Introduction to Software Testing

Before we get to the central parts of the book, let us introduce essential concepts of software testing. Why is it necessary to test software at all? How does one test software? How can one tell whether a test has been successful? How does one know if one has tested enough? In this chapter, let us recall the most important concepts, and at the same time get acquainted with Python and interactive notebooks.

Simple Testing

Let's start with a simple example. Your co-worker has been asked to implement a square root function $\sqrt{x}$. (Let's assume for a moment that the environment does not already have one.) After studying the Newton–Raphson method, she comes up with the following Python code, claiming that, in fact, this my_sqrt() function computes square roots.

In [1]:
def my_sqrt(x):
    """Computes the square root of x, using the Newton–Raphson method"""
    approx = None
    guess = x / 2
    while approx != guess:
        approx = guess
        guess = (approx + x / approx) / 2
    return approx

Your job is now to find out whether this function actually does what it claims to do.

Running a Function

To find out whether my_sqrt() works correctly, we can test it with a few values. For x = 4, for instance, it produces the correct value:

In [2]:

The upper part above my_sqrt(4) (a so-called cell) is an input to the Python interpreter, which by default evaluates it. The lower part (2.0) is its output. We can see that my_sqrt(4) produces the correct value.

The same holds for x = 2.0, apparently, too:

In [3]:

Interacting with Notebooks

If you are reading this in the interactive notebook, you can try out my_sqrt() with other values as well. Click on one of the above cells with invocations of my_sqrt() and change the value – say, to my_sqrt(1). Press Shift+Enter (or click on the play symbol) to execute it and see the result. If you get an error message, go to the above cell with the definition of my_sqrt() and execute this first. You can also run all cells at once; see the Notebook menu for details. (You can actually also change the text by clicking on it, and corect mistaks such as in this sentence.)

Executing a single cell does not execute other cells, so if your cell builds on a definition in another cell that you have not executed yet, you will get an error. You can select Run all cells above from the menu to ensure all definitions are set.

Also keep in mind that, unless overwritten, all definitions are kept across executions. Occasionally, it thus helps to restart the kernel (i.e. start the Python interpreter from scratch) to get rid of older, superfluous definitions.

Debugging a Function

To see how my_sqrt() operates, a simple strategy is to insert print() statements in critical places. You can, for instance, log the value of approx, to see how each loop iteration gets closer to the actual value:

In [4]:
def my_sqrt_with_log(x):
    """Computes the square root of x, using the Newton–Raphson method"""
    approx = None
    guess = x / 2
    while approx != guess:
        print("approx =", approx)  # <-- New
        approx = guess
        guess = (approx + x / approx) / 2
    return approx
In [5]:
approx = None
approx = 4.5
approx = 3.25
approx = 3.0096153846153846
approx = 3.000015360039322
approx = 3.0000000000393214

Interactive notebooks also allow to launch an interactive debugger – insert a "magic line" %%debug at the top of a cell and see what happens. Unfortunately, interactive debuggers interfere with our dynamic analysis techniques, so we mostly use logging and assertions for debugging.

Checking a Function

Let's get back to testing. We can read and run the code, but are the above values of my_sqrt(2) actually correct? We can easily verify by exploiting that $\sqrt{x}$ squared again has to be $x$, or in other words $\sqrt{x} \times \sqrt{x} = x$. Let's take a look:

In [6]:
my_sqrt(2) * my_sqrt(2)

Okay, we do have some rounding error, but otherwise, this seems just fine.

What we have done now is that we have tested the above program: We have executed it on a given input and checked its result whether it is correct or not. Such a test is the bare minimum of quality assurance before a program goes into production.

Automating Test Execution

So far, we have tested the above program manually, that is, running it by hand and checking its results by hand. This is a very flexible way of testing, but in the long run, it is rather inefficient:

  1. Manually, you can only check a very limited number of executions and their results
  2. After any change to the program, you have to repeat the testing process

This is why it is very useful to automate tests. One simple way of doing so is to let the computer first do the computation, and then have it check the results.

For instance, this piece of code automatically tests whether $\sqrt{4} = 2$ holds:

In [7]:
result = my_sqrt(4)
expected_result = 2.0
if result == expected_result:
    print("Test passed")
    print("Test failed")
Test passed

The nice thing about this test is that we can run it again and again, thus ensuring that at least the square root of 4 is computed correctly. But there are still a number of issues, though:

  1. We need five lines of code for a single test
  2. We do not care for rounding errors
  3. We only check a single input (and a single result)

Let us address these issues one by one. First, let's make the test a bit more compact. Almost all programming languages do have a means to automatically check whether a condition holds, and stop execution if it does not. This is called an assertion, and it is immensely useful for testing.

In Python, the assert statement takes a condition, and if the condition is true, nothing happens. (If everything works as it should, you should not be bothered.) If the condition evaluates to false, though, assert raises an exception, indicating that a test just failed.

In our example, we can use assert to easily check whether my_sqrt() yields the expected result as above:

In [8]:
assert my_sqrt(4) == 2

As you execute this line of code, nothing happens: We just have shown (or asserted) that our implementation indeed produces $\sqrt{4} = 2$.

Remember, though, that floating-point computations may induce rounding errors. So we cannot simply compare two floating-point values with equality; rather, we would ensure that the absolute difference between them stays below a certain threshold value, typically denoted as $\epsilon$ or epsilon. This is how we can do it:

In [9]:
EPSILON = 1e-8
In [10]:
assert abs(my_sqrt(4) - 2) < EPSILON

We can also introduce a special function for this purpose, and now do more tests for concrete values:

In [11]:
def assertEquals(x, y, epsilon=1e-8):
    assert abs(x - y) < epsilon
In [12]:
assertEquals(my_sqrt(4), 2)
assertEquals(my_sqrt(9), 3)
assertEquals(my_sqrt(100), 10)

Seems to work, right? If we know the expected results of a computation, we can use such assertions again and again to ensure our program works correctly.

Generating Tests

Remember that the property $\sqrt{x} \times \sqrt{x} = x$ universally holds? We can also explicitly test this with a few values:

In [13]:
assertEquals(my_sqrt(2) * my_sqrt(2), 2)
assertEquals(my_sqrt(3) * my_sqrt(3), 3)
assertEquals(my_sqrt(42.11) * my_sqrt(42.11), 42.11)

Still seems to work, right? Most importantly, though, $\sqrt{x} \times \sqrt{x} = x$ is something we can very easily test for thousands of values:

In [14]:
for n in range(1, 1000):
    assertEquals(my_sqrt(n) * my_sqrt(n), n)

How much time does it take to test my_sqrt() with 100 values? Let's see.

We use our own Timer module to measure elapsed time. To be able to use Timer, we first import our own utility module, which allows us to import other notebooks.

In [16]:
from Timer import Timer
In [17]:
with Timer() as t:
    for n in range(1, 10000):
        assertEquals(my_sqrt(n) * my_sqrt(n), n)

10,000 values take about a hundredth of a second, so a single execution of my_sqrt() takes 1/1000000 second, or about 1 microseconds.

Let's repeat this with 10,000 values picked at random. The Python random.random() function returns a random value between 0.0 and 1.0:

In [18]:
import random
In [19]:
with Timer() as t:
    for i in range(10000):
        x = 1 + random.random() * 1000000
        assertEquals(my_sqrt(x) * my_sqrt(x), x)

Within a second, we have now tested 10,000 random values, and each time, the square root was actually computed correctly. We can repeat this test with every single change to my_sqrt(), each time reinforcing our confidence that my_sqrt() works as it should. Note, though, that while a random function is unbiased in producing random values, it is unlikely to generate special values that drastically alter program behavior. We will discuss this later below.

Run-Time Verification

Instead of writing and running tests for my_sqrt(), we can also go and integrate the check right into the implementation. This way, each and every invocation of my_sqrt() will be automatically checked.

Such an automatic run-time check is very easy to implement:

In [20]:
def my_sqrt_checked(x):
    root = my_sqrt(x)
    assertEquals(root * root, x)
    return root

Now, whenever we compute a root with my_sqrt_checked()$\dots$

In [21]:

we already know that the result is correct, and will so for every new successful computation.

Automatic run-time checks, as above, assume two things, though:

  • One has to be able to formulate such run-time checks. Having concrete values to check against should always be possible, but formulating desired properties in an abstract fashion can be very complex. In practice, you need to decide which properties are most crucial, and design appropriate checks for them. Plus, run-time checks may depend not only on local properties, but on several properties of the program state, which all have to be identified.

  • One has to be able to afford such run-time checks. In the case of my_sqrt(), the check is not very expensive; but if we have to check, say, a large data structure even after a simple operation, the cost of the check may soon be prohibitive. In practice, run-time checks will typically be disabled during production, trading reliability for efficiency. On the other hand, a comprehensive suite of run-time checks is a great way to find errors and quickly debug them; you need to decide how many such capabilities you would still want during production.

System Input vs Function Input

At this point, we may make my_sqrt() available to other programmers, who may then embed it in their code. At some point, it will have to process input that comes from third parties, i.e. is not under control by the programmer.

Let us simulate this system input by assuming a program sqrt_program() whose input is a string under third-party control:

In [22]:
def sqrt_program(arg):
    x = int(arg)
    print('The root of', x, 'is', my_sqrt(x))

We assume that sqrt_program is a program which accepts system input from the command line, as in

$ sqrt_program 4

We can easily invoke sqrt_program() with some system input:

In [23]:
The root of 4 is 2.0

What's the problem? Well, the problem is that we do not check external inputs for validity. Try invoking sqrt_program(-1), for instance. What happens?

Indeed, if you invoke my_sqrt() with a negative number, it enters an infinite loop. For technical reasons, we cannot have infinite loops in this chapter (unless we'd want the code to run forever); so we use a special with ExpectTimeOut(1) construct to interrupt execution after one second.

In [24]:
from ExpectError import ExpectTimeout
In [25]:
with ExpectTimeout(1):
Traceback (most recent call last):
  File "<ipython-input-25-add01711282b>", line 2, in <module>
  File "<ipython-input-22-53e8ec8bb3ca>", line 3, in sqrt_program
    print('The root of', x, 'is', my_sqrt(x))
  File "<ipython-input-1-f91e5d79a1e7>", line 6, in my_sqrt
    approx = guess
  File "<ipython-input-1-f91e5d79a1e7>", line 6, in my_sqrt
    approx = guess
  File "<string>", line 16, in check_time
TimeoutError (expected)

The above message is an error message, indicating that something went wrong. It lists the call stack of functions and lines that were active at the time of the error. The line at the very bottom is the line last executed; the lines above represent function invocations – in our case, up to my_sqrt(x).

We don't want our code terminating with an exception. Consequently, when accepting external input, we must ensure that it is properly validated. We may write, for instance:

In [26]:
def sqrt_program(arg):
    x = int(arg)
    if x < 0:
        print("Illegal Input")
        print('The root of', x, 'is', my_sqrt(x))

and then we can be sure that my_sqrt() is only invoked according to its specification.

In [27]:
Illegal Input

But wait! What happens if sqrt_program() is not invoked with a number? Then we would try to convert a non-number string, which would also result in a runtime error:

In [28]:
from ExpectError import ExpectError
In [29]:
with ExpectError():
Traceback (most recent call last):
  File "<ipython-input-29-8c5aae65a938>", line 2, in <module>
  File "<ipython-input-26-ea86281b33cf>", line 2, in sqrt_program
    x = int(arg)
ValueError: invalid literal for int() with base 10: 'xyzzy' (expected)

Here's a version which also checks for bad inputs:

In [30]:
def sqrt_program(arg):
        x = float(arg)
    except ValueError:
        print("Illegal Input")
        if x < 0:
            print("Illegal Number")
            print('The root of', x, 'is', my_sqrt(x))
In [31]:
The root of 4.0 is 2.0
In [32]:
Illegal Number
In [33]:
Illegal Input

We have now seen that at the system level, the program must be able to handle any kind of input gracefully without ever entering an uncontrolled state. This, of course, is a burden for programmers, who must struggle to make their programs robust for all circumstances. This burden, however, becomes a benefit when generating software tests: If a program can handle any kind of input (possibly with well-defined error messages), we can also send it any kind of input. When calling a function with generated values, though, we have to know its precise preconditions.

The Limits of Testing

Despite best efforts in testing, keep in mind that you are always checking functionality for a finite set of inputs. Thus, there may always be untested inputs for which the function may still fail.

In the case of my_sqrt(), for instance, computing $\sqrt{0}$ results in a division by zero:

In [34]:
with ExpectError():
    root = my_sqrt(0)
Traceback (most recent call last):
  File "<ipython-input-34-24ede1f53910>", line 2, in <module>
    root = my_sqrt(0)
  File "<ipython-input-1-f91e5d79a1e7>", line 7, in my_sqrt
    guess = (approx + x / approx) / 2
ZeroDivisionError: float division by zero (expected)

In our tests so far, we have not checked this condition, meaning that a program which builds on $\sqrt{0} = 0$ will surprisingly fail. But even if we had set up our random generator to produce inputs in the range of 0–1000000 rather than 1–1000000, the chances of it producing a zero value by chance would still have been one in a million. If the behavior of a function is radically different for few individual values, plain random testing has few chances to produce these.

We can, of course, fix the function accordingly, documenting the accepted values for x and handling the special case x = 0:

In [35]:
def my_sqrt_fixed(x):
    assert 0 <= x
    if x == 0:
        return 0
    return my_sqrt(x)

With this, we can now correctly compute $\sqrt{0} = 0$:

In [36]:
assert my_sqrt_fixed(0) == 0

Illegal values now result in an exception:

In [37]:
with ExpectError():
    root = my_sqrt_fixed(-1)
Traceback (most recent call last):
  File "<ipython-input-37-55b1caf1586a>", line 2, in <module>
    root = my_sqrt_fixed(-1)
  File "<ipython-input-35-f3e21e80ddfb>", line 2, in my_sqrt_fixed
    assert 0 <= x
AssertionError (expected)

Still, we have to remember that while extensive testing may give us a high confidence into the correctness of a program, it does not provide a guarantee that all future executions will be correct. Even run-time verification, which checks every result, can only guarantee that if it produces a result, the result will be correct; but there is no guarantee that future executions may not lead to a failing check. As I am writing this, I believe that my_sqrt_fixed(x) is a correct implementation of $\sqrt{x}$, but I cannot be 100% certain.

With the Newton-Raphson method, we may still have a good chance of actually proving that the implementation is correct: The implementation is simple, the math is well-understood. Alas, this is only the case for few domains. If we do not want to go into full-fledged correctness proofs, our best chance with testing is to

  1. Test the program on several, well-chosen inputs; and
  2. Check results extensively and automatically.

This is what we do in the remainder of this course: Devise techniques that help us to thoroughly test a program, as well as techniques that help us checking its state for correctness. Enjoy!

Lessons Learned

  • The aim of testing is to execute a program such that we find bugs.
  • Test execution, test generation, and checking test results can be automated.
  • Testing is incomplete; it provides no 100% guarantee that the code is free of errors.


There is a large number of works on software testing and analysis. For this book, we are happy to recommend "Software Testing and Analysis" \cite{Pezze2008} as an introduction to the field; its strong technical focus very well fits our methodology. Other important must-reads with a comprehensive approach to software testing, including psychology and organization, include "The Art of Software Testing" \cite{Myers2004} as well as "Software Testing Techniques" \cite{Beizer1990}.


Exercise 1: Testing Shellsort

Consider the following implementation of a Shellsort function, taking a list of elements and (presumably) sorting it.

In [38]:
def shellsort(elems):
    sorted_elems = elems.copy()
    gaps = [701, 301, 132, 57, 23, 10, 4, 1]
    for gap in gaps:
        for i in range(gap, len(sorted_elems)):
            temp = sorted_elems[i]
            j = i
            while j >= gap and sorted_elems[j - gap] > temp:
                sorted_elems[j] = sorted_elems[j - gap]
                j -= gap
            sorted_elems[j] = temp

    return sorted_elems

A first test indicates that shellsort() might actually work:

In [39]:
shellsort([3, 2, 1])
[1, 2, 3]

The implementation uses a list as argument elems (which it copies into sorted_elems) as well as for the fixed list gaps. Lists work like arrays in other languages:

In [40]:
a = [5, 6, 99, 7]
print("First element:", a[0], "length:", len(a))
First element: 5 length: 4

The range() function returns an iterable list of elements. It is often used in conjunction with for loops, as in the above implementation.

In [41]:
for x in range(1, 5):

Part 1: Manual Test Cases

Your job is now to thoroughly test shellsort() with a variety of inputs.

First, set up assert statements with a number of manually written test cases. Select your test cases such that extreme cases are covered. Use == to compare two lists.

Part 2: Random Inputs

Second, create random lists as arguments to shellsort(). Make use of the following helper predicates to check whether the result is (a) sorted, and (b) a permutation of the original.

In [45]:
def is_sorted(elems):
    return all(elems[i] <= elems[i + 1] for i in range(len(elems) - 1))
In [46]:
is_sorted([3, 5, 9])
In [47]:
def is_permutation(a, b):
    return len(a) == len(b) and all(a.count(elem) == b.count(elem) for elem in a)
In [48]:
is_permutation([3, 2, 1], [1, 3, 2])

Start with a random list generator, using [] as the empty list and elems.append(x) to append an element x to the list elems. Use the above helper functions to assess the results. Generate and test 1,000 lists.

Exercise 2: Quadratic Solver

Given an equation $ax^2 + bx + c = 0$, we want to find solutions for $x$ given the values of $a$, $b$, and $c$. The following code is supposed to do this, using the equation $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In [55]:
def quadratic_solver(a, b, c):
    q = b * b - 4 * a * c
    solution_1 = (-b + my_sqrt_fixed(q)) / (2 * a)
    solution_2 = (-b - my_sqrt_fixed(q)) / (2 * a)
    return (solution_1, solution_2)
In [56]:
quadratic_solver(3, 4, 1)
(-0.3333333333333333, -1.0)

The above implementation is incomplete, though. You can trigger

  1. a division by zero; and
  2. violate the precondition of my_sqrt_fixed().

How does one do that, and how can one prevent this?

Part 1: Find bug-triggering inputs

For each of the two cases above, identify values for a, b, c that trigger the bug.

Part 2: Fix the problem

Extend the code appropriately such that the cases are handled. Return None for nonexistent values.

Part 3: Odds and Ends

What are the chances of discovering these conditions with random inputs? Assuming one can do a billion tests per second, how long would one have to wait on average until a bug gets triggered?