Types as Propositions

Some of the most meaningful mathematical realizations that I’ve had have been unexpected connections between two topics; that is, realizing that two concepts that first appeared quite distinct are in fact one and the same. In our first linear algebra courses, we learn that manipulations of matrices is, in fact, equivalent to solving systems of equations. In quantum mechanics, we see that physically observable quantities are, mathematically speaking, linear operators (I still don’t quite grok this one). And, my personal favorite example, we learn in functional analysis that the linear functionals in the dual space of a Hilbert space are themselves in perfect correspondence with the functions in the original space.1

Recently, I’ve stumbled upon another such result, which has captured my attention for a while. The result, often referred to as Curry-Howard correspondence, is the statement that propositions in a formal logical system are equivalent to types in the simply typed lambda calculus. Loosely, this means that logical statements are equivalent to data types!

Let’s unpack that a bit; “propositions” are just statements in a logical system.2 In mathematics, for example, one might put forward the proposition “no even numbers are prime,” or “14 is greater than 18”. Note that propositions need not be true; in fact, some logical systems support propositions that cannot even be determined to be true or false.3 “Types” can be though of as types in a computing language; Integer, Boolean, and so on. We will have much more to say about types as we move forward, but for now, hold in your mind the conventional notion of types as defined in a language such as Java or Python (or better yet, Haskell).

How on earth could these two be in correspondence? On the surface, they appear entirely separate concepts. In this post, I’ll spend some time unpacking what this equivalence is actually saying, using a simple example. I am far from a full understanding of it, but as usual, I write about it in the hopes that I’ll be forced to clarify what I do understand, or even better, be corrected by someone more knowledgable than myself.

Speaking of those more knowledgable than myself, there are various resources online that I found very helpful in understanding the correspondence: Philip Wadler’s talk on the subject is a great starting point, and there are a number of useful discussions available on StackExchange and various functional programming forums.

An Example

I was confused by the idea of propositions as types when I first encountered it, and after learning more, I believe that the root of my confusion lies in the fact that types such as Integer, Boolean, and String, which we are familiar with from programming, correspond to very trivial propositions, making them poor examples. We’ll have to introduce something a bit fancier; a conditional type. For example, OddInt might be odd Integers, and PrimeInt might be prime integers. We’ll approximate these conditional types with custom classes in Scala. Classes and types are different beasts, of course, but we will ignore that distinction in this post.4

Let’s consider one conditional type in particular: BigInteger. This type (actually a class in this example) is defined as follows:

One could then instantiate a BigInteger as follows:

Now the fundemanetal question: what proposition corresponds to this type? In simple scenarios like this, the corresponding proposition is that the type can be inhabited; that is, there exists a value that satisfies that type. For example, the type BigInteger corresponds to the claim “there exists an integer $$i$$ for which $$i > 10,000$$”. Obviously, such an integer exists, and the fact that we can instantiate this type indicates that it corresponds to a true proposition. Alternatively, consider a type WeirdInteger, which is an integer satisfying i < 3 && i > 5. We can define the type well enough, but there are no values which satisfy it; it is an uninhabitable type, and so corresponds to a false proposition.

Functions and Implication

Let’s make things a little more interesting. In programming languages, there are not only primitive types like Integer and Boolean, but there are also function types, which are the types of functions. For example, in Scala, the function def f(x: Int) = x.toString has type Int => String, which is to say it is a function that maps integers to strings.

What sort of propositions would functions correspond to? It turns out that functions naturally map to implication. In some ways, the correspondence here is very natural. Consider the conditional type BigInteger, and the conditional type BiggerInteger. The definition of the latter should look familiar, from above:

Now, we can write a function that maps BigInteger to BiggerInteger:

Recall that the proposition corresponding to the type BigInteger is the statement “there exists an integer greater than 10,000”, and the proposition corresponding to Bigger is the statement “there exists an integer greater than 20,000”; the proposition corresponding to the function type BigInteger => BiggerInteger is then just the statement “the existence of an integer above 10,000 implies the existence of an integer above 20,000”. And note that, as it should be for an implication, we do not care whether there actually does exist an integer above 10,000; we simply know that if one exists, then its existence implies the existence of an integer above 20,000.

To be a bit more explicit, the function that we wrote above can be thought of as a proof of the implication; in particular, if we suppose that there exists an $$i$$ such that $$i > 10,000$$, then clearly $$2i > 20,000$$, and so if we let $$j=2i$$, then we have proven the existence of a $$j$$ such that $$j > 20,000$$. This is what the theoretical computer scientists mean when they say that “programs are proofs”.

Of course, Scala is not a proof-checking language, and cannot tell during compilation that the function makeBigger is valid; we would need a much richer type system to be able to validate such functions. Consider that the following function compiles with no problem, although there are no input values for which it will not throw a (runtime) exception:

Wait… what?

If you think about it a bit more, it’s sort of a weird example; you could map any type to BiggerInteger, just by doing def f[A](a:A): BiggerInteger = new BiggerInteger(20001). This is because the proposition that corresponds to BiggerInteger is true (the type is inhabitable), and if B is true, then A implies B for any A at all.

Common languages such as Haskell only express very trivial propositions with their types; there does exist one uninhabitable type (void), but I have not found much use for it in practice. The benefit of using conditional types for these examples is that we can explore at least some types which have corresponding false propositions, such as WeirdInteger, which are integers i which satisfy i < 3 && i > 5.

In Conclusion

Seeing all this, you can begin to get a sense of how computer-assisted proof techniques might arise out of it. If the fact that a program compiles is equivalent to the truth the corrsponding proposition, then all we need is a language with a rich enough type system to express interesting statements. Examples of languages used in this way include Coq and Agda. A thorough discussion of such languages is beyond both the scope of this post and my understanding.

I think what keeps me interested in this subject is that it still remains quite opaque to me; I’ve struggled to even come up with these simple (and flawed) examples of how Curry-Howard correspondence plays out in practice. I hope that anyone reading this who understand the subject better than I do will leave a detailed list of my misunderstandings, so that I can better grasp this mysterious and fascinating topic.

1. This statement is difficult to understand without background in functional analysis, but it is in fact one of the most beautiful examples of such an equivalence result.

2. I’m being a bit sloppy here. The type of logic we’re talking about here is not classical logic, but rather in the sense of natural deduction

3. Such systems are called undecidable; see the wiki entry on decidability for more information.

4. We won’t be careful about whether the idea of conditional types presented here corresponds well with conditional types as they are actually implemented in programming languages such as Typescript

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