How does Djinn work?

Question

OK, so I realise that I will probably regret this for the rest of my life, but... How does Djinn actually work?

The documentation says that it uses an algorithm whic

Answer 1

In the most general terms, according to the Curry-Howard isomorphism there is a correspondence between types and propositions and also values and proofs. Djinn uses this correspondence.

Do be more concrete, say that you want to find a Haskell term of type (a, b) -> (b, a). First you translate the type to a statement in logic (Djinn uses propositional logic, i.e., no quantifiers). The logic statement goes (A and B) is true implies (B and A) is true. The next step is to prove this. For propositional logic it is always possible to mechanically prove or disprove a statement. If we can disprove it, then that means that there can be no corresponding term in (terminating) Haskell. If we can prove it, then there is a Haskell term of that type, and furthermore, the Haskell term has the exact same structure as the proof.

The last statement has to be qualified. There are different sets of axioms and inference rules you can choose use to prove the statement. There is only a correspondence between the proof and the Haskell term if you pick a constructive logic. The "normal", i.e., classical logic has things like A or (not A) as an axiom. That would correspond to the Haskell type Either a (a -> Void), but there's no Haskell term of this type so we can't use classical logic. It is still true that any propositional statement can be proven or disproven in constructive propositional logic, but it's considerably more involved doing so than in classical logic.

So to recap, Djinn works by translating the type to a proposition in logic, then it uses a decision procedure for constructive logic to prove the proposition (if possible), and finally the proof is translated back to a Haskell term.

(It's too painful to illustrate how this works here, but give me 10 minutes at a white board and it will be crystal clear to you.)

As a final comment for you to ponder: Either a (a -> Void) can be implemented if you have Scheme's call/cc. Pick a more concrete type like Either a (a -> Int) and figure out how.