# Pure Functions and Total functions

Tags: functional programming.

If you ask a programmer “do you use functions?” he’ll probably give you a funny look. “Of course I use functions!” he’ll reply. “I use a function to tell me what time it is; I use a function to save data to a database; I use a function to tell me the position of the mouse cursor; etc. I use functions all the time!”

If a mathematician happened to be within earshot during this conversation, she’ll probably give him a funny look. “None of those things are functions!” she’ll point out. “A function is a mapping from a set of inputs to a set of outputs. For example, $f(x) = x^2 + 2x$.”

# Pure Functions

Because I’m a fan of descriptivism, I’m not interested in trying to argue that one of them is “right” and the other person is “wrong”. Instead, I’ll point out that because of this ambiguity, we humans have come up with the term “pure function” (and “impure function”) to disambiguate which function we mean. A “pure” function is a function in the mathematician’s sense: a mapping from a set of inputs to a set of outputs.

The output of a pure function is determined completely by its inputs: If you pass the value 3 to a pure function and it returns 15, then by virtue of the fact that that function is pure, you are guaranteed that any time you invoke the function with 3 again in the future, it will always return 15.

This is in contrast to a function whose output may depend on the current time. Or a function which reads from a global variable as part of how it determines its output. Passing in the same inputs might yield a different output for two different invocations.

## Side Effects

A pure function by definition does not have any “side effects”. So not only are we guaranteed that a pure function will always return the same output given the same input, but we’re also guaranteed that the function will “do nothing” except return its output.

This is in contrast to a function which writes something to disk, or which sets a global variable, or throws an exception, or anything else beyond returning a value that is in some way observable to the caller.

It is always safe to replace a given invocation of a pure function with the result of that invocation. For example, given the code snippet x = f(1) + f(1), if you know that f is a pure function, then it is safe to refactor this code to x = 2 * f(1). On the other hand, if f was impure, then this refactoring might have introduced a bug. For example, perhaps f incremented a counter, and the original implementation would increment the counter twice (since f was invoked twice), whereas the refactored implementation would only increment the counter once.

(Incidentally, a bit of jargon for you: we refer to the concept “it is always safe to replace a given invocation of a pure function with the result of that invocation” as “Referential Transparency”).

## Pure Functions are Easy to Test

One nice property of pure functions is that they are much easier to test than impure functions. A pure function simply returns some result given an input and is guaranteed to do nothing else but return that result. Therefore, to test a pure function, you simply need to provide it an input, and then check that the returned result is what you expected it to be.

assertThat(Math.sin(Math.pi * 0.0), is(0));
assertThat(Math.sin(Math.pi * 0.5), is(1));
assertThat(Math.sin(Math.pi * 1.0), is(0));
assertThat(Math.sin(Math.pi * 1.5), is(-1));
assertThat(Math.sin(Math.pi * 2.0), is(0));

If you were testing a function for which its output depended on something other than the parameters you pass to it, then you’d first have to know what those hidden dependencies were (which would possibly lead to your tests becoming coupled to the implementation of the function instead of its contract), and then you’d need to see if you can intercept and control those hidden dependencies somehow.

If you were testing a function which produced a side effect, you often don’t want the side effect to actually happen during testing. For example, let’s say you have a function deleteUserFromDB that you wanted to test. You want to check that the function is indeed capable of deleting users, but you definitely don’t want your tests to connect to the real production database. What if there is a bug in the implementation and it deletes the wrong user? I mean, you’re testing this function because you’re not sure that it’s bug free, right?

In practice, this usually means creating one or more mock objects, but this assumes that the function under test provides some sort of hook so that you can inject in your mocks. If the function is hardcoded to connected to a database, interjecting your mocks may be difficult or impossible, depending on what programming language you’re using and what facilities it provides.

If you’re testing a function which does not necessarily return the same value given the same inputs, and you check that it’s output is correct for a given invocation, how can you be sure that it’ll be right on the next invocation? Maybe that’s too complicated to check, so you’ll just be satisfied if you can verify that you get the results you want on the first invocation of the function. But then how can you make sure your test is the first one to invoke the function? What if somebody adds a test to the test suite that runs before your test, and happens to invoke the function you want to test? Does your test runner even guarantee a deterministic run-order for your tests?

## Pure Functions are Easy to Parallelize

Speaking of non-determinism, the behavior of a pure function is determined entirely by its inputs. So in particular, the behavior of a pure function does not depend on the time or timing at which it is executed, or what thread it is executed upon.

This makes them extremely easy to parallelize; in fact, the compiler can automatically parallelize any sequence of pure functions without any guidance from the programmer, without the need to perform any locking or any other form of synchronization. Race conditions and other multithreading bugs are only possible in the presence of side effects, which pure functions are guaranteed to not have.

# Total Functions

A related term is “Total Function”. A function is “total” if it is defined for all of its possible inputs. As a counter example, consider the function tail that takes a list as its input and returns a new list which is equal to the original list, except with its first element removed. tail is not a total function (it is a “partial function”), because it is not defined for the empty list.

Note that “total” and “pure” are orthogonal concepts: A function can be both, neither, or any combination in between. For example:

• not(x: boolean): boolean, which returns the negation of its input, is both pure and total.
• f(x: int): int = { callCount++; return 100 / x; } is neither pure nor total. It is not defined for the input 0, and it performs a side effect.
• f(x: int): int = { callCount++; return x; } is total but not pure.

Things get tricky when we want to see a non-total function that happens to also be pure…

## Pure partial functions

If a partial function expresses its non-totality via a mechanism such as throwing an exceptions, then the caller can usually observe the fact that an exception was thrown (e.g. via a catch statement). In such a case, we would say that the partial function is impure, because it triggered an observable side-effect.

On the other hand, if a partial function expressed its non-totality by not returning (e.g. by going into an infinite loop), then the caller would never have the opportunity to observe any side effect (because control never returns to the caller). Hence the function is pure.

Notice that if you’re working in a programming language which allows you to throw exceptions, but provides no facility to catch exceptions, then from the caller’s perspective, this is the same as the infinite-loop example: The caller has no opportunity to observe any side effect, and thus the non-total function is also pure.

Another variant on this idea is if a programming language provides two types of throwables; for example, an “error” can be thrown and caught, but a “fatal” can only be thrown but never caught. In such a language, a partial function that threw “errors” (or a mix of “errors” and “fatals” depending on some business logic) would be impure, but a partial function that threw only “fatals” could be pure.

Originally published on Aug 5, 2015