## The Y Combinator and strict positivity

One of the most mind-bending features of the untyped lambda calculus is the fixed-point combinator, which is a function `fix` with the property that `fix f == f (fix f)`. Writing these combinators requires nothing besides lambdas; one of the most famous of which is the Y combinator `λf.(λx.f (x x)) (λx.f (x x))`.

Now, if you’re like me, you saw this and tried to implement it in a typed functional programming language like Haskell:

Prelude> let y = \f -> (\x -> f (x x)) (\x -> f (x x)) <interactive>:2:43: Occurs check: cannot construct the infinite type: t1 = t1 -> t0 In the first argument of `x', namely `x' In the first argument of `f', namely `(x x)' In the expression: f (x x)

Oops! It doesn’t typecheck.

There is a solution floating around, which you might have encountered via a Wikipedia article or Russell O'Connor's blog, which works by breaking the infinite type by defining a newtype:

Prelude> newtype Rec a = In { out :: Rec a -> a } Prelude> let y = \f -> (\x -> f (out x x)) (In (\x -> f (out x x))) Prelude> :t y y :: (a -> a) -> a

There is something very strange going on here, which Russell alludes to when he refers to `Rec` as “non-monotonic”. Indeed, any reasonable dependently typed language will reject this definition (here it is in Coq):

Inductive Rec (A : Type) := In : (Rec A -> A) -> Rec A. (* Error: Non strictly positive occurrence of "Rec" in "(Rec A -> A) -> Rec A". *)

What is a “non strictly positive occurrence”? It is reminiscent to “covariance” and “contravariance” from subtyping, but more stringent (it is strict, after all!) Essentially, a recursive occurrence of the type (e.g. `Rec`) may not occur to the left of a function arrow of a constructor argument. `newtype Rec a = In (Rec a)` would have been OK, but `Rec a -> a` is not. (`(Rec a -> a) -> a` is not OK either, despite `Rec a` being in a positive position.)

There are good reasons for rejecting such definitions. The most important of these is excluding the possibility of defining the Y Combinator (party poopers!) which would allow us to create a non-terminating term without explicitly using a fixpoint. This is not a big deal in Haskell (where non-termination abounds), but in a language for theorem proving, everything is expected to be terminating, since non-terminating terms are valid proofs (via the Curry-Howard isomorphism) for any proposition! Thus, adding a way to sneak in non-termination with the Y Combinator would make the type system very unsound. Additionally, there is a sense in which types that are non-strictly positive are “too big”, in that they do not have set theoretic interpretations (a set cannot contain its own powerset, which is essentially what `newtype Rec = In (Rec -> Bool)` claims).

To conclude, types like `newtype Rec a = In { out :: Rec a -> a }` look quite innocuous, but they’re actually quite nasty and should be used with some care. This is a bit of a bother for proponents of higher-order abstract syntax (HOAS), who want to write types like:

data Term = Lambda (Term -> Term) | App Term Term

Eek! Non-positive occurrence of `Term` in `Lambda` strikes again! (One can feel the Pittsburgh-trained type theorists in the audience tensing up.) Fortunately, we have things like parametric higher-order abstract syntax (PHOAS) to save the day. But that’s another post...

Thanks to Adam Chlipala for first introducing me to the positivity condition way back last fall during his Coq class, Conor McBride for making the offhand comment which made me actually understand what was going on here, and Dan Doel for telling me non-strictly positive data types don’t have set theoretic models.