A reckless introduction to Hindley-Milner type inference (2019)
reasonableapproximation.net
reasonableapproximation.net
Proving that a program terminates eventually is a very weak guarantee, since it could be a program that terminates after a million years. This matters mostly for proof languages, where it’s useful to write proofs about functions that you never run. If the function provably terminates, you can assume that whatever value it returns exists, without calculating it.
For a possibly long-running job like a SQL query, a static guarantee that it terminates eventually is almost certainly not what you want. You’re going to need to test its performance, and ideally have a timeout and a way of handling the timeout. If you want to get fancy, a progress bar and cancel button might be nice.
Edit:
Compilers need to run fast for the programs we actually write. A type system that possibly doesn’t terminate under unusual conditions isn’t that big a deal, as long as people don’t write code like that. Hopefully you get a decent error message.
Compare with memory use. We don’t want compilers to use excessive amounts of memory, but don’t need static guarantees about memory use.
At work we use a home-grown Haskell-like language with “row types” which can be split into disjoint sets of fields in order to represent joins and the like.
It works well in most cases, but it’s very easy to forget a type annotation and trigger unmanageable build-times. Often times, it is not obvious where something needs to be annotated. Sure, we can act as human time-outs, but the productivity loss can be substantial, especially for a newcomer to the language.
There's no practical difference between exponential and infinite time, you need to do this for bad exponential time cases anyways.
Many langauges type systems (Java, Typescript, C++, Rust, ...) really aren't guaranteed to terminate in the absence of a timeout either. They're (the type systems) turing complete, you can write infinite loops that the compiler can't know are infinite loops. This makes no practical difference relative to languages which "merely" have exponential time type systems, because with an exponential time type system you're alread accepting the risk that typechecking might not terminate before the heat death of the universe.
A termination guarantee for type checking doesn’t really help.
In practical algorithms of significance, we can often do much better than this, proving both termination, and the time complexity of that termination. For instance, a DFA of a regular language will terminate in O(n) of the length of the haystack, and we also have a bound for the time it takes to construct that DFA.
Sometimes, as you're alluding to, we know something terminates, but the time complexity is impractical: exhaustive naïve search for Traveling Salesman will, in principle, halt. This is, itself, useful information.
The article claims (I believe correctly) not just that H-M inference halts, but that it does so in "near-linear" time. That's a much tighter bound than merely proving termination.
However, without a precise cost model of your operations (say, you just know each operation has a bounded time and such they are each O(1 ) but nothing beyond that), that's often the best you can do
There's something called resource aware computation that can do more finer grained bounds on the runtime and memory usage of functions (and other resource usage like stack space so you can prove you always use less than 8MB or whatever number) not only complexity but also the constants. With the right cost model you could model CPU and memory latencies and other things like CPU port usage, cache usage etc, but there's an interplay with the optimizer here (this could even be used to spot performance regressions of compiler codegen even without running any benchmark)
I don't understand why people talk so much about eventual termination.
Having static types is a low cost solution that catches a lot of possible bugs. Purity catches another bunch of bugs. No-nulls catches even more. At this point you can kinda start trusting function names and type signatures to do what they say on the box (often said about Haskell). Likewise, if it compiles, it probably does the right thing.
So where's the next gap in Haskell that lets bugs through? I'd argue that (eventual) termination would be that next step. If you say a function has type a->a (for all a), it only admits one possible implementation... except for bad ones, like 'undefined' or an infinite recursion - both of which termination would catch for you.
To be super-pedantic, it only admits one possible denotation, assuming termination. There are an infinite number of implementations that waste finite time on something useless before returning the argument. We usually don't talk about those, because... Why would you write code that does that? But it's worth keeping in mind as an edge case in this sort of discussion.
Given:
verySilly :: Int -> (a -> a)
verySilly = go . abs . toInteger
where
go n
| n == 1 = id
| even n = go (n `div` 2)
| otherwise = go (n * 3 + 1)
I agree; that part went on so long that I got bored and stopped reading before the article actually got around to saying anything about type inference.
The intro here flags that the author i) is not happy with the post in its current form, ii) was resynthesized from wikipedia "without particularly putting in the effort to check my own understanding", which doesn't inspire confidence.
And then when you do get into the content, stuff looks incorrect, but a non-expert reader, given the caveats in the intro, should be on the lookout for errors but isn't equipped to identify them with confidence. E.g. the very first inference rule included is described as:
> if you have a variable x declared to have some polytype π, and if the monotype μ is a specialisation of π, then x can be judged to have type μ
... but I think this should be "π is a specialization of μ".
Basically, I think this is the kind of material where detail matters, where errors seriously impede understanding, and a resource where the reader/learner has to be on guard second-guessing the author is a bad intro resource. But surely there are better ones out there.
f: ∀A, B. A -> B -> A
Then we have poly type π = "∀A, B. A -> B -> A".
The monotype μ = Str -> Int -> Str is a specialization of π, so we are also permitted to judge that f has type μ.
The type specialization operator "looks backwards", where we write sorta "π ≤ μ" because there's a kind of "subtyping" relationship (though most HM type systems don't actually contain subtyping) - anywhere that a term of type μ is used, a term of type π can be used instead because it can be specialized to μ.
I think this is a good complementary article: https://boxbase.org/entries/2018/mar/5/hindley-milner/ (it also links to three more academic sources for more context/detail)
This is good reading - it demystifies the funny typing rules in their mathematical form. https://legacy-blog.akgupta.ca/blog/2013/05/14/so-you-still-...
(Neither of these tell you how to implement the checker yourself though)
So I'm more confident in the essay than I was at the time. But yeah, getting put on guard by those caveats doesn't seem ideal for intro material. (But if I hadn't had them and I'd gotten something wrong, that would also not have been ideal...)
For writing the type checker, "write you a haskell" that someone else linked was really helpful, and so was a paper called "typing haskell in haskell": https://web.cecs.pdx.edu/~mpj/thih/thih.pdf
Can we reflect on how this person was so wary of reading the article for fear of being misled (instead of just reading it and afterwards reflecting/evaluating) because they were so sure it must be wrong (for whatever reasons) and it turns they themselves were wrong. This is called irony and dunning krueger.
From the article:
> HM type systems can't implement heterogenous lists [...] > [well it does when wrapping the elements in a sum type, but then] > it can only accept types you know about in advance. > [which makes it] a massive pain to work with.
Sorry, I totally disagree... Heterogeneous lists without the sum types are a massive pain! Maybe it is easier for the initial development effort, but if you need to maintain code with heterogeneous lists (or dynamically typed code in general) the pain will come at some point and likely when already in production.
I much rather fix compile errors than runtime errors. And a massive thanks to Hindley and Milner for advancing type systems!
Found their photos in gratitude just now:
Well, if you have an interface that all entries adhere to it is not a heterogeneous list in my book. In Haskell that could be achieve without a sum type wrapper, but with a type class.
Native heterogeneous lists will often be convenient, and sometimes when you've used them they're going to turn out in retrospect to have been a bad idea but sometimes they'll be just fine, especially if they stay fairly locally scoped. I haven't done any serious work in a language that supports them for years, so I don't have a strong opinion about how often each of those cases happens.
But if you decide you want heterogeneous lists, but your language doesn't support them, so you try to implement that with a sum type, then that's basically always going to be a massive pain.
Is this not silently admitting that strong types to some degree have won?
Don't get me wrong: I like Clojure/LISPs/Ruby. But I would not choose them for a new project these days.
(and I do not like JS, which has them too)
But I don't expect that my own experience says much about the language ecosystem in general. I don't particularly have an opinion on whether or not strong types have "won", and I didn't intend to comment on that question.
In Haskell, any time a heterogeneous list turns out to be fine, I expect to be able to model it. Often it'll look like "I'm applying a polymorphic function to every one of these list elements", and then you can either do a sum type (as discussed in the post) or an existential (which doesn't need you to list up front all the types you might use). If the function is "is negative?", it'll look something like (untested)
data SomeNum = forall a. Num a => SomeNum a
isNegative :: SomeNum -> Bool
isNegative (SomeNum n) = n < 0
numbers = [SomeNum (3 :: Int), SomeNum (5.2 :: Double), SomeNum valOfUnexpectedNumericType]
anyNegative = any isNegative numbers
So you can model heterogeneous lists in this case, and it's safer (because you can't put in a value that can't be compared to 0) but also less convenient. Whether that's an improvement or not will depend on the situation.
It's a slightly confusing name, though; it makes me think of a difference list, which seems to be a completely unrelated data structure (basically a rope).
You could construct a basically-equivalent data structure in Haskell, but I think normally you'd use an HList (defined in many places but e.g. https://hackage.haskell.org/package/HList-0.5.2.0/docs/Data-...). I've only occasionally had use for them myself, at any rate I don't think they're convenient for the "apply a function to all these elements" case.
This is the kind of real-world usage where difflists or heterogeneous lists shine. The same technique is used by this library to define type-safe GraphQL resolvers: https://github.com/andreas/ocaml-graphql-server