This is part of the book Category Theory for Programmers. The previous instalment was Category: The Essence of Composition. See the Table of Contents.

The category of types and functions plays an important role in programming, so let’s talk about what types are and why we need them.

Who Needs Types?

There seems to be some controversy about the advantages of static vs. dynamic and strong vs. weak typing. Let me illustrate these choices with a thought experiment. Imagine millions of monkeys at computer keyboards happily hitting random keys, producing programs, compiling, and running them.

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With machine language, any combination of bytes produced by monkeys would be accepted and run. But with higher level languages, we do appreciate the fact that a compiler is able to detect lexical and grammatical errors. Lots of monkeys will go without bananas, but the remaining programs will have a better chance of being useful. Type checking provides yet another barrier against nonsensical programs. Moreover, whereas in a dynamically typed language, type mismatches would be discovered at runtime, in strongly typed statically checked languages type mismatches are discovered at compile time, eliminating lots of incorrect programs before they have a chance to run.

So the question is, do we want to make monkeys happy, or do we want to produce correct programs?

The usual goal in the typing monkeys thought experiment is the production of the complete works of Shakespeare. Having a spell checker and a grammar checker in the loop would drastically increase the odds. The analog of a type checker would go even further by making sure that, once Romeo is declared a human being, he doesn’t sprout leaves or trap photons in his powerful gravitational field.

Types Are About Composability

Category theory is about composing arrows. But not any two arrows can be composed. The target object of one arrow must be the same as the source source object of the next arrow. In programming we pass the results on one function to another. The program will not work if the target function is not able to correctly interpret the data produced by the source function. The two ends must fit for the composition to work. The stronger the type system of the language, the better this match can be described and mechanically verified.

The only serious argument I hear against strong static type checking is that it might eliminate some programs that are semantically correct. In practice, this happens extremely rarely and, in any case, every language provides some kind of a backdoor to bypass the type system when that’s really necessary. Even Haskell has unsafeCoerce. But such devices should by used judiciously. Franz Kafka’s character, Gregor Samsa, breaks the type system when he metamorphoses into a giant bug, and we all know how it ends.

Another argument I hear a lot is that dealing with types imposes too much burden on the programmer. I could sympathize with this sentiment after having to write a few declarations of iterators in C++ myself, except that there is a technology called type inference that lets the compiler deduce most of the types from the context in which they are used. In C++, you can now declare a variable auto and let the compiler figure out its type.

In Haskell, except on rare occasions, type annotations are purely optional. Programmers tend to use them anyway, because they can tell a lot about the semantics of code, and they make compilation errors easier to understand. It’s a common practice in Haskell to start a project by designing the types. Later, type annotations drive the implementation and become compiler-enforced comments.

Strong static typing is often used as an excuse for not testing the code. You may sometimes hear Haskell programmers saying, “If it compiles, it must be correct.” Of course, there is no guarantee that a type-correct program is correct in the sense of producing the right ouput. The result of this cavalier attitude is that in several studies Haskell didn’t come as strongly ahead of the pack in code quality as one would expect. It seems that, in the commercial setting, the pressure to fix bugs is applied only up to a certain quality level, which has everything to do with the economics of software development and the tolerance of the end user, and very little to do with the programming language or methodology. A better criterion would be to measure how many projects fall behind schedule or are delivered with drastically reduced functionality.

As for the argument that unit testing can replace strong typing, consider the common refactoring practice in strongly typed languages: changing the type of an argument of a particular function. In a strongly typed language, it’s enough to modify the declaration of that function and then fix all the build breaks. In a weakly typed language, the fact that a function now expects different data cannot be propagated to call sites. Unit testing may catch some of the mismatches, but testing is almost always a probabilistic rather than a deterministic process. Testing is a poor substitute for proof.

What Are Types?

The simplest intuition for types is that they are sets of values. The type Bool (remember, concrete types start with a capital letter in Haskell) is a two-element set of True and False. Type Char is a set of all Unicode characters like 'a' or 'ą'.

Sets can be finite or infinite. The type of String, which is a synonym for a list of Char, is an example of an infinite set.

When we declare x to be an Integer:

x :: Integer

we are saying that it’s an element of the set of integers. Integer in Haskell is an infinite set, and it can be used to do arbitrary precision arithmetic. There is also a finite-set Int that corresponds to machine type, just like the C++ int.

There are some subtleties that make this identification of types and sets tricky. There are problems with polymorphic functions that involve circular definitions, and with the fact that you can’t have a set of all sets; but as I promised, I won’t be a stickler for math. The great thing is that there is a category of sets, which is called Set, and we’ll just work with it. In Set, objects are sets and morphisms (arrows) are functions.

Set is a very special category, because we can actually peek inside its objects and get a lot of intuitions from doing that. For instance, we know that an empty set has no elements. We know that there are special one-element sets. We know that functions map elements of one set to elements of another set. They can map two elements to one, but not one element to two. We know that an identity function maps each element of a set to itself, and so on. The plan is to gradually forget all this information and instead express all those notions in purely categorical terms, that is in terms of objects and arrows.

In the ideal world we would just say that Haskell types are sets and Haskell functions are mathematical functions between sets. There is just one little problem: A mathematical function does not execute any code — it just knows the answer. A Haskell function has to calculate the answer. It’s not a problem if the answer can be obtained in a finite number of steps — however big that number might be. But there are some calculations that involve recursion, and those might never terminate. We can’t just ban non-terminating functions from Haskell because distinguishing between terminating and non-terminating functions is undecidable — the famous halting problem. That’s why computer scientists came up with a brilliant idea, or a major hack, depending on your point of view, to extend every type by one more special value called the bottom and denoted by _|_, or Unicode ⊥. This “value” corresponds to a non-terminating computation. So a function declared as:

f :: Bool -> Bool

may return True, False, or _|_; the latter meaning that it would never terminate.

Interestingly, once you accept the bottom as part of the type system, it is convenient to treat every runtime error as a bottom, and even allow functions to return the bottom explicitly. The latter is usually done using the expression undefined, as in:

f :: Bool -> Bool
f x = undefined

This definition type checks because undefined evaluates to bottom, which is a member of any type, including Bool. You can even write:

f :: Bool -> Bool
f = undefined

(without the x) because the bottom is also a member of the type Bool->Bool.

Functions that may return bottom are called partial, as opposed to total functions, which return valid results for every possible argument.

Because of the bottom, you’ll see the category of Haskell types and functions referred to as Hask rather than Set. From the theoretical point of view, this is the source of never-ending complications, so at this point I will use my butcher’s knife and terminate this line of reasoning. From the pragmatic point of view, it’s okay to ignore non-terminating functions and bottoms, and treat Hask as bona fide Set (see Bibliography at the end).

Why Do We Need a Mathematical Model?

As a programmer you are intimately familiar with the syntax and grammar of your programming language. These aspects of the language are usually described using formal notation at the very beginning of the language spec. But the meaning, or semantics, of the language is much harder to describe; it takes many more pages, is rarely formal enough, and almost never complete. Hence the never ending discussions among language lawyers, and a whole cottage industry of books dedicated to the exegesis of the finer points of language standards.

There are formal tools for describing the semantics of a language but, because of their complexity, they are mostly used with simplified academic languages, not real-life programming behemoths. One such tool called operational semantics describes the mechanics of program execution. It defines a formalized idealized interpreter. The semantics of industrial languages, such as C++, is usually described using informal operational reasoning, often in terms of an “abstract machine.”

The problem is that it’s very hard to prove things about programs using operational semantics. To show a property of a program you essentially have to “run it” through the idealized interpreter.

It doesn’t matter that programmers never perform formal proofs of correctness. We always “think” that we write correct programs. Nobody sits at the keyboard saying, “Oh, I’ll just throw a few lines of code and see what happens.” We think that the code we write will perform certain actions that will produce desired results. We are usually quite surprised when it doesn’t. That means we do reason about programs we write, and we usually do it by running an interpreter in our heads. It’s just really hard to keep track of all the variables. Computers are good at running programs — humans are not! If we were, we wouldn’t need computers.

But there is an alternative. It’s called denotational semantics and it’s based on math. In denotational semantics every programing construct is given its mathematical interpretation. With that, if you want to prove a property of a program, you just prove a mathematical theorem. You might think that theorem proving is hard, but the fact is that we humans have been building up mathematical methods for thousands of years, so there is a wealth of accumulated knowledge to tap into. Also, as compared to the kind of theorems that professional mathematicians prove, the problems that we encounter in programming are usually quite simple, if not trivial.

Consider the definition of a factorial function in Haskell, which is a language quite amenable to denotational semantics:

fact n = product [1..n]

The expression [1..n] is a list of integers from 1 to n. The function product multiplies all elements of a list. That’s just like a definition of factorial taken from a math text. Compare this with C:

int fact(int n) {
    int i;
    int result = 1;
    for (i = 2; i <= n; ++i)
        result *= i;
    return result;
}

Need I say more?

Okay, I’ll be the first to admit that this was a cheap shot! A factorial function has an obvious mathematical denotation. An astute reader might ask: What’s the mathematical model for reading a character from the keyboard or sending a packet across the network? For the longest time that would have been an awkward question leading to a rather convoluted explanation. It seemed like denotational semantics wasn’t the best fit for a considerable number of important tasks that were essential for writing useful programs, and which could be easily tackled by operational semantics. The breakthrough came from category theory. Eugenio Moggi discovered that computational effect can be mapped to monads. This turned out to be an important observation that not only gave denotational semantics a new lease on life and made pure functional programs more usable, but also shed new light on traditional programming. I’ll talk about monads later, when we develop more categorical tools.

One of the important advantages of having a mathematical model for programming is that it’s possible to perform formal proofs of correctness of software. This might not seem so important when you’re writing consumer software, but there are areas of programming where the price of failure may be exorbitant, or where human life is at stake. But even when writing web applications for the health system, you may appreciate the thought that functions and algorithms from the Haskell standard library come with proofs of correctness.

Pure and Dirty Functions

The things we call functions in C++ or any other imperative language, are not the same things mathematicians call functions. A mathematical function is just a mapping of values to values.

We can implement a mathematical function in a programming language: Such a function, given an input value will calculate the output value. A function to produce a square of a number will probably multiply the input value by itself. It will do it every time it’s called, and it’s guaranteed to produce the same output every time it’s called with the same input. The square of a number doesn’t change with the phases of the Moon.

Also, calculating the square of a number should not have a side effect of dispensing a tasty treat for your dog. A “function” that does that cannot be easily modelled as a mathematical function.

In programming languages, functions that always produce the same result given the same input and have no side effects are called pure functions. In a pure functional language like Haskell all functions are pure. Because of that, it’s easier to give these languages denotational semantics and model them using category theory. As for other languages, it’s always possible to restrict yourself to a pure subset, or reason about side effects separately. Later we’ll see how monads let us model all kinds of effects using only pure functions. So we really don’t lose anything by restricting ourselves to mathematical functions.

Examples of Types

Once you realize that types are sets, you can think of some rather exotic types. For instance, what’s the type corresponding to an empty set? No, it’s not C++ void, although this type is called Void in Haskell. It’s a type that’s not inhabited by any values. You can define a function that takes Void, but you can never call it. To call it, you would have to provide a value of the type Void, and there just aren’t any. As for what this function can return, there are no restrictions whatsoever. It can return any type (although it never will, because it can’t be called). In other words it’s a function that’s polymorphic in the return type. Haskellers have a name for it:

absurd :: Void -> a

(Remember, a is a type variable that can stand for any type.) The name is not coincidental. There is deeper interpretation of types and functions in terms of logic called the Curry-Howard isomorphism. The type Void represents falsity, and the type of the function absurd corresponds to the statement that from falsity follows anything, as in the Latin adage “ex falso sequitur quodlibet.”

Next is the type that corresponds to a singleton set. It’s a type that has only one possible value. This value just “is.” You might not immediately recognise it as such, but that is the C++ void. Think of functions from and to this type. A function from void can always be called. If it’s a pure function, it will always return the same result. Here’s an example of such a function:

int f44() { return 44; }

You might think of this function as taking “nothing”, but as we’ve just seen, a function that takes “nothing” can never be called because there is no value representing “nothing.” So what does this function take? Conceptually, it takes a dummy value of which there is only one instance ever, so we don’t have to mention it explicitly. In Haskell, however, there is a symbol for this value: an empty pair of parentheses, (). So, by a funny coincidence (or is it a coincidence?), the call to a function of void looks the same in C++ and in Haskell. Also, because of the Haskell’s love of terseness, the same symbol () is used for the type, the constructor, and the only value corresponding to a singleton set. So here’s this function in Haskell:

f44 :: () -> Integer
f44 () = 44

The first line declares that f44 takes the type (), pronounced “unit,” into the type Integer. The second line defines f44 by pattern matching the only constructor for unit, namely (), and producing the number 44. You call this function by providing the unit value ():

f44 ()

Notice that every function of unit is equivalent to picking a single element from the target type (here, picking the Integer 44). In fact you could think of f44 as a different representation for the number 44. This is an example of how we can replace explicit mention of elements of a set by talking about functions (arrows) instead. Functions from unit to any type A are in one-to-one correspondence with the elements of that set A.

What about functions with the void return type, or, in Haskell, with the unit return type? In C++ such functions are used for side effects, but we know that these are not real functions in the mathematical sense of the word. A pure function that returns unit does nothing: it discards its argument.

Mathematically, a function from a set A to a singleton set maps every element of A to the single element of that singleton set. For every A there is exactly one such function. Here’s this function for Integer:

fInt :: Integer -> ()
fInt x = ()

You give it any integer, and it gives you back a unit. In the spirit of terseness, Haskell lets you use the wildcard pattern, the underscore, for an argument that is discarded. This way you don’t have to invent a name for it. So the above can be rewritten as:

fInt :: Integer -> ()
fInt _ = ()

Notice that the implementation of this function not only doesn’t depend on the value passed to it, but it doesn’t even depend on the type of the argument.

Functions that can be implemented with the same formula for any type are called parametrically polymorphic. You can implement a whole family of such functions with one equation using a type parameter instead of a concrete type. What should we call a polymorphic function from any type to unit type? Of course we’ll call it unit:

unit :: a -> ()
unit _ = ()

In C++ you would write this function as:

template<class T>
void unit(T) {}

Next in the typology of types is a two-element set. In C++ it’s called bool and in Haskell, predictably, Bool. The difference is that in C++ bool is a built-in type, whereas in Haskell it can be defined as follows:

data Bool = True | False

(The way to read this definition is that Bool is either True or False.) In principle, one should also be able to define a Boolean type in C++ as an enumeration:

enum bool {
    true,
    false
};

but C++ enum is secretly an integer. The C++11 “enum class” could have been used instead, but then you would have to qualify its values with the class name, as in bool::true and bool::false, not to mention having to include the appropriate header in every file that uses it.

Pure functions from Bool just pick two values from the target type, one corresponding to True and another to False.

Functions to Bool are called predicates. For instance, the Haskell library Data.Char is full of predicates like isAlpha or isDigit. In C++ there is a similar library that defines, among others, isalpha and isdigit, but these return an int rather than a Boolean. The actual predicates are defined in std::ctype and have the form ctype::is(alpha, c), ctype::is(digit, c), etc.

Challenges

  1. Define a higher-order function (or a function object) memoize in your favorite language. This function takes a pure function f as an argument and returns a function that behaves almost exactly like f, except that it only calls the original function once for every argument, stores the result internally, and subsequently returns this stored result every time it’s called with the same argument. You can tell the memoized function from the original by watching its performance. For instance, try to memoize a function that takes a long time to evaluate. You’ll have to wait for the result the first time you call it, but on subsequent calls, with the same argument, you should get the result immediately.
  2. Try to memoize a function from your standard library that you normally use to produce random numbers. Does it work?
  3. Most random number generators can be initialized with a seed. Implement a function that takes a seed, calls the random number generator with that seed, and returns the result. Memoize that function. Does it work?
  4. Which of these C++ functions are pure? Try to memoize them and observe what happens when you call them multiple times: memoized and not.
    1. The factorial function from the example in the text.
    2. std::getchar()
    3. bool f() { 
          std::cout << "Hello!" << std::endl;
          return true; 
      }
    4. int f(int x)
      {
          static int y = 0;
          y += x;
          return y;
      }
  5. How many different functions are there from Bool to Bool? Can you implement them all?
  6. Draw a picture of a category whose only objects are the types Void, () (unit), and Bool; with arrows corresponding to all possible functions between these types. Label the arrows with the names of the functions.

Bibliography

  1. Nils Anders Danielsson, John Hughes, Patrik Jansson, Jeremy Gibbons, Fast and Loose Reasoning is Morally Correct. This paper provides justification for ignoring bottoms in most contexts.

Next: Categories Great and Small