# Lab 2 Solutions

# Introduction

In the last lab, you learned some basic expressions and wrote some python code. In this lab, we will introduce lists and explore higher order functions (HOFs).

# Lists

In Data 8, you have recently started working with `Tables`

. Tables are an extremely useful and powerful data type. In CS88 we will work with other data types. Python provides
several important built-in data types that we can build from. So far, you have met numberical data types (ints, floats, and booleans) and one sequence type (strings). Lists, tuples, and dictionaries are other sequence data types in Python. Here, we will take a closer look at lists. **A list can contain a sequence of values of any type.**

You can create a list just by placing the values, separated by commas, within square brackets. Here are some examples. As you will see in one of the examples, lists can contain other lists.

```
>>> [1,2,3]
[1, 2, 3]
>>> ["frog", 3, 3.1415]
['frog', 3, 3.1415]
>>> [True, [1, 2], 42]
[True, [1, 2], 42]
```

Open up your python interpreter and create some lists of your own.

You learned last week that what really makes a data type useful is the operations that you can
perform on it. What can you do with lists?

```
>>> x = [1,2,3] # assign them to variables
>>> len(x) # get their length, i.e., the number of elements in them
3
>>> x + [4,5] # + is concatenation
[1, 2, 3, 4, 5]
>>> [1,2] * 3 # * is replication
[1, 2, 1, 2, 1, 2]
>>> len([1,2] * 3)
6
>>> [1,2] * [3,4] # what's this?
TypeError: can't multiply sequence by non-int of type 'list'
```

The `in`

operator is very useful when working with lists. It operates on the entire list and produces a boolean that answers the question, "Is this item in the list?".

```
>>> 2 in [1,2,3]
True
>>> "frog" in [1,2,3]
False
>>> [1,2] in [1,2,3]
False
>>> [1,2] in [[1,2],3]
True
```

### Question 1: Second Max

Write a function that finds the second highest number in a list of positive integers. You can assume that the list always has at least two integers.

```
def second_max(lst):
"""
Return the second highest number in a list of positive integers.
>>> second_max([3, 2, 1, 0])
2
>>> second_max([2, 3, 3, 4, 5, 6, 7, 2, 3])
6
>>> second_max([1, 5, 5, 5, 1])
5
>>> second_max([5, 6, 6, 7, 1])
6
>>> second_max([5, 6, 7, 7, 1])
7
"""
"*** YOUR CODE HERE ***"
highest = 0
second_highest = 0
for num in lst:
if num >= highest:
second_highest = highest
highest = num
elif num < highest and num > second_highest:
second_highest = num
return second_highest
```

Use OK to test your code:

`python3 ok -q second_max`

## List Comprehensions

Now that we can create lists, assign variables, write expressions, and
define functions, we can compose these concepts to do lots of interesting
things. Python's `list comprehensions`

open a beautiful world of data-centric
programming.

The comprehension is in brackets, just like a list, but rather than a static
sequence of literals, it is a dynamically computed list.

```
>>> somelist = [1, 2, 9, -1, 0]
>>> [x+1 for x in somelist]
[2, 3, 10, 0, 1]
>>> [x*x for x in somelist]
[1, 4, 81, 1, 0]
```

In general, the expression just inside the `[`

is evaluated for each element
in the list, using the variable between the `for`

and the `in`

to name each
element in succession. The result is the transformed list.

```
>>> def square(x):
... return x*x
...
>>> def squares(s):
... return [square(x) for x in s]
...
>>> squares([0,1,2,4])
[0, 1, 4, 16]
>>>x, y = 2, 3
>>> x+y
5
>>> [x+y for x,y in [[1,2], [2,3], [3,4]]
[3, 5, 7]
```

This is a powerful design pattern, called `map`

, that you will use in often
in analyzing data. It *maps*, or transforms, one data structure into another
under some expression, often by applying a function to each of the elements.

Do you remember the Table.apply( ) function from Data 8? The Table.apply function is another great example of the `map`

design pattern as it applies a "transformation" or a function to a row or column.

Sometimes you need a sequence to get started, and Python provides handy tools
for that. One of them is `range`

.

```
>>> [x*x for x in range(10)]
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81]
```

You can review `range`

in Section 2.3
of Composing Programs.

### Question 2: Perfect squares

Implement the function `squares`

, which takes in a list of positive
integers, and returns a new list which contains only elements of the original
list that are perfect squares. Use a list comprehension.

```
from math import sqrt
def is_square(n):
return float(sqrt(n)) == int(sqrt(n))
def squares(seq):
"""Returns a new list containing elements of the original list that are
perfect squares.
>>> seq = [49, 8, 2, 1, 102]
>>> squares(seq)
[49, 1]
>>> seq = [500, 30]
>>> squares(seq)
[]
"""
"*** YOUR CODE HERE ***"
return [n for n in seq if is_square(n)]
```

Use OK to test your code:

`python3 ok -q squares`

### Question 3: Perfect Pairs

Implement the function `pairs`

, which takes in an integer n,
and returns a new list of lists which contains pairs of numbers from 1 to n.
Use a list comprehension.

```
def pairs(n):
"""Returns a new list containing two element lists from values 1 to n
>>> pairs(1)
[[1, 1]]
>>> x = pairs(2)
>>> x
[[1, 1], [2, 2]]
>>> pairs(5)
[[1, 1], [2, 2], [3, 3], [4, 4], [5, 5]]
>>> pairs(-1)
[]
"""
"*** YOUR CODE HERE ***"
return [[i, i] for i in range(1, n + 1)]
```

Use OK to test your code:

`python3 ok -q pairs`

# Functions as Arguments (Funargs)

So far we have used several types of data - ints, floats, booleans,
strings, lists, tuples, and numpy.arrays. We perform operations on them by
constructing expressions; we assign them to variables; we pass them to functions
and return them as results. So what about functions themselves? So far we have
*called them*, that is we applied them to arguments. Sometimes we *compose* them -
just like in math; apply a function to the result of applying a function. You
did that several times above.

In modern programming languages like Python, functions are *first class citizens*;
we can pass them around and put them in data structures. Take a look at the following
and try it out for various functions that you have available in the `.py`

file for this
lab.

```
>>> square(square(3))
81
>>> square
<function square at 0x102033d90>
>>> x = square
>>> x(3)
9
>>> x(x(2))
16
>>>
```

## Introduction to 'Map'

Higher order functions fit into a domain of programming known as "functional" or "functional form" programming, centered around this idea of passing and returning functions as parameters and arguments. In class, you learned the command `map`

that is a fundamental example of higher order functions.

Let's take a closer look at how `map`

works. At its core, `map`

applies a function to all items in an input list. It takes in a function as the first parameter and a series of inputs as the second parameter.

`map(function_to_apply, list_of_inputs)`

A potentially easier way to think about `map`

is to draw an equivalent with a list comprehension! Given the `func`

(function to apply) and `inputs`

(list of inputs), a map is similar to this:

`[func(x) for x in inputs]`

Keep in mind that the `map`

function actually returns a `map`

object, not a list. We need to convert this object to a `list`

by passing it into the `list()`

function.

Let's do a Python Tutor example to understand how map works.

Open Python Tutor in a new tab.

This code should already be there:

```
INCR = 2
def inc(x):
return x+INCR
def mymap(fun, seq):
return [fun(x) for x in seq]
result = mymap(inc, [5, 6, 7])
print(result)
```

So what's happening here? In the first 3 lines, we're defining a function `inc`

which increments an input `x`

by a certain amount, `INCR`

.

Notice that `INCR`

is defined once in the Global frame. This is a nice review of how Python resolves references when there are both local and global variables. When the `inc`

method executes, python needs to find the value `INCR`

. Since the `INCR`

variable isn't declared locally, within the `inc`

function, Python will look at the parent frame, the frame in which `inc`

was declared, for the value of `INCR`

. In this case, since the `inc`

function was declared in the Global frame, the global `INC`

variable value will be used.

The second function, `mymap`

, is an example of how map works in the form of a list comprehension! Notice that `mymap`

takes in a function as its first argument and a sequence as its second. Just like `map`

, this list comprehension runs each element of `seq`

through the `fun`

method.

As you run through the program in Python Tutor, notice how the list comprehension in `mymap`

will repeatedly call the `inc`

function. The functional anatomy of how `map`

works is exactly encapsulated by the `mymap`

function.

### Question 4: Converter

Given a list of temperatures in Celsius format, convert each temperature value in the list from Celsius to Fahrenheit.A couple tips:

- Make sure to use the
`map`

keyword for this solution! - The temperature converter function will be passed in as a method, so there is no need for you to write it again!

If you're feeling stuck, think about the parameters of `map`

. This is meant to be a simple problem that provides hands-on experience of understanding what `map`

does.

```
def converter(temperatures, convert):
"""Returns a sequence that converts each Celsius temperature to Fahrenheit
>>> def convert(x):
... return 9.0*x/5.0 + 32
>>> temperatures = [10, 20, 30, 40, 50]
>>> converter(temperatures, convert)
[50.0, 68.0, 86.0, 104.0, 122.0]
"""
"*** YOUR CODE HERE ***"
return list(map(convert, temperatures))
```

Use OK to test your code:

`python3 ok -q converter`

## Introduction to 'Filter'

The `filter`

keyword is similar in nature to `map`

with a very important distinction. In `map`

, the function we pass in is being applied to every item in our sequence. In `filter`

, the function we pass in *filters* the elements, only leaving the elements for which the function returns true. For example, if I wanted to remove all negative numbers from a list, I could use the `filter`

function to identify values that are greater than or equal to 0, and filter out the rest.

```
def isPositive(number):
return number >= 0
numbers = [-1, 1, -2, 2, -3, 3, -4, 4]
positive_nums = list(filter(isPositive, numbers))
```

Again, similar to `map`

, the output of the `filter`

function is a `filter`

object, not a list, so you need to call `list()`

. The equivalent for filter in the form of a list comprehension would look something along the lines of this:

`positive_nums = [number for number in numbers if isPositive(number)]`

## Introduction to 'Reduce'

`Reduce`

takes in three different parameters: A function, a sequence, and an identity. The function and sequence are the same parameters that we saw in `map`

and `filter`

. The identity can be thought of as the container where you are going to store all of your results. In the above case, the identity would be the `product`

variable.

`Reduce`

is very useful for performing computations on lists that involve every element in the list. Computations are performed in a rolling fashion, where the function acts upon each element one at a time.

Let's say I wanted to calculate the product of the square roots of a list of numbers. The non-`reduce`

version of this code would look something along the lines of this:

```
product = 1
numbers = [4, 9, 16, 25, 36]
for num in numbers:
product = product * num**.5
```

Here's the `reduce`

version

```
multiplicative_identity = 1
nums = [4, 9, 16, 25, 36]
def sqrtProd(x, y):
return x * y ** .5
reduce(sqrtProd, nums, multiplicative_identity)
```

### Question 5: reduce

Write the higher order function `reduce`

which takes

- reducer - a two-argument function that reduces elements to a single value
- s - a sequence of values
- base - the starting value in the reduction. This is usually the identity of the reducer

If you're feeling stuck, think about the parameters of `reduce`

. This is meant to be a simple problem that provides hands-on experience of understanding what `reduce`

does.

```
from operator import add, mul
def reduce(reducer, s, base):
"""Reduce a sequence under a two-argument function starting from a base value.
>>> def add(x, y):
... return x + y
>>> def mul(x, y):
... return x*y
>>> reduce(add, [1,2,3,4], 0)
10
>>> reduce(mul, [1,2,3,4], 0)
0
>>> reduce(mul, [1,2,3,4], 1)
24
"""
"*** YOUR CODE HERE ***"
for x in s:
base = reducer(base, x)
return base
```

Use OK to test your code:

`python3 ok -q reduce`

## Higher Order Functions

Thus far, in Python Tutor, we’ve visualized Python programs in the form of environment diagrams that display which variables are tied to which values within different frames. However, as we noted when introducing Python, values are not necessarily just primitive expressions or types like float, string, integer, and boolean.

In a nutshell, a higher order function is any function that takes a function as a parameter or provides a function has a return value. We will be exploring many applications of higher order functions.

Let's think about a more practical use of higher order functions. Pretend you’re a math teacher, and you want to teach your students how coefficients affect the shape of a parabola.

Open Python Tutor in a new tab

Paste this code into the interpreter:

```
def define_parabola(a, b, c):
def parabola(x):
return a*(x**2) + b*x + c
return parabola
parabola = define_parabola(-2, 3, -4)
y1 = parabola(1)
y2 = parabola(10)
print(y1, y2)
```

Now step through the code. In the `define_parabola`

function, the coefficient values of 'a', 'b', and 'c' are taken in, and in return, a parabolic function with those coefficient values is returned.

As you step through the second half of the code, notice how the value of `parabola`

points at a function object! The `define_parabola`

higher order nature comes from the fact that its return value is a function.

Another thing noting is where the pointer moves after the `parabola`

function is called. Notice that the pointer goes to line 2, where `parabola`

was originally defined. In a nutshell, this example is meant to show how a closure is returned from the `define_parabola`

function.

### Question 6: Piecewise

Implement `piecewise`

, which takes two one-argument functions, `f`

and `g`

,
along with a number `b`

. It returns a new function that takes a number `x`

and
returns either `f(x)`

if `x`

is less than `b`

, or `g(x)`

if `x`

is greater than
or equal to `b`

.

```
def piecewise(f, g, b):
"""Returns the piecewise function h where:
h(x) = f(x) if x < b,
g(x) otherwise
>>> def negate(x):
... return -x
>>> def identity(x):
... return x
>>> abs_value = piecewise(negate, identity, 0)
>>> abs_value(6)
6
>>> abs_value(-1)
1
"""
"*** YOUR CODE HERE ***"
def h(x):
if x < b:
return f(x)
return g(x)
return h
```

Use OK to test your code:

`python3 ok -q piecewise`

### Question 7: Intersect

Two functions intersect at an argument `x`

if they return equal values.
Implement `intersects`

, which takes a one-argument functions `f`

and a value
`x`

. It returns a function that takes another function `g`

and returns whether
`f`

and `g`

intersect at `x`

.

```
def intersects(f, x):
"""Returns a function that returns whether f intersects g at x.
>>> def square(x):
... return x * x
>>> def triple(x):
... return x * 3
>>> def increment(x):
... return x + 1
>>> def identity(x):
... return x
>>> at_three = intersects(square, 3)
>>> at_three(triple) # triple(3) == square(3)
True
>>> at_three(increment)
False
>>> at_one = intersects(identity, 1)
>>> at_one(square)
True
>>> at_one(triple)
False
"""
"*** YOUR CODE HERE ***"
def at_x(g):
return f(x) == g(x)
return at_x
```

Use OK to test your code:

`python3 ok -q intersects`

## Submit

Make sure to submit this assignment by running:

`python3 ok --submit`

# Extra Credit Practice Open in a new window

These questions are new this semester. They're a mix of Parsons Problems, Code Tracing questions, and Code Writing questions.

Confused about how to use the tool? Check out https://codestyle.herokuapp.com/cs88-lab01 for some problems designed to demonstrate how to solve these types of problems.

These cover some similar material to lab, so can be helpful to further review or try to learn the material. Unlike lab and homework, after you've worked for long enough and tested your code enough times on any of these questions, you'll have the option to view an instructor solution. You'll unlock each question one at a time, either by correctly answering the previous question or by viewing an instructor solution.

**Starting from lab 2 onward, each set of questions are worth half (0.5) a point per lab, for a total opportunity of 4-5 points throughout the semester.**

Use OK to test your code:

`python3 ok -q extra_credit`

# More FUN with FUNctions (Optional)

The questions in `lab02_extra.py`

are optional problems for you to solve if you want more practice with functions. Be warned—question 9 is quite a challenge!

### Question 8: Applying Function Arguments

```
def square(x):
return x * x
def twice(f,x):
"""Apply f to the result of applying f to x"
>>> twice(square,3)
81
"""
"*** YOUR CODE HERE ***"
return f(f(x))
```

Use OK to test your code:

`python3 ok -q twice`

```
def increment(x):
return x + 1
def apply_n(f, x, n):
"""Apply function f to x n times.
>>> apply_n(increment, 2, 10)
12
"""
"*** YOUR CODE HERE ***"
res = x
for i in range(n):
res = f(res)
return res
```

Use OK to test your code:

`python3 ok -q apply_n`

### Question 9: 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*. Here are the definitions of `zero`

, as well as a function that
returns one more than its argument:

```
def zero(f):
def _zero(x):
return x
return _zero
def successor(n):
def _succ(f):
def t(x):
return f(n(f)(x))
return t
return _succ
```

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*.

```
def one(f):
"""Church numeral 1: same as successor(zero)"""
"*** YOUR CODE HERE ***"
def _one(x):
return f(x)
return _one
def two(f):
"""Church numeral 2: same as successor(successor(zero))"""
"*** YOUR CODE HERE ***"
def _two(x):
return f(f(x))
return _two
```

Next, implement a function `church_to_int`

that converts a church numeral
argument to a regular Python integer.

```
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(lambda x: x + 1)(0)
```

Use OK to test your code:

`python3 ok -q church_to_int`

Finally, implement functions `add_church`

, `mul_church`

, and `pow_church`

that
perform addition, multiplication, and exponentiation on church numerals.

```
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)
```

Use OK to test your code:

```
python3 ok -q add_church
python3 ok -q mul_church
python3 ok -q pow_church
```

Church numerals are a way to represent non-negative integers via
repeated function application. The definitions of `zero`

, `one`

, and
`two`

show that each numeral is a function that takes a function and
repeats it a number of times on some argument `x`

.

The `church_to_int`

function reveals how a Church numeral can be mapped
to our normal notion of non-negative integers using the increment
function.

Addition of Church numerals is function composition of the functions of
`x`

, while multiplication (added to the question for these solutions)
is composition of the functions of `f`

.