Re: [Haskell-cafe] Reifying case expressions [was: Re: On stream processing, and a new release of timeplot coming]

26 Dec 2011


      2011/12/26 Gábor Lehel 
...
On Sun, Dec 25, 2011 at 9:19 PM, Eugene Kirpichov 
wrote:
...
Hello Heinrich,
Thanks, that's sure some food for thought!
A few notes:
* This is indeed analogous to Iteratees. I tried doing the same with
Iteratees but failed, so I decided to put together something simple of my
own.
* The Applicative structure over this stuff is very nice. I was thinking,
what structure to put on - and Applicative seems the perfect fit. It's
also
possible to implement Arrows - but I once tried and failed; however, I
was
trying that for a more complex stream transformer datatype (a hybrid of
Iteratee and Enumerator).
* StreamSummary is trivially a bifunctor. I actually wanted to make it an
instance of Bifunctor, but it was in the category-extras package and I
hesitated to reference this giant just for this purpose :) Probably
bifunctors should be in prelude.
Edward Kmett has been splitting that up into a variety of smaller
packages, for instance:
http://hackage.haskell.org/package/bifunctors
Actually it's not a bifunctor - it's a functor in one argument and
contrafunctor in the other.
Is there a name for such a structure?
...
...
* Whereas StreamSummary a r abstracts deconstruction of lists, the dual
datatype (StreamSummary a r ->) abstracts construction; however I just
now
(after looking at your first definition of length) understood that it is
trivially isomorphic to the regular list datatype - you just need to be
non-strict in the state - listify :: ListTo a [a] = CaseOf [] (\x -> fmap
(x:) listify). So you don't need functions of the form (forall r .
ListTo a
r -> ListTo b r) - you just need (ListTo b [a]). This is a revelation for
me.
On Sun, Dec 25, 2011 at 2:25 PM, Heinrich Apfelmus
 wrote:
...
Eugene Kirpichov wrote:
...
In the last couple of days I completed my quest of making my graphing
utility timeplot ( http://jkff.info/software/timeplotters ) not load
...
...
whole input dataset into memory and consequently be able to deal with
datasets of any size, provided however that the amount of data to
*draw*
is
not so large. On the go it also got a huge speedup - previously
visualizing
a cluster activity dataset with a million events took around 15 minutes
and
a gig of memory, now it takes 20 seconds and 6 Mb max residence.
(I haven't yet uploaded to hackage as I have to give it a bit more
testing)
The refactoring involved a number of interesting programming patterns
that
I'd like to share with you and ask for feedback - perhaps something can
be
simplified.
The source is at http://github.com/jkff/timeplot
The datatype of incremental computations is at
https://github.com/jkff/timeplot/blob/master/Tools/TimePlot/Incremental.hs.
...
Strictness is extremely important here - the last memory leak I
eliminated
was lack of bang patterns in teeSummary.
Your  StreamSummary  type has a really nice interpretation: it's a
reification of  case  expressions.
For instance, consider the following simple function from lists to
integers
length :: [a] -> Int
   length xs = case xs of
       []     -> 0
       (y:ys) -> 1 + length ys
We want to reify the case expression as constructor of a data type. What
type should it have? Well, a case expression maps a list  xs  to a
result,
here of type Int, via two cases: the first case gives a result and the
other
maps a value of type  a  to a function from lists to results again. This
explanation was probably confusing, so I'll just go ahead and define a
data
type that represents functions from lists  [a] to some result of type  r
data ListTo a r = CaseOf r (a -> ListTo a r)
interpret :: ListTo a r -> ([a] -> r)
   interpret (CaseOf nil cons) xs =
       case xs of
           []     -> nil
           (y:ys) -> interpret (cons y) ys
As you can see, we are just mapping each  CaseOf  constructor back to a
built-in case expression.
Likewise, each function from lists can be represented in terms of our
new
data type: simply replace all built-in case expressions with the new
constructor
length' :: ListTo a Int
   length' = CaseOf
       (0)
       (\x -> fmap (1+) length')
length = interpret length'
The CaseOf may look a bit weird, but it's really just a straightforward
translation of the case expression you would use to define the function
 go
 instead.
Ok, this length function is really inefficient because it builds a huge
expression of the form  (1+(1+...)). Let's implement a strict variant
instead
lengthL :: ListTo a Int
   lengthL = go 0
       where
       go !n = CaseOf (n) (\x -> go (n+1))
While we're at it, let's translate two more list functions
foldL' :: (b -> a -> b) -> b -> ListTo a b
   foldL' f b = Case b (\a -> foldL' f $! f b a)
sumL    :: ListTo Int Int
   sumL    = foldL' (\b a -> a+b) 0
And now we can go for the point of this message: unlike ordinary
functions
from lists, we can compose these in lock-step! In particular, the
following
applicative instance
instance Applicative (ListTo a) where
       pure b = CaseOf b (const $ pure b)
       (CaseOf f fs) <*> (CaseOf x xs) =
           CaseOf (f x) (\a -> fs a <*> xs a)
allows us to write a function
average :: ListTo Int Double
   average = divide <$> sumL <*> lengthL
       where
       divide a b = fromIntegral a / fromIntegral b
that runs in constant space! Why does this work? Well, since we can now
inspect case expressions, we can choose to evaluate them in lock-step,
essentially computing  sum  and  length  with just one pass over the
input
list. Remember that the original definition
average xs = sum xs / length xs
has a space leak because the input list xs is being shared.
Remarks:
1. Reified case expressions are, of course, the same thing as Iteratees,
modulo chunking and weird naming.
2. My point is topped by scathing irony: if Haskell had a form of
*partial
evaluation*, we could write applicative combinators for *ordinary*
functions
 [a] -> r  and express  average  in constant space.  In other words,
the
partial
...
evaluation would make it unnecessary to reify case expressions for the
purpose of controlling performance / space leaks.
Best regards,
Heinrich Apfelmus
--
http://apfelmus.nfshost.com
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--
Eugene Kirpichov
Principal Engineer, Mirantis Inc. http://www.mirantis.com/
Editor, http://fprog.ru/
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Eugene Kirpichov
Principal Engineer, Mirantis Inc. http://www.mirantis.com/
Editor, http://fprog.ru/