Discussion 2: Environment Diagrams, Higher-Order Functions

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Call Expressions

Call expressions, such as square(2), apply functions to arguments. When executing call expressions, we create a new frame in our diagram to keep track of local variables:

  1. Evaluate the operator, which should evaluate to a function.
  2. Evaluate the operands from left to right.
  3. Draw a new frame, labelling it with the following:

    • A unique index (f1, f2, f3, ...).
    • The intrinsic name of the function, which is the name of the function object itself. For example, if the function object is func square(x) [parent=Global], the intrinsic name is square.
    • The parent frame ([parent=Global]).
  4. Bind the formal parameters to the argument values obtained in step 2 (e.g. bind x to 3).
  5. Evaluate the body of the function in this new frame until a return value is obtained. Write down the return value in the frame.

If a function does not have a return value, it implicitly returns None. In that case, the “Return value” box should contain None.

Note: Since we do not know how built-in functions like min(...) or imported functions like add(...) are implemented, we do not draw a new frame when we call them, since we would not be able to fill it out accurately.

Q1: Call Diagram

Let’s put it all together! Draw an environment diagram for the following code. You may not have to use all of the blanks provided to you.

def double(x):
    return x * 2

hmmm = double
wow = double(3)
Global frame
f1: [parent=]
Return value
f2: [parent=]
Return value

Q2: Nested Calls Diagrams

Draw the environment diagram that results from executing the code below. You may not need to use all of the frames and blanks provided to you.

def f(x):
    return x

def g(x, y):
    if x(y):
        return not y
    return y

x = 3
x = g(f, x)
f = g(f, 0)
Global frame
f1: [parent=]
Return value
f2: [parent=]
Return value
f3: [parent=]
Return value
f4: [parent=]
Return value

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)
>>> (lambda f, x: f(x))(lambda y: y + 1, 10)

Q3: Lambda the Environment Diagram

Draw the environment diagram for the following code and predict what Python will output.

a = lambda x: x * 2 + 1
def b(b, x):
    return b(x + a(x))
x = 3
x = b(a, x)
Global frame
f1: [parent=]
Return value
f2: [parent=]
Return value
f3: [parent=]
Return value

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 compose below takes in two functions as arguments and returns a function that is the composition of the two arguments.

def composer(func1, func2):
    """Return a function f, such that f(x) = func1(func2(x))."""
    def f(x):
        return func1(func2(x))
    return f

HOFs are powerful abstraction tools that allow us to express certain general patterns as named concepts in our programs.

HOFs in Environment Diagrams

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.

Q4: Make Adder

Draw the environment diagram for the following code:

n = 9
def make_adder(n):
    return lambda k: k + n
add_ten = make_adder(n+1)
result = add_ten(n)
Global frame
f1: [parent=]
Return value
f2: [parent=]
Return value

There are 3 frames total (including the Global frame). In addition, consider the following questions:

  1. In the Global frame, the name add_ten points to a function object. What is the intrinsic name of that function object, and what frame is its parent?
  2. What name is frame f2 labeled with (add_ten or λ)? Which frame is the parent of f2?
  3. What value is the variable result bound to in the Global frame?

Q5: Make Keeper

Write a function that takes in a number n and returns a function that can take in a single parameter cond. When we pass in some condition function cond into this returned function, it will print out numbers from 1 to n where calling cond on that number returns True.

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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)

This is useful if, say, you needed to calculate a lot of powers of 2. Using the normal pow function, you would have to put in 2 as the first argument for every function call:

>>> pow(2, 3)
>>> pow(2, 4)
>>> pow(2, 10)

With curried_pow, however, you can create a one-argument function specialized for taking powers of 2 one time, and then keep using that function for taking powers of 2:

>>> pow_2 = curried_pow(2)
>>> pow_2(3)
>>> pow_2(4)
>>> pow_2(10)

This way, you don't have to put 2 in as an argument for every call! If instead you wanted to take powers of 3, you could quickly make a similar function specialized in taking powers of 3 using curried_pow(3).

Another point that will be relevant once you learn about sequences: Currying is also helpful in contexts where only one-argument functions are allowed, such as with the map function. The map applies a one-argument function to every term in a sequence. If we wanted to take the power of 2 for every number in a sequence using map, we would be forced to use the one-argument function pow_2 (as defined in the example above) instead of using pow with 2 as the first argument.

Q6: Currying

Write a function curry that will curry any two argument function.

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HOFs and Lambdas

Q7: Make Your Own Lambdas

For each of the following expressions, write functions f1, f2, f3, and f4 such that the evaluation of each expression succeeds, without causing an error. Be sure to use lambdas in your function definition instead of nested def statements. Each function should have a one line solution.

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Q8: Lambdas and Currying

Write a function lambda_curry2 that will curry any two argument function like with curry, but this time using lambdas.

Your solution to this problem should only be one line.

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Extra Practice

This question is particularly challenging, so it is recommended to attempt if you are feeling confident on the previous questions or are studying for the exam.

Q9: Match Maker

Implement match_k, which takes in an integer k and returns a function that takes in a variable x and returns True if all the digits in x that are k apart are the same.

For example, match_k(2) returns a one argument function that takes in x and checks if digits that are 2 away in x are the same.

match_k(2)(1010) has the value of x = 1010 and digits 1, 0, 1, 0 going from left to right. 1 == 1 and 0 == 0, so the match_k(2)(1010) results in True.

match_k(2)(2010) has the value of x = 2010 and digits 2, 0, 1, 0 going from left to right. 2 != 1 and 0 == 0, so the match_k(2)(2010) results in False.

Important: You may not use strings or indexing for this problem. You do not have to use all the lines, one staff solution does not use the line directly above the while loop.

Hint: Floor dividing by powers of 10 gets rid of the rightmost digits.

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