Homework 2: Higher Order Functions

hw02.zip (hw02.zip)

Required questions

Several doctests refer to these functions:

from operator import add, mul square = lambda x: x * x identity = lambda x: x triple = lambda x: 3 * x increment = lambda x: x + 1

Getting Started Videos

Higher Order Functions

Q1: Product

The summation(n, term) function from the higher-order functions lecture adds up term(1)

+ term(n) . Write a similar function called product that returns term(1) * * term(n) .

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

n: a positive integer

term: a function that takes one argument to produce the term

>>> product(3, identity) # 1 * 2 * 3

6

>>> product(5, identity) # 1 * 2 * 3 * 4 * 5

120

>>> product(3, square) # 1^2 * 2^2 * 3^2

36

>>> product(5, square) # 1^2 * 2^2 * 3^2 * 4^2 * 5^2

14400

>>> product(3, increment) # (1+1) * (2+1) * (3+1)

24

>>> product(3, triple) # 1*3 * 2*3 * 3*3

162

*** YOUR CODE HERE ***

Use Ok to test your code:

python3 ok -q product ✂

Q2: Accumulate

Lets take a look at how summation and product are instances of a more general function called accumulate , which we would like to implement:

accumulate(merger, base, n, term): Return the result of merging the first n terms in a sequence and base.

The terms to be merged are term(1), term(2), , term(n). merger is a two-argument commutative function.

>>> accumulate(add, 0, 5, identity) # 0 + 1 + 2 + 3 + 4 + 5

15

>>> accumulate(add, 11, 5, identity) # 11 + 1 + 2 + 3 + 4 + 5

26

>>> accumulate(add, 11, 0, identity) # 11

11

>>> accumulate(add, 11, 3, square) # 11 + 1^2 + 2^2 + 3^2

25

>>> accumulate(mul, 2, 3, square) # 2 * 1^2 * 2^2 * 3^2

72

>>> # 2 + (1^2 + 1) + (2^2 + 1) + (3^2 + 1)

>>> accumulate(lambda x, y: x + y + 1, 2, 3, square)

19

>>> # ((2 * 1^2 * 2) * 2^2 * 2) * 3^2 * 2

>>> accumulate(lambda x, y: 2 * x * y, 2, 3, square)

576

>>> accumulate(lambda x, y: (x + y) % 17, 19, 20, square)

16

*** YOUR CODE HERE *** accumulate has the following parameters:

term and n : the same parameters as in summation and product

merger : a two-argument function that speci es how the current term is merged with

the previously accumulated terms. base : value at which to start the accumulation.

For example, the result of accumulate(add, 11, 3, square) is

11 + square(1) + square(2) + square(3) = 25

Note: You may assume that merger is commutative. That is, merger(a, b) == merger(b, a) for all a , b , and c . However, you may not assume merger is chosen from a xed function set and hard-code the solution.

After implementing accumulate , show how summation and product can both be de ned as function calls to accumulate .

Important: You should have a single line of code (which should be a return statement) in each of your implementations for summation_using_accumulate and product_using_accumulate , which the syntax check will check for.

summation_using_accumulate(n, term): Returns the sum: term(1) + + term(n), using accumulate.

>>> summation_using_accumulate(5, square)

55

>>> summation_using_accumulate(5, triple)

45

*** YOUR CODE HERE ***

def product_using_accumulate(n, term):

Returns the product: term(1) * * term(n), using accumulate.

>>> product_using_accumulate(4, square)

576

>>> product_using_accumulate(6, triple)

524880

*** YOUR CODE HERE ***

Use Ok to test your code:

python3 ok -q accumulate python3 ok -q summation_using_accumulate

python3 ok -q product_using_accumulate ✂

The syntax check will run automatically when you submit the assignment, but you can also run the check directly by running the following command.

Use Ok to test your code:

python3 ok -q accumulate_syntax_check ✂

Submit

Make sure to submit this assignment by running:

python3 ok submit

Just for fun Question

This question is out of scope for 61A. You can try it if you want an extra challenge, but its just a puzzle that is not required or recommended at all. Almost all students will skip it, and thats ne.

If youre interested in learning more about this, feel free to attend the Extra Topics (https://cs61a.org/extra/) lectures.

Q3: 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 su cient 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 de nitions 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, de ne 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 ***

Use Ok to test your code:

python3 ok -q church_to_int python3 ok -q add_church python3 ok -q mul_church

python3 ok -q pow_church ✂