Rao Discussion 2: Environment Diagrams, Higher-Order Functions

• disc02.pdf

This discussion worksheet is for the Rao offering of CS 61A. Your work is not graded and you do not need to submit anything.

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Last week, you learned about the environment diagram, which is a visual model for keeping track of the state of a computer program. We have already gone over rules for variable assignment and def statements. This week, we will go over several more rules for constructing environment diagrams.

Scope

Scope refers to the idea that names have meaning only within a specific context. For example, the following program produces an error because the scope of the variable b is local—limited to the call to f(). Outside of that function call, b has no meaning, so the line print(b) errors.

def f():
    b = 2
f()
print(b)

On the other hand, some variables are defined with a broader scope. The following code prints 2. b is accessible within the call to f() because b is defined in the global scope, outside of any function.

b = 2
def f():
    print(b)
f()

Scope can also be relevant in disambiguating variables. In the following code, there are two variables named b: one is defined in the global scope and the other is defined in the local scope of the call to f(3). The line print(b) prints 3 instead of 2 because the local variable takes precedence over the global variable.

b = 2
def f(b):
    print(b)
f(3)

Call Expressions

In Python, local variables are created whenever a function is called. To formalize this idea of scope, we have rules for how call expressions, such as square(2), should be treated in environment diagrams. Specifically, 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 as it was named in its def statement. For example, if the function object is func square(x) [parent=Global], the intrinsic name is square.
   • The parent frame (e.g. [parent=Global]).
4. In the new frame, bind the formal parameters to the argument values obtained in step 2 (e.g. bind x to 2).
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 you never hit a return statement in your function, 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 because we would not be able to fill it out accurately.

When we want to look up the value of a variable, we follow these rules:

1. Search for the variable in the current frame.
2. If the variable is not bound in the current frame, look up to the parent frame. If the variable is not bound in that frame, look up to its parent frame, etc.
3. If you look up all the way to the global frame and still cannot find the variable, throw a NameError.

Q1: Call Diagram

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

def double(x):
    return x * 2

hmmm = double
wow = double(3)
hmmm(wow)

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)

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 function 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

Lambda functions do not have an intrinsic name, so we use the Greek letter lambda as a placeholder wherever we would usually use a function's intrinsic name.

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)

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

HOFs are powerful abstraction tools because they allow us to express certain general patterns (functions) as named concepts in our programs.

Environment diagrams can model more complex programs that utilize higher order functions.

def delete_num(x):
    """Returns a lambda function that takes in y and deletes x digits from y."""
    return lambda y: y // (10 ** x) # Note that ** means exponent (^) in Python

delete_two = delete_num(2)
delete_two(614)

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 delete_two (which is really the lambda function), we need to know what x is in order to compute y // (10 ** x). 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)

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 lambda)? Which frame is the parent of f2?
3. What value is the variable result bound to in the Global frame?

Q5: Make Keeper

Implement make_keeper, which takes a positive integer n and returns a function f that takes as its argument another one-argument function cond. When f is called on cond, it prints out the integers from 1 to n (including n) for which cond returns a true value when called on each of those integers. Each integer is printed on a separate line.

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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, here is a curried version of the pow function:

>>> def curried_pow(x):
...     def h(y):
...         return pow(x, y)
...     return h

>>> curried_pow(2)(3)
8
>>> pow(2, 3) == curried_pow(2)(3)
True

This is useful if, say, you need 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)
8
>>> pow(2, 4)
16
>>> pow(2, 10)
1024

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)
8
>>> pow_2(4)
16
>>> pow_2(10)
1024

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).

Q6: Currying

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

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First, try implementing curry with def statements. Then attempt to implement curry in a single line using lambda expressions.

HOFs and Lambdas

Q7: Make Your Own Lambdas

For the following problem, first read the doctests for functions f1, f2, f3, and f4. Then, implement the functions to conform to the doctests without causing any errors. 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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Extra Practice

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

Q8: 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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