Re: Eta Reduction

Can we throw some whole program partial evaluation with a termination decision oracle into the mix? On Tuesday, April 1, 2014, John Lato wrote:

I think this is a great idea and should become a top priority. I would probably start by switching to a type-class-based seq, after which perhaps the next step forward would become more clear.

John L. On Apr 1, 2014 2:54 AM, "Dan Doel" javascript:_e(%7B%7D,'cvml','dan.doel@gmail.com');> wrote:

...

In the past year or two, there have been multiple performance problems in various areas related to the fact that lambda abstraction is not free, though we tend to think of it as so. A major example of this was deriving of Functor. If we were to derive Functor for lists, we would end up with something like:

instance Functor [] where fmap _ [] = [] fmap f (x:xs) = f x : fmap (\y -> f y) xs

This definition is O(n^2) when fully evaluated,, because it causes O(n) eta expansions of f, so we spend time following indirections proportional to the depth of the element in the list. This has been fixed in 7.8, but there are other examples. I believe lens, [1] for instance, has some stuff in it that works very hard to avoid this sort of cost; and it's not always as easy to avoid as the above example. Composing with a newtype wrapper, for instance, causes an eta expansion that can only be seen as such at the core level.

The obvious solution is: do eta reduction. However, this is not operationally sound currently. The problem is that seq is capable of telling the difference between the following two expressions:

undefined \x -> undefined x

The former causes seq to throw an exception, while the latter is considered defined enough to not do so. So, if we eta reduce, we can cause terminating programs to diverge if they make use of this feature.

Luckily, there is a solution.

Semantically one would usually identify the above two expressions. While I do believe one could construct a semantics that does distinguish them, it is not the usual practice. This suggests that there is a way to not distinguish them, perhaps even including seq. After all, the specification of seq is monotone and continuous regardless of whether we unify ⊥ with \x -> ⊥ x or insert an extra element for the latter.

The currently problematic case is function spaces, so I'll focus on it. How should:

seq : (a -> b) -> c -> c

act? Well, other than an obvious bottom, we need to emit bottom whenever our given function is itself bottom at every input. This may first seem like a problem, but it is actually quite simple. Without loss of generality, let us assume that we can enumerate the type a. Then, we can feed these values to the function, and check their results for bottom. Conal Elliot has prior art for this sort of thing with his unamb [2] package. For each value x :: a, simply compute 'f x `seq` ()' in parallel, and look for any successes. If we ever find one, we know the function is non-bottom, and we can return our value of c. If we never finish searching, then the function must be bottom, and seq should not terminate, so we have satisfied the specification.

Now, some may complain about the enumeration above. But this, too, is a simple matter. It is well known that Haskell programs are denumerable. So it is quite easy to enumerate all Haskell programs that produce a value, check whether that value has the type we're interested in, and compute said value. All of this can be done in Haskell. Thus, every Haskell type is programatically enumerable in Haskell, and we can use said enumeration in our implementation of seq for function types. I have discussed this with Russell O'Connor [3], and he assures me that this argument should be sufficient even if we consider semantic models of Haskell that contain values not denoted by any Haskell program, so we should be safe there.

The one possible caveat is that this implementation of seq is not operationally uniform across all types, so the current fully polymorphic type of seq may not make sense. But moving back to a type class based approach isn't so bad, and this time we will have a semantically sound backing, instead of just having a class with the equivalent of the current magic function in it.

Once this machinery is in place, we can eta reduce to our hearts' content, and not have to worry about breaking semantics. We'd no longer put the burden on programmers to use potentially unsafe hacks to avoid eta expansions. I apologize for not sketching an implementation of the above algorithm, but I'm sure it should be elementary enough to make it into GHC in the next couple versions. Everyone learns about this type of thing in university computer science programs, no?

Thoughts? Comments? Questions?

Cheers, -- Dan

[1] http://hackage.haskell.org/package/lens [2] http://hackage.haskell.org/package/unamb [3] http://r6.ca/

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