(g(x))
    return h

Q5. (Optional extra for experts) The logician Alonzo Church invented a system of representing non-negative integers entirely using functions. Here are the definitions of 0, and a function that returns 1 more than its argument:

def zero(f):
    return lambda x: x

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

This representation is known as Church numerals. Define one and two directly, which are also functions. To get started, apply successor to zero. Then, give a direct definition of the add function (not in terms of repeated application of successor) over Church numerals. Finally, implement a function church_to_int that converts a church numeral argument to a regular Python int:

def one(f):
    """Church numeral 1."""
    "*** YOUR CODE HERE ***"

def two(f):
    """Church numeral 2."""
    "*** YOUR CODE HERE ***"

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

    >>> church_to_int(zero)
    >>> church_to_int(one)
    >>> church_to_int(two)
    "*** YOUR CODE HERE ***"

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))
    "*** YOUR CODE HERE ***"

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))
    >>> church_to_int(mul_church(three, four))
    "*** YOUR CODE HERE ***"

It is easy to find answers to this question on the Internet. Try to solve it on your own before looking it up!

Note: "Extra for Experts" problems are completely optional. You are encouraged to try these questions, but certainly don't be discouraged if you don't solve them.