Discussion 2: Higher-Order Functions, Self Reference, Lambda Expressions
Higher Order Functions
A higher order function (HOF) is a function that manipulates other functions by taking in functions as arguments, returning a function, or both. For example, the function compose1
below takes in two functions as arguments and returns a function that is the composition of the two arguments.
def compose1(f, g):
def h(x):
return f(g(x))
return h
HOFs are powerful abstraction tools that allow us to express certain general patterns as named concepts in our programs.
A Note on Lambda Expressions
A lambda expression evaluates to a function, called a lambda function. For example, lambda y: x + y
is a lambda expression, and can be read as “a function that takes in one parameter y
and returns x + y
.”
A lambda expression by itself evaluates to a function but does not bind it to a name. Also note that the return expression of this function is not evaluated until the lambda is called. This is similar to how defining a new function using a def statement does not execute the function’s body until it is later called.
>>> what = lambda x : x + 5
>>> what
<function <lambda> at 0xf3f490>
Unlike def
statements, lambda expressions can be used as an operator or an operand to a call expression. This is because they are simply one-line expressions that evaluate to functions. In the example below, (lambda y: y + 5)
is the operator and 4
is the operand.
>>> (lambda y: y + 5)(4)
9
>>> (lambda f, x: f(x))(lambda y: y + 1, 10)
11
Currying
One important application of HOFs is converting a function that takes multiple arguments into a chain of functions that each take a single argument. This is known as currying. For example, the function below converts the pow
function into its curried form:
>>> def curried_pow(x):
def h(y):
return pow(x, y)
return h
>>> curried_pow(2)(3)
8
Q1: Keep Ints
Write a function that takes in a function cond
and a number n
and prints numbers from 1 to n
where calling cond
on that number returns True
.
Q2: (Tutorial) Make Keeper
Write a function similar to keep_ints
like in Question 1 #n
and returns a function that has one parameter cond
. The returned function prints out numbers from 1 to n
where calling cond
on that number returns True
.
HOFs in Environment Diagrams
Recall that an environment diagram keeps track of all the variables that have been defined and the values they are bound to. However, values are not necessarily only integers and strings. Environment diagrams can model more complex programs that utilize higher order functions.
Lambdas are represented similarly to functions in environment diagrams, but since they lack instrinsic names, the lambda symbol (λ) is used instead.
The parent of any function (including lambdas) is always the frame in which the function is defined. It is useful to include the parent in environment diagrams in order to find variables that are not defined in the current frame. In the previous example, when we call add_two
(which is really the lambda function), we need to know what x
is in order to compute x + y
. Since x
is not in the frame f2
, we look at the frame’s parent, which is f1
. There, we find x
is bound to 2.
As illustrated above, higher order functions that return a function have their return value represented with a pointer to the function object.
Q3: Curry2 Diagram
Draw the environment diagram that results from executing the code below.
def curry2(h):
def f(x):
def g(y):
return h(x, y)
return g
return f
make_adder = curry2(lambda x, y: x + y)
add_three = make_adder(3)
add_four = make_adder(4)
five = add_three(2)
Return value |
Return value |
Return value |
Return value |
Return value |
Q4: Curry2 Lambda
Write curry2
as a lambda function.
Q5: (Tutorial) HOF Diagram Practice
Draw the environment diagram that results from executing the code below.
n = 7
def f(x):
n = 8
return x + 1
def g(x):
n = 9
def h():
return x + 1
return h
def f(f, x):
return f(x + n)
f = f(g, n)
g = (lambda y: y())(f)
Return value |
Return value |
Return value |
Return value |
Q6: YY Diagram
The following question is more challenging than the previous ones. Nonetheless, it’s a fun problem to try.
Draw the environment diagram that results from executing the code below.
Tip: Using the
+
operator with two strings results in the second string being appended to the first. For example"C"
+"S"
concatenates the two strings into one string"CS"
.
y = "y"
h = y
def y(y):
h = "h"
if y == h:
return y + "i"
y = lambda y: y(h)
return lambda h: y(h)
y = y(y)(y)
Return value |
Return value |
Return value |
Return value |
Self Reference
Self-reference refers to a particular design of HOF, where a function eventually returns itself. In particular, a self-referencing function will not return a function call, but rather the function object itself. As an example, take a look at the print_all
function:
def print_all(x):
print(x)
return print_all
Self-referencing functions will oftentimes employ helper functions that reference the outer function, such as the example below, print_sums
.
def print_sums(n):
print(n)
def next_sum(k):
return print_sums(n + k)
return next_sum
A call to print_sums
returns next_sum
. A call to next_sum
will return the result of calling print_sums
which will, in turn, return another function next_sum
. This type of pattern is common in self-referencing functions.
A Note on Recursion: This differs from recursion because typically each new call returns a new function rather than a function call. We have not yet covered recursion so don’t worry too much about what this means!
Q7: (Tutorial) Warm Up: Make Keeper Redux
These exercises are meant to help refresh your memory of topics covered in lecture and/or lab this week before tackling more challenging problems.
In this question, we will explore the execution of a self-reference function, make_keeper_redux
, based off Question 2, make_keeper
. The function make_keeper_redux
is similar to make_keeper
, but now the returned function also returns another function with the same behavior. Feel free to paste and modify your code for make_keeper
below.
(Hint: you only need to add one line to your make_keeper
solution. What is currently missing from make_keeper_redux
?)
Q8: Print Delayed
Write a function print_delayed
that delays printing its argument until the next function call. print_delayed
takes in an argument x
and returns a new function delay_print
. When delay_print
is called, it prints out x
and returns another delay_print
.
Q9: (Tutorial) Print N
Write a function print_n
that can take in an integer n
and returns a repeatable print function that can print the next n
parameters. After the nth parameter, it just prints "done".