[TYPES] Basic principal type terminology

[Sean McDirmid]

Hi Gabriel,

Ah, I see! I think my question was a bit confusing (or rather I was confused) since I think I conflated principal type computation vs. what a principal type is. I'm just computing the principal type differently, but it is still a principal type rather than something else, just because it came from one direction and not the other. 

Thanks for the clarification!

Sean

From: Gabriel Scherer [mailto:gabriel.scherer at gmail.com] 
Sent: Thursday, March 5, 2015 5:28 PM
To: Sean McDirmid
Cc: types-list at lists.seas.upenn.edu
Subject: Re: [TYPES] Basic principal type terminology

I think you need to be more precise about the typing of assignment and method calls. Assuming that A.DoHousePetThing() requires A to have trait HousePet to be well-typed, Mammal cannot be the principal type of A because it is not a valid type for A (A.DoHousePetThing() is not well-typed if we only have (A : Mammal)). Conversely, (Mammal + Housepet) does not seem to be a valid type for A, because intuitively (A = C) would not be well-typed.
In usual type system, there is a strong relation between what you call "types of a term" and "required types for a time": if a type T is required (imposed by a usage context) for a term A, then any "type for A" should be instantiable into T, otherwise the whole program is not well-typed. In particular, the principal type should be instantiable into any "required type": the two notions you distinguish are equivalent.
(I suspect you've got the order reversed when you say "a type for a term such that it is an instance of all types required of this term", my understanding is that you rather talk about "a type which is *instantiable* into all types required for this term". These ordering questions are always tricky, but I think both required types HousePet and Mammal are instances of your candidate HousePet+Mammal, not the other way around.)

On Thu, Mar 5, 2015 at 2:50 AM, Sean McDirmid <smcdirm at microsoft.com> wrote:
[ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list ]

Hi,

A principal type is defined in wiki as "a type for a term such that all other types for this term are an instance of it." What about a type defined as "a type for a term such that it is an instance of all types required of this term?" Whereas a principal type seems to be an intersection over bindings, such a "usage" type is a union over uses. Please forgive this OO example, but it is the best I could think of to exemplify the difference in my head:

trait HousePet
  def DoHousePetThing()
trait Mammal:
  def DoMammalThing()
trait Dog : Mammal
trait Cat : Mammal

val B : Dog
val C : Cat
A = B
A = C
// the principal type of A is Mammal
A.DoHousePetThing()
A.DoMammalThing()
// the usage type of A is HousePet + Mammal

The principal type of A is the intersection of Dog and Cat (say Mammal). The X type I'm computing in my system is based on usage, so it is just "HousePet" + "Mammal." So while a principle type starts at top and becomes more specific with each bind (top -> Dog -> Mammal), a usage type starts at bottom and becomes more general with each use (bottom -> HousePet -> HousePet + Mammal).

Could this "usage" type be the opposite of a principal type, and if so, what has it been called in the literature? Or maybe I'm just looking at this all wrong: would such the usage type "HousePet + Mammal" still be a principle type if it was propagated backwards during type inference to bindings so that B is "Dog + HousePet" and C is "Cat + HousePet?"

Thanks,

Sean