Comment by _hardwaregeek | Hacker News Reader

Comment by _hardwaregeek | Hacker News Reader

I'm nowhere near qualified enough to critique this article in a meaningful manner, but I've been thinking recently that types and compile time functions go hand in hand.

Type systems are essentially an attempt to embed a logic that can prove various invariants about the code. What varies across type system to type system is A. what you can express and B. how much you can check. The issue is, most programming language designers decide these two things for you. That's why stuff like reification vs type erasure bites designers; the decision is irreversible.

But what if it wasn't? And this seems to be what this article is proposing. That if you can write compile time functions that deal with types as a construct, you can leave decisions like type erasure and reification to the programmer. Granted, that essentially forces programmers to act as language designers. And I doubt most programmers want to learn stuff like type rules or how to prove soundness.

However it does have some really cool implications. Like right now, Ruby has this whole debate over adding types. And sure, Ruby could just add normal static typing with some annotations and it'd work. But it kind of doesn't make sense for such a dynamic language to use such a static construct. What'd make more sense (to me) is some sort of metaprogramming on code that allows for checking invariants. Like being able to load a class, then run it through a checker at runtime. I've been reading a book on dependent types (The Little Typer) and the whole idea of types having types with a meta-type of sorts (universes in dependent types, meta classes in Ruby) seems very applicable to Ruby.

The main issue is that designing a good type system is a dark art.

The more powerful a type system, the easier it is to manipulate terms but the harder it is to reason about types. That's the fundamental tradeoff.

Being able to specify your own type language is like having an untyped language governing your term language.

There are promising directions though. In 20 years they might even have matured.

   That's the fundamental tradeoff.

I'd like to refine this to: the fundamental tradeoff is between expressivity of the typing system and computational complexity of type-inference. Hindley-Milner's polymorphic typing system "has hit a sweetspot" (B. Pierce) between expressivity, type-inference. Almost every extension to Hindley-Milner makes full type inference undecidable.

Type inference is just one part of it. That's taking a term and asking "what type could this have?" Another thing you can do is turn it around and ask "what terms have this type?". That question also gets complicated easily.

And that's principled type systems. If you put ad hoc garbage into the types it might be inferrable and whatnot but still be ridiculous to reason about.

   what terms have this type?

This is a very interesting question, but one that is rarely asked by compilers or by programmers, or indeed by theoreticians.

I conjecture that the complexity of both questions is related, although I admit that I'd be hard-pressed even to define properly what the complexity of "what type could this have?" might be.

Oh, it's more common a question than you suggest. If I am writing a block of code and I come to some hole in context, for example `foo (bar _) baz`, and I know what type of thing has to go in that hole, then the very next thing I ask is "What term do I need there?".

This is a big question for theoreticians working on new languages. The problem is called canonicity in that context. Here is an example of such a proof of "What terms have this type [in this context]?": https://arxiv.org/abs/1607.04156

And finally, compilers will ask this question if they're nice to you :) Here is GHC recently gaining the ability to ask this: https://phabricator.haskell.org/D3361

The questions are related by proofs of cut elimination / normalization, which is the compatibility between a term and its context. Complex types describe more complex interactions between terms and contexts. Higher rank polymorphism has undecidable inference because a higher rank type can be used in very complex contexts.

I know that this question is being asked and reseached -- indeed I do research in this direction. But it's not well understood, and it's not nearly as widely asked a question as the type-inference question which the java compiler asks every time you type something like

   int n = 2 + 3;

Note that 2+3 needs to be type-inferred.

   cut elimination / normalization,

I'm not sure this can be reduced to cut-elimination, because the question "What terms have this type?" can be asked for non-terminating programs as well -- indeed it becomes much more interesting.

   https://arxiv.org/abs/1607.04156

Aside: I'm not following HoTT closely, I guess the problem whether canonicity of the natural number type holds in HoTT in general (= can be derived from the axioms of HoTT alone) is still open?

> I'm not sure this can be reduced to cut-elimination, because the question "What terms have this type?" can be asked for non-terminating programs as well -- indeed it becomes much more interesting.

Oh please share some references :) I think in that context what matters is the progress and preservation parts of cut elimination.

> Aside: I'm not following HoTT closely, I guess the problem whether canonicity of the natural number type holds in HoTT in general (= can be derived from the axioms of HoTT alone) is still open?

I haven't either, but I think that's the case.

   share some references

The PBE / IPS (= programming-by-example / inductive program synthesis) literature cares about this a great deal. The approach to PBE / IPS in the tradition of [1], essentially seeks a unique representative of each equivalence class of programs w.r.t. to the usual typed notion of program equivalence.

Since most interesting examples of PBE / IPS use recursion, [1] goes through a hierarchy of recursion principles, e.g. map, filter, fold, full recursion. The order of choosing recursion principles matters a great deal for controlling overfitting, the bête noire of machine learning.

The basic problem is this: the answer to the question "What terms have this type?" is almost always an infinite set with a great deal of redundancy. E.g. 2+3 and 3+2 are both an answer to "What terms have type Int?". But both answers are contextually indistinguishable, so in practise we only one to see one of them. Since contexual equivalence is typically undecidable there are limits to how clever you can be in a computer program to avoid redundancies. Moreover, in PBE / IPS you typically don't want to have just values, you are interested essentially in short programs inhabiting a type. So cut-eliminiation, at least as conventionally understood, doesn't quite do the job.

[1] J. K. Feser, S. Chaudhuri, I. Dillig, Synthesizing Data Structure Transformations from Input-Output Examples.

Thanks! This looks like a really interesting part of PL theory that I haven't looked at before. The problem you describe immediately reminds me of Equality Saturation; I wonder if it's been applied in this space: http://www.cs.cornell.edu/~ross/publications/eqsat/

That's interesting, although from scanning the abstract, I don't immediately see the connection between ES and PBE.

Well the common technical problem is that of representing program fragments along with their equalities so that they can be utilized in the case I linked, and pruned in the case you suggested. It's just a musing though. I look forward to reading up on PBE in depth.

That's why you go with bidirectional type checking instead. This scales much better to more interesting language constructs like rank-n polymorphism, and dependent types, gives better localised error messages, and the downside of having to have some top level annotations is worth it, because that's best practice anyway.

Bidirectional is a nice, pragmatic compromise, but it is a reaction to the difficulty with taking full type inference beyond let-polymorphism. We wouldn't be using bidirectional, if Damas-Hindley-Milner would scale.

I dunno, I do prefer having type directed editing (eg. auto case splitting, and autocomplete), so full type inference is less of a holy grail to me.

Damas-Hindley-Milner inference is also hyper-optimised to a specific point on the design space, and does an excellent job there, but (to echo Conor McBride) I would love to see less conflation of the ideas of execution phase, parametericity, and implicitness going forward.

    less conflation of

That's interesting. But I'm not sure what you mean. Would you be able to explain this more, or point me to something I can read on this?

> I've been thinking recently that types and compile time functions go hand in hand.

Type checking is a stage: http://lambda-the-ultimate.org/node/2575

Type systems as macros: http://lambda-the-ultimate.org/node/5426

Dependently typed languages do use the same evaluation mechanism for the type level and value level. In Java you have List<Int> and that's different than foo(bar), but in dependently typed languages Int is just a value of type Type, and List is a function that takes a Type and returns a Type. The difference is that this evaluation takes place at compile time.

For run-time checking there is the work on higher order contracts. That's a systematic way to create wrappers around functions that check the input and output values against a contract. For example, the contract A -> B applied to a function f wraps f with a check on the input and output. The A and B itself can be primitive contracts such as Int which just checks if a value is an int, or function contracts, so that you can have (Int -> Int) -> Int. The neat thing about contracts is that they can tell you whose fault it is if something goes wrong. If you wrap f with the contract (Int -> Int) -> Int, and you call f(g) with some function g, then the contract will ensure that f can only pass Int to g, and if it passes something else the contract will tell you that's f's fault. On the other hand, if g returns something that's not an Int, then the contract can tell you that's g's fault. In general for some value v wrapped with a contract C evaluated in a context E(v wrapped with C), the contract can tell you if it's v's fault if something goes wrong, or E's fault. This is very helpful if v is a module and E is code that uses the module: the contracts can tell you if the module is at fault or whether the user of the module is at fault.

Your next step on this journey is to learn a LISP.

Actually the source for this thought process was going to Racket School and seeing them explain Turnstile and typechecking via syntax-parse and compile time functions :D. It's a fascinating idea but I'm skeptical of the whole "every library is a language" philosophy of Racket scaling. Bugs like infinite compile times are fine when you're writing all the code, but it breaks down when you have libraries.

As opposed to the current state of the art in more "normal" languages where bugs result in infinite loops / deadlocks at runtime.

I've imported many a library which deadlocked, crashed, or infinite looped. In reality it's not that big a deal; it happens and you figure out the bug.

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