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Gradual Type Precision as Retraction

2016-12-15 Max New

Notes by Ben Greenman


In the 60s and 70s, Dana Scott modeled the typed lambda calculus with projections from the untyped lambda calculus [1][2]. In the 90s, Robby Findler and Matthias Blume explored the idea of contracts as "pairs of projections" [3].

[1] "Relating Theories of the Lambda Calculus", 1980.
[2] "Data Types as Lattices", 1975.
[3] "Contracts as Pairs of Projections", FLOPS 2006.

M: note that Robby stopped working on contracts as pairs of projections because he wasn't able to use it to describe dependent contracts
Max: well I'm not working on contracts in general,

In the 00s, people started working on gradual typing. This talk will outline how Dana Scott's ideas can provide a semantic foundation for gradual typing,

Let's take this connection seriously and see if it helps us design contract systems / gradually typed languages better.

TLDR; Scott

A retraction between domains A and B is a pair of functions:

  • s : A >--> B
  • r : B -->> A
  • such that s is injective and r is surjective
  • and composing r ∘ s gives the identity function

The function s is called a section and the function r is called a retraction. Together, they are a section/retraction pair.

Here is a picture:

            s    *-------*
    *---*  >-->  |       |
    | A |        |   B   |
    *---*  <<--  |       |
            r    *-------*

Will: okay, a retraction implies an isomorphism between A and a subset of B

Composing s ∘ r may not be the identity on B, but it is an idempotent on B. An idempotent on a domain X is a function e : X -> X such that e ∘ e = e.

Here is a proof that s ∘ r is an idempotent on B:

    (s ∘ r) ∘ (s ∘ r) =
    s ∘ (r ∘ s) ∘ r =
    s ∘ id_A ∘ r =
    s ∘ r

Scott proved that idempotents in a typed λ calculus are in close correspondence with untyped λ calculi. More precisely, Scott proved [1]:

Every untyped theory is the theory of a reflexive domain in a typed theory.

Connection to gradual typing: every type is the retract of a dynamic type. (Every untyped term can be typed at a dynamic type and every typed term can be type-erased.)

TLDR; Siek/Taha

A gradually typed languages allows typed and untyped terms in the same program. The type system accepts any terms that might possibly be well-typed. The runtime system inserts casts / coercions that check whether the runtime value of untyped terms matches the types that typed terms expect.

The Idea

Contracts / casts / coercions in a gradually typed language have a natural interpretation in terms of sections and retractions.

M: a minute ago, you threw in a sub-clause "the contracts we can check at runtime", so you're assuming a topological space where everything is computable. When you pick such spaces, you usually get some junk that doesn't have the right properties.
Max: I don't care about denotational semantics, I care about programming languages, so I'm not worried about the junk. It doesn't come up.
M: at some point you're going to show us a concrete category, rigth?
Max: yes, coming soon


There's a problem with this picture, blame doesn't have an obvious Dana Scott interpretation. How do we relate blame?

Does blame have to be a structure I define on the language, or can we have semantic properties that blame needs to satisfy? I'm looking for emantic concepts that inform how we implement blame in a programming language.

M: read Robby's tech. report [3], there are arrows that represent blame
Christos: CPCF [4] has a nice model for blame

[4] "Correct Blame for Contracts", POPL 2011.

Wadler and Findler[5] defined gradual type precision in terms of blame. I'm going to derive type precision, and we'll see that it's nearly the same definition but disagrees at one important point.

M: good!

[5] "Well Typed Programs Can't be Blamed", ESOP 2009.


  • always start with the middle whiteboard
  • you can always move whiteboards as you write!

Lets get more concrete

Types in our gradually typed language include ground types, function types, and the dynamic type.

  τ := Bool | Num | τ → τ | Dyn

The simply-typed rule for checking an application mostly works, but you have to do a little extra:

    Γ ⊢ t : A → B
    Γ ⊢ u : C
    Γ ⊢ t u : B

The idea is,

  • if A and C are static types, they must be equal. (be strict on static types!)
  • if A or C is Dyn, this should typecheck

M: what if Γ ⊢ t : Dyn ?
Max: it typechecks, and a runtime cast coerces t to a function type

The source language has implicit casts and typechecking makes these casts explicit.

And so, the full application rule uses a type compatibility relation, τ ~ τ

    Γ ⊢ t : A → B
    Γ ⊢ u : C
    A ~ C
    Γ ⊢ t u : B

Here are two rules for τ ~ τ:

   A ~ Dyn

   A ~ A'
   B ~ B'
   A → B ~ A' → B'

Wadler/Findler also define a type precision relation A ⊑ B to say when A is "more precise" (i.e. has fewer Dyn) than B.

"A is a subset of B" is a fair intuition.

Here's one surprising rule from the type precision relation you see in the literature:

   A ⊑ A'
   B ⊑ B'
   A → B ⊑ A' → B'

Scary right? Wadler/Findler call this "naive subtyping". But it makes sense of you think of as "has fewer Dyn".

(Max comments:

Subset is possibly a misleading intuition for A ⊑ B. However, it agrees with a slightly different notion which Mellies-Zeilberger [6] call refinement in which you think of A -> B ⊑ A' -> B' as meaning that we are seeing A -> B as the subset of A' -> B' functions that happen to, when given things in the subset A ⊑ A', give you values that land in the subset B ⊑ B'.

[6] )

With type intersection A ⊓ B, such that:

   A ⊓ B ⊑ A
   A ⊓ B ⊑ B

Type compatibility falls out:

   A ~ B := ( A ⊓ B /= ∅ )

Will: τ ~ τ is not transitive?
Max: right, for the same reason that non-empty intersection isn't transitive

Here's the typing rule for an explicit cast from type A to type B:

    Γ ⊢ t : A
    A ~ B
    Γ ⊢ <B ⇐ A>p t : B

To implement a gradually typed language, you need to introduce tags that stick around at runtime. Use these to implement τ ~ τ.

    G := Bool | Num | Dyn→Dyn

We'll use (Dyn_G v) to represent a value v that's tagged with the ground type G.

Here's the dynamic semantics of casts from Wadler / Findler. These p are "blame labels" that are used to report errors to the programmer.

    <B ⇐ B>p v ↦ v

    <Dyn ⇐ G>p v ↦ (Dyn_G v)

    <Dyn ⇐ A→B>p v ↦ Dyn_(Dyn→Dyn) (<Dyn→Dyn ⇐ A→B>p v)

    <A'→B' ⇐ A→B> v ↦ (λ (x : A') <B' ⇐ B> (v (<A ⇐ A'>p' x)))

(BG: there are 2 more rules incoming)

And p' is p with blame reversed.

M: the p' operations assumes you have a closed system, in the real world, programs are open and you can link new types in. How are you going to compute p'?
Max: at link-time, just wait until you have the whole program
M: what's cheating about p' is that in principle you should write these rules with an evaluation context and the ' operation would take a second argument, like p' = reverse_blame(p, e), where e is the original context where p was generated. (when the program is loaded)
Christos: right, the continuation that really matters is the context where you attach the contract
Max: exactly, you can't forget it and attach it later
Will: we get the point, using e and a context would clutter the presentation
M: theoreticians say "let there be a fresh variable", we all know things go horribly wrong if you're not careful about that

The last rule is where checking happens:

    <A ⇐ Dyn>p (Dyn_G v) ↦ <A ⇐ G>p v   -- if `G ~ A`
    <A ⇐ Dyn>p (Dyn_G v) ↦ blame p      -- otherwise

If G is compatible with A, just keep checking the underlying value if not compatible, type error. Who's to blame? The dynamic value, for not satisfying the type.

I like my syntax to mean something, I want a semantic definition in terms of these things here. And I'll give one in terms of section/retraction pairs.

M: in between today and Wadler/Findler, Christos gave a semantics in terms of blame assignment
Christos: I disagree with that statement
M: you could easily add this chapter to your dissertation

Type Precision as Section/Retraction

Rule #1 in a gradually typed language is that you can cast from any type to Dyn and back again. In other words:

    A >--> Dyn -->> A

is the identity function. This makes a section/retraction pair for any non-function type A.

Might work for A→B with an eta law. Not sure.

Now type precision is a statement about how these casts to/from Dyn commute:

    A ⊑ B


    A >--> Dyn

    |       =

    B >--> Dyn

(BG: the vertical arrow means A >--> B)


    A <<-- Dyn

    A       =

    B <<-- Dyn

(BG: the vertical arrow is supposed to represent A <<-- B)

A is more precise than B if casting from Dyn to B and then casting to A is the same as casting directly from Dyn to A. You can say, the cast to A checks more things, so it's more likely to fail.

All of these casts that happen at runtime in a gradually typed language factor into uses of the above two diagrams.

Christos: this is exactly whaat you're doing with the third rule, right? by first casting to Dyn→Dyn and then casting to Dyn
Max: right, and the fact that these arrows commute is what justifies removing extra casts in your implementation

Another notion from Wadler / Findler is positive and negative subtyping. They define type precision in terms of <:+ and <:-

    A ⊑ B := A <:+ B and B <:- A

    A <:+ B := <B ⇐ A>p never blames the "positive" party in `p`

    B <:- A := <A ⇐ B>p never blames the "negative" party in `p`

Unlike naive subtyping, the definitions of <:+ and <:- are contravariant for functions. Contravariance regained! For example:

    A' <:+ A
    B <:- B'
    (A → B) <:- (A' → B')

In Wadler/Findler, G <:- Dyn. This means casting from the ground type never blames the negative party.

M: so that's asymmetric
Max: well it's not the case that a cast from Dyn to anything never blames

OK, so Wadler / Findler got type precision from these 2 judgments. If we follow the section/retraction intuition we get 2 judgments too, one for sections and one for retractions. In general:

  A <:+ B iff A >--> B

positive subtyping implies we can extract a section,

  B <:- A iff B -->> A

negative subtyping implies we can extract a retraction.

Since every (ground) type comes with a section/retraction pair into Dyn,

    A <:+ Dyn

    Dyn <:- A

we can derive <:+ and <:- Here's one example:

    A' <:- A
    B <:+ B'
    A→B <:+ A'→B

The term that computes this is:

    λ(f : A→B) s_B ∘ f ∘ r_A

Christos: isn't this just what Robby and Matthias Blume did in their implementation?
Max: yes
M: Christos, you did this too.
M: in our world (the tech. report version) we use a pair of projections. Does one of them correspond to your sections?
Max: no it's different
M: if we can figure that out, we can use topology instead of ...
Christos: what's the difference?
Max: This one's typed, you can only get one projection (one idempotent) from this. But there, you have 2 projections. So strictly more information. Maybe this one is insufficent to justify blame.
M: exactly, you need the 2 for the blame part
Max: Well you still have 2 functions, so still possible to distinguish But there you really need 2 contracts
M: maybe we can build more
Max: maybe you have more junk
M: In Scott's world, any retract is a type. Just because the type terrorists haven't written them down doesn't mean they won't be useful
Max: agreed

We get mostly the same rules for positive and negative subtyping, but:

    G <:- Dyn

Is NOT true. If it was true, that would mean that casting

    <Boolean ⇐ Dyn> (<Dyn ⇐ Boolean> v)

would have to be the identity for any v. That's obviously false.

Let's look more at this disagreement. Maybe Wadler and Findler defined blame wrong. Maybe this cast should maybe raise negative blame.

Look at the cast from ground types to Dyn. It turns out, G <:- Dyn was implied by the dynamic semantics of casts:

    <Dyn ⇐ G>p v ↦ Dyn_G v

because this rule throws the blame label p away. So it says no party is ever blamed.

We could change the dynamic semantics to keep p.

  <Dyn ⇐ G>p v ↦ Dyn_G(v, p)

And we also need to change the rule that performs a cast:

  <A ⇐ Dyn>p (Dyn_G (v, q)) ↦ <A ⇐ G>p v      --- if G ~ A
  <A ⇐ Dyn>p (Dyn_G (v, q)) ↦ blame(p, q')    --- otherwise

Before, we always blamed p. Reasoning was, v is a bad dynamic value. Why not blame q', the context, for performing the cast on v ?

Christos: Let me see if I understand correctly. If I have a function with a contract that says it expects booleans, then what does this contract mean? Is it:

  • my callers have an obligation to produce booleans?
  • I have an obligation to use my argument as a boolean? Is this the distinction you're making?
    Max: yes, I'm saying it's reasonable to blame the function for not using its argument as a boolean

Here's an example, imagine we have two typed printing functions.

    printN : Num → ⊥
    printB : Bool → ⊥

And a small program that does these casts:

    ⊤ -> B ⇒ Dyn ⇒ N -> ⊥

(BG: maybe,

   ((λ (x : Bool)
     (printN (<Nat ⇐ Dyn>(<Dyn ⇐ Bool> x))))


Racket would blame true when the cast fails. It tells you where you can fix the program to prevent the error. But how does the language know that printN is the correct function? Maybe it's a mistake. Only the programmer knows.

Will: I doubt it

M: As you know, for 5 years I have said to the amusement of conference audiences that "Phil Wadler is wrong". When two things crash, either one could be at fault.
Max: In my language, you still have the proeprty that well-typed programs don't get blamed
M: The BOUNDARY gets blamed. The untyped guy was innocent and suddently got confronted with a brutal type
Max: yes, mismatch between type and a dynamic term
BenS: could you actually write this program?
Max: yeah, type systems for gradually typed languages are very stupid

This is non-transitivity of compatibility coming through. It also shows that type compatibility doesn't give you strong safety guarantees.

In a big program, the two parties could be very far apart. It's not just applying true immediately to a function. The Racket implementation is throwing away good debugging information!

M: This message has been lost at ICFP, they are praying to the god of specifications. They never want to hear that specifications can be wrong. Very few places in the world that admit specs can be wrong.
Max: And you're all in one of them :)
Christos: What's wrong is using types to generate specifications on untyped code.

My system doesn't blame more often than the original, but gives more information when it does blame.

Is there a connection to complete monitoring?

Christos: This isn't about complete monitoring, but it may violate "correct blame" as defined in the papers. The very shallow structure for blame (because of the "blame-swapping" p') make it incomparable to CPCF [4].
M: The direction of flow is very important, if I have no control over someone stuffing a value down my throat, I can't be blamed
Max: The old system just doesn't blame enough parties. I guess it's a "completeness of blame" problem.
Christos: Completeness of monitoring means you covered all the channels. So this system doesn't affect complete monitoring. Correctness of blame means that wherever I point you to, you can fix something there and move the blame somewhere else. Seems like you want to blame the place with the "least distance" to a "correct" program.
Max: no no, it's about blaming all the places that you could make a change to fix the issue

Ryan: What is p? What does p mean? What is a party? Is p a party? We haven't seen a definition yet. Seems more that p should be a boundary with 2 sides
M: p is the boundary and is preferential of one side as "the party"
Ryan: Do the authors of the original system think of p as a pair of parties?

Christos: I've been looking at contracts for information flow, your idea is related. You want a taint-tracking system for base values.

The meta-theory definition of correct blame tracks provenance. The Racket contract system doesn't. I've talked to Matthew about implementing this, and he says it will not happen, for a very practical reason. The issue is that you need proxies for every base value and every primitive operation needs to dispatch on these proxies.
M: In smalltalk, where 5 is an object, you can maybe do this. In our world it is way too expensive.
Max: You don't need to proxy, just add the blame information to the dynamic tag you're already keeping on values, to implement integer? etc.
M: Does the metadata have unbounded size?
Max: No, each p is just a source location. That's a fixed size if you know the number of locations
Will: Fixed number of bits seems easy
M: But that's a lot of bits
Will: It's logarithmic.
Max: For provenance, you just need the original location. That's one source location for each value.
M: There are 2 philosophies of gradual typing, in the Siek/Taha style every function application is a program boundary.
Will: Still a small number of bits
M: Still a few thousand
Will: That's nothing. The size you need is logarithmic in the number of boundaries.
Will: Here's what I'd do. Instead of representing all values by 64 bits, I'd just use 50 bits and keep the rest for tracking this stuff. Just keep enough extra bits to have overflow space in case there's dynamic linking. Shouldn't be hard to accomodate the space you need in practice.
M: Okay cool, it's just a matter of implementation. Your dissertation is spelled out.