Property-based testing

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Unit testing. We all do it. Some of us even practice TDD (although I wish I had a penny for each time I see a company claiming that they practice TDD when what they actually mean is “We write unit tests.”).

Test Driven Design means that your development is actually driven by your tests. It doesn’t mean that you write unit tests (although that is required), it doesn’t mean that you write your tests first (although that is also required), but it means that the desired behaviour is implemented incrementally, one test at a time. This normally means that, for each cycle of this method, a single test is written and then the simplest possible implementation is written that will make that test pass. After each test is added and passes (along with the rest of the test suite covering that particular piece of functionality) the developer is free to refactor the code–after all, up until this point the code has grown quite organically and is likely of poor quality. This repeated process is often referred to as Red-Green-Refactor: first the test is added and fails (the test is “red”), then the implementation is augmented until the test passes (the test “goes green”) and then the dev can refactor to improve code quality.

There are many, many different aspects to writing tests, each of which has a hand in how readable, maintainable, comprehensive, useful and correct your test suite is. These include (but are far from limited to): black-box / gray-box / white-box philosophy, mocking strategies, outside-in or inside-out…

But one aspect I don’t hear many people talking about (outside of the functional programming community at least) is how you pick your test cases. It might not seem that important–just throw some example data at the test and assert that for each of those inputs the expected output is produced. Simple, right?

A ha, dear reader. Allow me to throw a spanner in the works.

This doesn’t always work (at least, not well). Let’s go for the most basic of examples:

How would you unit test the + operator?

Examples? 1 + 0? 2 + 2? 423798 + 278? Where do you stop? How would a developer, following TDD by the book, incrementally implement that function?

When designing tests it’s often helpful to pretend that the person writing the tests and the person completing the implementation of the function are different people. Let us play the part of the coder who has been tasked with implementing the “add” function as above. The signature of the function is agreed and the tester throws their first test case at us: 0 + 0. So, we dutifully write the most simple case that will solve that:

let add x y = 0

The test passes. Yay! Now, here comes the second test case: 1, 0 which should equal 1. So:

let add x y =
    if x = 1 then 1 else 0

Okay, that’s fine. It’s early days yet! So. 2, 2 = 4?

let add x y = 
    if x = 1 then 1
    elif x = 2 then 4
    else 0

Hmmm, it doesn’t look like we’re being driven towards a real implementation here. Maybe let’s try one more test case?

let add x y = 
    if x = 1 then 1
    elif x = 2 then 4
    elif x = 423798 && y = 278 then 424076
    else 0

No, this isn’t working. We’re solving each case as they come in, but we’re not getting anywhere: the if/else statement is just going to grow and grow and grow. We’re stuck in a loop of: tester adds an example, implementation solves that one particular example.

So, how do you choose your test examples? Wouldn’t it be better if you could randomly generate inputs to your test? It would certainly stop the implementation from devolving into a series of hard-coded values, but then what would you assert? This is something that has fascinated me for a while now.

Allow me to introduce Property-Based Testing.

In this context the word property is used in the mathematical way; a property means some quality, attribute, feature or characteristic that a function, or its input and output, has. In short, property-based testing doesn’t assert what the output is, it instead asserts the the output has some property / characteristic. Sometimes the property you are checking is dependent on the input having some property as well (this may or may not be the same property as you expect to see in the output). An example of this might be:

For any two positive numbers x and y, x + y must be positive

You can see here that we are now able to write an assertion that is true for all inputs to the program. For this particular property–that the output of the function is positive–we have placed a constraint on the input (x and y must be positive). We are now able to automatically generate the input data for our tests because we don’t need to know what the data is in advance.

To see how this might work, let’s write the above property in a more formal language:

For all x, y
where x > 0
and y > 0
it follows that: x + y > 0

It almost looks as if we are now following a Gherkin-style BDD syntax for our tests! Okay, so now we understand what a property is we need to come up with a way of discovering more properties.

Continuing with the mathematical example of addition, your function might have the concept of *identity*. From https://en.wikipedia.org/wiki/Function_(mathematics)#Identity_function, “The unique function over a set X that maps each element to itself is called the identity function for X”, meaning “a function who’s output is always equal to its input”. For addition this is simply “+ 0”, and is a property that we can check.

For all x
it follows that x + 0 = x

There are more mathematical properties that would be useful to check, such as commutativity:

For all x, y:
it follows that x + y = y + x

Associativity:

For all x, y, z
it follows that (x + y) + z = x + (y + z)

Given these properties, it’s now much harder to write an implementation of (+) that is incorrect yet passes all our tests.

This is all very good in theory, but how does this actually work? How do we actually implement this? In part 2 of this series we’ll go through a worked example with something a bit harder than addition: the Diamond Kata.

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