Study Guide: Mutation


This is a study guide with links to past lectures, assignments, and handouts, as well as additional practice problems to assist you in learning the concepts.


Important: For solutions to these assignments once they have been released, see the main website





Mutable Values

Mutation is the act of changing a value's attributes. Previously, we've seen immutable values like numbers and strings. Every time we work with numbers and strings, Python creates an entirely new number or string to represent the result.

n = 0
def no_effect(n):
    n += 1

Invoking no_effect(n) doesn't affect the value of n in the global frame.

In constrast, mutable values can change existing attributes. Changes to mutable values are globally-visible: any name which references that value will see the same changes. The list l can be mutated which will be seen in the global frame.

l = [0]
def effective(l):

Mutation is dangerous! It's not necessarily clear just from the function call effective(l) that our own l will be changed. When designing systems, we often prefer to write in a more functional style that makes it clear when the value of a name might change.

l = [0]
def add_one(l):
    return l + [1]
l = add_one(l)

The re-assignment to l makes it clear that the value of l might change, whereas, in the previous example, we would need to be a lot more careful with remembering whether or not effective mutates the values we pass to it.

What further complicates the situation is that not all operations are mutative! Some operations on lists, like append, will mutate the original list. But many other operations, like the + operator, will create a new list.


Lists are a type of mutable sequence. Because lists are also compound values, we represent them in environment diagrams with a pointer just like we do with function values.

>>> a = [1, 2, 3]
>>> b = a
>>> b.append(4)
>>> a
[1, 2, 3, 4]
>>> b
[1, 2, 3, 4]

In this example, both a and b refer to the same underlying list so changes to the list are visible to both names.


Two separate lists that have the same contents are equal but not necessarily identical. It could be the case that we happen to have two copies of a list with the exact same elements.

>>> a = [1, 2, 3, 4]
>>> b = [1, 2, 3]
>>> b.append(4)
>>> b
[1, 2, 3, 4]
>>> a == b
>>> a is b


The following is a list of common expressions which mutate the original lst. While nowhere near complete, it covers the majority of the cases you should be familiar with. Come up with examples in Python Tutor to learn how each expression affects the list.

  • Index and slice replacement

    • lst[i] = elem
    • lst[i:j] = seq
  • Element insertion

    • lst.append(elem)
    • lst.insert(i, elem)
  • Sequence insertion

    • lst.extend(seq)
    • lst += lst but not lst = lst + lst
  • Element removal

    • lst.pop() and lst.pop(i)
    • lst.remove(elem)

And a few expressions which create new lists.

  • lst + lst
  • lst * n
  • lst[i:j]
  • list(lst)

Mutable Functions

The nonlocal keyword tells Python to modify the binding in the nearest non-global parent frame rather than the current local frame. More formally, our environment diagram rules change slightly.

Assignment Statements

  1. Evaluate the expression to the right of the assignment operator =.
  2. If nonlocal, find the nearest parent frame which contains the name but not including the global frame. If not nonlocal, use the current local frame. (Note that it is a syntax error if the nonlocal keyword is used and the name doesn't exist in a non-global parent frame.)
  3. Bind the variable name to the value of the expression in the frame. Be sure you override the variable name if it had a previous binding.

Local State

While nonlocal only slightly modifies Python's execution rules, it has a big effect on how we reason about programs. Before nonlocal and mutable values, we've only written pure functions, or functions that, when called, have no effects other than returning a value. The most useful consequence of writing pure functions is that they are referentially transparency. In other words, they can be entirely replaced by the value that they return, without any change in the behavior of the program.

def add(x, y):
    return x + y

add(1, add(2, 3)) == add(1, 5) == 6

Referential transparency is important because of how we've approached concepts in this course. We can trust that a recursive call will return the right result because of how we can imagine replacing the recursive call with its return value.

With nonlocal, we can no longer rely upon this behavior. The same call to a function at different times in a program can produce different results.

def cumulative_adder(x):
    def add(y):
        nonlocal x
        x += y
        return x
    return add

add = cumulative_adder(1)

We can now start to view functions as objects that can change over time, which allows for greater control, but makes our programs harder to understand.

Isomorphic Quiz Questions

Q1: Nonlocal Environment Diagram

Draw the environment diagram that results from running the following code.

def moon(f):
    sun = 0
    moon = [sun]
    def run(x):
        nonlocal sun, moon
        def sun(sun):
            return [sun]
        y = f(x)
        return moon[0] and moon[1]
    return run

moon(lambda x: moon)(1)

After you've done it on your own, generate an environment diagram in python tutor to check your answer.

Q2: Digits

Draw the environment diagram that results from executing the following code.

def three(x):
    three = [x]*3
    nine = [three]*3
    def what(x):
        nonlocal what, three, nine
        three[x] += 1
        three = [[x]]*3
        nine[x] = three[x]
        def what(b):
            if b:
                return sum(nine[0])
            return sum(nine[2])
        return x + what(nine[x] is three[x-1])
    return what


Practice Problems


Q3: Funny Adder

Predict what Python will display when the following lines are typed into the interpreter:

>>> def make_funny_adder(n):
...     def adder(x):
...         if x == 'new':
...             nonlocal n
...             n = n + 1
...         else:
...             return x + n
...     return adder
>>> h = make_funny_adder(3)
>>> h(5)
>>> j = make_funny_adder(7) >>> j(5)
>>> h('new') >>> h(5)

Q4: List Accumulator

Although we can't change variables outside of our frame without a nonlocal statement, we can update values stored in mutatable objects in parent frames. Define a function make_accumulator that returns an accumulator function, which takes one numerical argument and returns the sum of all arguments ever passed to accumulator. Use a list and not a nonlocal statement:

def make_accumulator():
    """Return an accumulator function that takes a single numeric argument and
    accumulates that argument into total, then returns total.

    >>> acc = make_accumulator()
    >>> acc(15)
    >>> acc(10)
    >>> acc2 = make_accumulator()
    >>> acc2(7)
    >>> acc3 = acc2
    >>> acc3(6)
    >>> acc2(5)
    >>> acc(4)
"*** YOUR CODE HERE ***"
total = [0] def accumulator(amount): total[0] += amount return total[0] return accumulator

Q5: Nonlocal Accumulator

Now, define a function make_accumulator_nonlocal that returns an accumulator function, which takes one numerical argument and returns the sum of all arguments ever passed to accumulator. Use a nonlocal statement.

def make_accumulator_nonlocal():
    """Return an accumulator function that takes a single numeric argument and
    accumulates that argument into total, then returns total.

    >>> acc = make_accumulator_nonlocal()
    >>> acc(15)
    >>> acc(10)
    >>> acc2 = make_accumulator_nonlocal()
    >>> acc2(7)
    >>> acc3 = acc2
    >>> acc3(6)
    >>> acc2(5)
    >>> acc(4)
"*** YOUR CODE HERE ***"
total = 0 def accumulator(amount): nonlocal total total += amount return total return accumulator


Q6: Dice

Recall make_test_dice from the Hog project. make_test_dice takes in a sequence of numbers and returns a zero-argument function. This zero-argument function will cycle through the list, returning one element from the list every time. Implement make_test_dice.

def make_test_dice(seq):
    """Makes deterministic dice.

    >>> dice = make_test_dice([2, 6, 1])
    >>> dice()
    >>> dice()
    >>> dice()
    >>> dice()
    >>> other = make_test_dice([1])
    >>> other()
    >>> dice()
"*** YOUR CODE HERE ***"
count = 0 def dice(): nonlocal count result = seq[count] count = (count + 1) % len(seq) return result return dice

Q7: Mutants

Draw environment diagrams to determine what Python would display.

>>> wolf = [1, 2, 3]
>>> def dog(lst):
...     def animal(ele):
...         ele = [ele] + lst
...         return [ele] + [beast[0]]
...     beast = [2, 3, animal]
...     return beast
>>> x = dog(wolf)[2](4)
>>> x
[[4, 1, 2, 3], 2]
>>> x = 18
>>> def it(i):
...     i = x
...     def shifty(getting):
...         nonlocal i
...         i = getting + x
...         def shiftier(y):
...             nonlocal getting
...             gettting = y*i
...             return i
...         return shiftier
...     return shifty
>>> shift = it('is')(x)(4)
>>> shift
>>> def piper(chapman):
...     chapman.append('state')
...     def alex(vause):
...         nonlocal chapman
...         chapman[1] = vause[1]
...         return chapman
...     return alex
>>> orange = piper(['litchfield', 'new york'])(['federal', 'prison'])
>>> orange
['litchfield', 'prison', 'state']