In bidirectional type checkers
Okay, let me start off by saying Iʼm not a huge fan of bidirectional type theory. It is not a satisfying explanation of typechecking for me, I feel like you run into its limitations way too quickly.I feel like I can say this now that Iʼve worked on getting PolyTT off the ground!
However, it is quite useful, especially for quickly getting started with a type theory implementation. It has a specific philosophical tack, which is commendable (“redexes need annotations”). And it supports some nice features.
In particular it lets you hack in cheap overloads and coercions (both fake and real) – but in a limited manner.By fake coercion, I mean a coercion that happens only in the elaborator for specific syntax – “elaboration hacks” in other words. Real coercions operate on arbitary terms, when actual and expected types are different but coercible, but these require more systematic incorporation into the theory.
Overloads, in this context, include operators that morally take implicit arguments. Of course, for builtins, the typechecker can do whatever it want, but a common feature is that it is used for resolving types that would be implemented as implicit arguments.
However, these overloads and coercions are limited to builtin types and functions. They need to be baked into the implementation.
In particular, if youʼre implementing a dependent type theory and choose a naïve bidirectional type checking algorithm, youʼre limiting implicit arguments to be only for builtins.
And any user of that theory is going to be frustrated at the verbosity of it. And maybe a little miffed at the unfairness of it.
How come the implementer keeps the cool tricks to herself?!
Letʼs try to fix this. How can we give the user access to implicit arguments for their own functions too?
As usual with type theory, it requires being more disciplined and systematic. Formalizing the unstated conventions underpinning common practice – this is the bread and butter of type theorists!
Our first observation comes from a page stolen out of the builtinsʼ book:
Require that user-defined operators are fully applied enough, in order to infer implicits.
What does this mean, “enough”? Weʼll need to do some work to figure it out.
But what it accomplishes is clear:
Knowing that we have some arguments applied already lets us syn them, and use the resulting type information to fill in the implicits.
I will be using syn and chk as verbs freely. These are the synthesize and check modes of bidirectional type theory.
Thus we would say that we need to be able to syn enough arguments to determine every implicit the user asked for. The remaining arguments can (and should) be chked – and since normal function arguments are also chked, this means they can be unapplied arguments.
I wonʼt be addressing how to parse user-defined operators. Thatʼs a difficult enough problem on its own.
Just note that Iʼll include prefix operators that look like normal function application (separated by a space), but differ in requiring a certain number of arguments to be applied. In fact, these are the easiest to parse, and the best source of examples.
We can also flip the tables and infer implicits if we start in chk mode in the first place!
In fact, this is strictly better: if we are typing the (partially-applied, or not) binding in chk, we know the types of the arguments and the return type.
Thus the main improvement in this case is (wait for it): emulating mixed-mode typing for functions.
(Having some arguments in syn, and determining some information from the expected type T of the whole partially applied function f a b : T.)
Which is why you shouldnʼt even be using bidirectional typing in the first place! Urgh! (Unification handles this exact problem more elegantly. And it forces you to make your coercions coherent!)
Thus weʼve already gone from “letʼs add implicits to bidirectional typing” to “letʼs emulate unification except worse”.
In particular, one way that it is worse is that we have to commit to modes up-front: we canʼt decide to try infering this argument as syn, then fall back to infering something else and returning to chk it, etc.
The other way that it is worse is that we canʼt provide partial types to anything.
So weʼll see what weʼre able to make of it.
We write ~mode1:I → ~mode0:O for a non-dependent function I → O that, when typechecked in mode0, typechecks its argument in mode1.
We write the dependent version as Π ~mode1:(i : I), ~mode0:O(i).
Since normal function applications chk their argument, we have this equation for reasoning about modes (the only equation I can think of?):
~syn:(A → B)
===============
~chk:A → ~syn:B
~syn:(Π (a : A), B(a))
===============
Π ~chk:(a : A), ~syn:B(a)That is, if we can syn a function type (in result position), it is equivalent to chking its arguments and syning its result. Thus the argument does not need to be provided, and so we will prefer the first notation to indicate that.
Itʼs instructive to consider a bunch of examples to get our bearings:
id {A : Type} : ~syn:A → ~syn:A
iter {A : Type} : Π ~chk:(n : Nat), ~syn:(A → A) → ~syn:(A → A)
const_Tt {A : Type} : Π ~chk:(B : Type), ~syn:A → ~syn:(B → A)
const_tT {A : Type} : ~syn:A → ~syn:(Π (B : Type), (B → A))
const_t {A B : Type} : ~chk:(A → B → A)
const_tt {A B : Type} : ~syn:A → ~syn:B → ~syn:A
compose {A B C : Type} : ~syn:(B → C) → ~syn:(A → B) → ~syn:(A → C)
compose {P Q R : Poly} : ~syn:(Q ⇒ R) → ~syn:(P ⇒ Q) → ~syn:(P ⇒ R)
compose : ~syn:Poly → ~chk:Poly → ~syn:PolyAnd we come up with some rules for figuring out the modes of each argument:
An important edge case to keep in mind: there will be some times where an argument needs to be synthesized, because we donʼt know all of its implicits, but yet it does not uniquely determine any implicits (e.g. if there is a complex type function based on the implicit).
If the type of the whole application is in syn, then we have to determine the implicits by synthesizing some arguments, and the rest can be in chk.
As mentioned above, we have more flexibility in chk mode.
First of all, it should be the case that you can chk the whole type (with nothing applied) to fill in implicits.
One word: injectivity.
Injective type constructors uniquely determine their arguments.
Eurgh.
Actually I donʼt think itʼs quite so bad.
But still have the obvious issue of data in types.
For some implicits, they may not be used directly.
E.g. (a -> b) -> (fin n -> a) -> (fin n -> b).
Ugh but that gets weird semantically though.
Parameters.
Sounds like open research.
Or QTT (Quantitative Type Theory), with quantity 0.
Now that weʼre picking apart types, we might as well throw in overloads.
compose {A B C : Type}
: ~syn:(B → C) → ~syn:(A → B) → ~syn:(A → C)
: ~syn:(B → C) → ~chk:(A → B) → ~chk:(A → C)
: ~chk:((B → C) → (A → B) → (A → C))
compose {P Q R : Poly}
: ~syn:(Q ⇒ R) → ~syn:(P ⇒ Q) → ~syn:(P ⇒ R)
: ~syn:(Q ⇒ R) → ~chk:(P ⇒ Q) → ~chk:(P ⇒ R)
: ~chk:((Q ⇒ R) → (P ⇒ Q) ⇒ (P ⇒ R))
compose
: ~syn:Poly → ~chk:Poly → ~syn:Poly
: ~chk:Poly → ~chk:Poly → ~chk:Poly
: ~chk:(Poly → Poly → Poly)The main rule is that before we are able to narrow down the overload, we must be able to tell what mode to run in.
Then we want to narrow it down in a sensible way. In particular, we want to outlaw overlapping instances. (It would be feasible to match in order of most specific to least (toposort the partially-order DAG), but this means that new overloads could change the meaning of code checked under previously-defined overloads.) Certainly by the end of matching, we want to be left with one unambiguous thing.
In this case, if we are checking a partially-applied compose in syn mode, we know that the first argument needs to be syned, regardless of which overload.
If it ends up having type Poly, we can actually chk the next argument against Poly (and synthesize Poly for the return type).
If it is one of the other overloads, the next argument needs to be syned as well (to cover the missing implicit A or P),This is where mixed-mode would be especially handy lol: we could tell the typechecker we are expecting a function or polynomial morphism respectively (one bit of info), and in particular a function into type B or polynomial morphism into poly Q respectively (second bit of info). at which point we know enough to synthesize the return type.
Where I rant about unification.
Like, consider overloaded numeric literals and operators.
0 + n is going to fail without an annotation, but n + 0 would succeed.
Thus the operator is commutative, but typechecking it isnʼt!
(Yeah, yeah, depending on how you implement + thatʼll be a beta-redex, but the point stands.
The same issue occurs with = {A : Type} : A → A → Type, for example.)
Itʼs really tempting to do speculative typechecking, and trying to syn an argument but fall back to syning another if it is not a synable argument. But that just gets to be ridiculous.