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Comment by Sharlin | Hacker News Reader

One thing I’ve been wondering is how to interpret the function types `Void -> a` and `a -> Void`, where Void is the uninhabited type (unique up to isomorphism). The number of inhabitants of those types should be |a|^0=1 and 0^|a|=0, respectively, but what does that mean? In real world, function(s) of type a->Void certainly exist, such functions being those that diverge (loop forever or halt via some non-local means).

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`Void` being the uninhabited type, in the light of the Curry-Howard isomorphism stands for a false proposition.

`a -> Void` get interpreted as "not a" or "from a follows contradiction" or equivalently "a is uninhabited". Combinatorially it's `0 ^ a` which for non-empty a is zero but is equal to 1 when a is empty (0^0=1). In other words there are no functions of type `a -> Void` for non-empty a and there's exactly one such function for uninhabited a (id :: Void -> Void).

`Void -> a` is interpreted "from falsehood, anything (follows)" https://en.wikipedia.org/wiki/Principle_of_explosion. Combinatorially a^0 = 1 for all a so there's exactly one such function. An easy way to define it is by induction on Void (which has no cases and you're done).

Yeah, Void makes for great for inductive definitions ;)

In mathematics, assuming 'void' is the empty set, the function void -> a exists, but is kind of trivial because it never returns anything, however functions a -> void can't exist because they'd have to return an element of void, which don't exist, the one exception being the function void -> void. Denoting A -> B as B^A this translates to:

    a^0 = 1  
    0^a = 0  
    0^0 = 1

The fact that there's precisely 1 function void -> a is considered rather special in category theory, and means that 'void' is the so called 'initial object' for the category (which is automatically unique up to isomorphism).

Yeah, it makes sense now. Thanks.

AFAIUI:

"Void -> a" means that the function cannot be called, because there no way to conjure a Void value. In Haskell, you can do it by (ab)using 'undefined', but that's kind of cheating. However, if you're using Idris with totality checking, I don't think you'll actually get any code calling such a function compile.

The "a -> Void" means that the function can never return.

`Void -> a` is inhabited by the slightly weird empty-case construct in (GHC) Haskell: `\x -> case x of {}` has type `Void -> a`

Also known as the `absurd` function IIRC?

Such functions can’t exist.

The “C” void is expressed by the unit singleton.

So it is |a|^1.

Yes, C void is a kinda-sorta unit type. But I’m talking about the real uninhabited type, ie. Void in Haskell, Nothing in Scala, ! (Never) in Rust. Real-world languages including Haskell are not total, so all functions a->b are in reality a->(b|Void). And for a good reason, as totality means loss of Turing completeness. In a total language, a function `a->Void` does not exist, but what about `Void->a`, then? What is the single inhabitant of that type?

This one typechecks but loops forever:

  f :: Void -> a
  f _ = let x = x in x

It is the so-called empty function, which diverges, e.g. throws an exception if called.

Since one can prove anything if you're assuming nonsense, a function of the type 'Void -> a' should be able to just return its argument.

And assuming the theory is consistent(so no infinite loops etc.) this is the only instance of that function, so the cardinality is 1.

That makes sense, thanks!

Where does an instance of a come from in that case? A function of that type has no way to construct one.

Yes, there are no possible return values. But there is a unique vacuous total function because no return values are ever needed. A function can be considered as a set of ordered pairs, one for each value in the domain. Since there are no values in the domain, no pairs need exist.

Another way to look at it is as a constrained subset of the Cartesian product of domain and codomain. In this case the domain is the empty set, so the Cartesian product is empty too.

The key is that you can (in theory, I don't know if Haskell has a proper bottom that can act as a type) give a value of Void/Bottom to anything, analogous to logic in that a proof of nonsense can be used as a proof of anything.

Take this Haskell code that passes the type checker:

  bottom = bottom

  f :: Int -> a
  f _ = bottom

Here 'bottom' has the conceptual type Void/Bottom and checks successfully against the 'a'.

So assuming we have this Void type in Haskell (I don't know if Haskell has it or not), this should also type check:

  f :: Void -> a
  f x = x

Just like bottom (of type Void) checked against the 'a' earlier, an argument of that type should also type check.

And correspondingly `a->Void` must be empty because you can’t prove a contradiction given a true proposition in a consistent system? Makes sense from the Curry-Howard point of view.

The "should be able to just return its argument" part is wrong; the types pretty clearly don't match, yeah? It is true that there is a unique function of type `Void -> a` for each `a`, but that isn't it.

When the Bottom type is included in a language it usually exhibits sub typing behavior. That can be expressed in Haskell, too:

  {-# LANGUAGE RankNTypes #-}

  type Bottom = forall a. a -- zero inhabitants

  f :: Bottom -> a
  f x = x

That passes the type checker just fine.

> That passes the type checker just fine.

I think you are running into some weird haskellisms with your Bottom-type.

The normal way of defining that in haskell is using the "EmptyDataDecls" pragma, like this:

    {-# LANGUAGE EmptyDataDecls #-}

    data Empty 

    g :: Empty -> a
    g x = x

Which doesn't pass the type checker.

(From a theoretic standpoint, I would have thought your Bottom was a essentially a type-level identity function..)

Again, languages with a built in bottom usually have sub typing behavior that makes bottom a subtype of everything (so you can give a value of type bottom to everything).

'forall a. a' is just how you define the bottom type in System F, which Haskell is somewhat based on. As a type it describes that a term of that type can just conjure a value of any type out of thin air, which is obviously nonsense. That what makes it the bottom type.

The sub typing behavior associated with that is just the normal subsumption rule of polymorphic functions. The same reason why you can pass a function of type 'forall a. a -> a' to something expecting a function of type 'Int -> Int'. The types don't 'match' directly, but they do under the subsumption rule.

Yeah, I just wanted to show that (GHC) haskell has a way of getting a type with "zero" inhabitants that actually gets caught by the type checker. (I say "zero" since of course my Empty-type above does contain non-terminating programs and (undefined :: a).)

With "forall a. a" I was coming from the persepctive of intuisionistic type theory where the usual parametric polymorphism just get subsumed by Π-types and universal quantification is usually represented with Π-types, so you would have:

         forall a.a       (Universal Quantification)
    <==> Π(a : Type) a    (Π-type)
    <==> (a : Type) -> a  (Agda-notation)
     ==> a -> a           (Π-type as regular function type)

where Type represents any type. That's what I meant by "type-level identity function".

Interesting, thanks! I hadn't considered using "forall a. a" as the bottom type, but it makes sense.

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