How to Prove Programs: Understanding the Curry-Howard Correspondence

host-to-prove-programs.md

How to Prove Programs: Understanding the Curry-Howard Correspondence

The Curry Howard correspondence is the relationship between computer programs and mathematical proofs. Specifically:

1. Types can be interpreted as logical claims.
2. Values can be interpreted as proofs of logical claims.
3. Evaluating code can be seen as making a proof simpler.

This is tricky to think about, much like the optical illusion of a dancer, where, if you concentrate really hard, you can reverse the direction you see them spinning.

Programs are Proofs

Let's take the Idris program:

total thinger : Nat
thinger = 1 + 2

(Ignore total for a moment.)

Its interpretation as a functional program is,

“let’s define a name thinger which represents a Nat (a natural number, or integer ≥ 0);

further, let’s define thinger to be equivalent to the expression 1+2.”

We often say that thinger "has type" Nat.

Its logic interpretation is,

“let’s propose that there is a natural number of some kind;

further, let’s offer the expression 1+2 as a witness – a kind of proof – that this claim is true.”

A note on logic: In classical logic we can sometimes prove things without an explicit witness, but we must abandon this ability to make our correspondence work. This restricted logic is called constructivist or intuitionistic. Classically we can prove the statement, "Either England has a king, or it does not," without knowing which part is true – or, in fact, anything about England. A constructivist proof of this statement requires we either present a proof of England having a king, or a proof of England not having a king.

Idris rejects incorrect proofs (witnesses) primarily through type checking and totality checking. (Idris also performs other necessary checks that won’t be covered here.)

Above, we claimed that thinger produces a Nat. Idris looks up + and sees that it has multiple versions for different kinds of numbers, but only one can produce a Nat, and this one needs two different Nats as input. It then checks whether it can interpret 1 and 2 as Nat, which it can. By assigning types to every sub-expression and verifying that the types connect correctly, Idris ensures the result of the expression will be a proof of the claim. This is type checking.

But not all correctly-typed programs work! Some loop forever, going nowhere. Some produce a forever-larger data structure and never stop to return anything – in practice, crashing when memory runs out. Some crash because they fail to account for a particular case. If any of these things could happen, we can't trust our witness. The keyword total tells Idris to ensure the computation will always succeed. It could take eons to calculate, which is fine – its value as a proof does not come from actually running it, but from knowing we could always get an answer from it if we wanted.

Evaluation

In the program above, we (or the compiler) could replace 1 + 2 with 3, making the proof simpler. This is what we mean when we say that the evaluation of programs corresponds to the simplification of proofs.

A More Logical Example

Above, we showed how to interpret Nat as a claim. In practice, when we are intentionally using Curry-Howard to prove things, we'll use types which were designed for the purpose.

Given two arguments, = produces a type that can only be witnessed if the two arguments are equivalent. Therefore, we can prove that 3 = 3:

total threeEqualsItself : 3 = 3
threeEqualsItself = Refl

Refl means "reflection"; it is the value that witnesses that any value is equivalent to itself. Since 3=3 is a reflection, the proof is satisfied.

We can pass two arbitrary things to =, e.g. True = 42, but this type has no values we can present. It's valid, we just say that it's uninhabited. Here's what happens if we try:

total badEquivalance : True = 42
badEquivalence = Refl

(input):1:50:When checking right hand side of badEquivalence with expected type
        True = 42

Type mismatch between
        x = x (Type of Refl)
and
        True = 42 (Expected type)
 
Specifically:
        Type mismatch between
                Bool
        and
                Integer

Proving Programs

We now have the foundation to prove properties of programs. Since programs are proofs and proofs are programs, nothing stops us from putting code in our proofs – or even making code the object of our proofs.

For example, when writing a function parse to turn some text into a value, we can prove a relationship between our parse and show, the standard library's function for turning a value into text:

total showAndParseAreSymmetric : (v : MyValueType) -> parse (show v) = v

This means, “turning any value of MyValueType into text and back yields a value equivalent to the original.”

A proof of this property would rule out all sorts of bugs: precision problems representing floating-point numbers, cases missed or coded incorrectly, or ambiguous textual representations (distinct values having the same textual representation). A proof ensures that any value could be written and read back without unexpectedly changing.

We get much more confidence from proving than from unit testing, since unit testing requires us to have explicitly thought of all the cases. We get a lot more confidence from proving than from generative testing, since generative testing over a large state space can fail to find super-rare corner cases that proving forces us to consider. It requires learning and more effort, but the result is confidence you can’t get any other way.

I have a question: What aspects of a language are necessary in order to form programs as proofs in this way? Obviously, they need to be typed, but do they need to be strongly typed? I'm unfamiliar with Idris and having trouble imagining how to do such a thing in programs I am more familiar with.

@eraserhd The part on witnesses has this:

Classically we can prove the statement, "Either England has a king, or it does not,"
without knowing which part is true – or, in fact, anything about England. A constructivist
proof of this statement requires we either present a proof of England having a king, or a
proof of England not having a king.

Now, I understand classically you can have a valid argument about a England having a King without having factual premisses:

 Kings cannot live where they do not rule
 Jason lives in England
 Jason is a King
 Therefore, England has a King.

Is this what you mean by not having a witness? There is no proof that Jason is King of England; only logical inference.

I've just learned there's no way to be notified for comments on a gist, so I've just seen these (@glovguy @s33dunda).

In case yous happen back:

@glovguy If you mean as opposed to a language where you can cast from any type to another by telling the compiler to trust you, I would say that using that feature probably breaks your proof. But you can write proofs in that language by avoiding that feature. Even Idris has this kind of footgun (it's called believe_me).

@s33dunda It's not about whether the premises are factual, but about whether you can compute the result with a program. For example, double negation works in classical logic: from (Not (Not A)) we can extract the value of A. In classical logic, this works because propositions are boolean-valued - called truth-functional. In constructive logic, propositions are proof values - called proof-functional. We can evaluate !!A and know the value of A, but (Not (Not A)) constructively is not a value in A, but a proof that there must be some value in a (without knowing what that value is).

I'm not confident that was helpful.