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</haskell>
 
</haskell>
   
These functions perform the same operation, however, the former is more
+
These functions perform the same operation, however, the former is more compact, and is considered cleaner. This is closely related to function pipelines (and to [http://www.vex.net/~trebla/weblog/pointfree.html unix shell scripting]): it is clearer to write <hask>let fn = f . g . h</hask> than to write <hask>let fn x = f (g (h x))</hask>.
compact, and is considered cleaner. This is closely related to function
 
pipelines (and to [http://www.vex.net/~trebla/weblog/pointfree.html unix shell scripting]
 
): it is clearer to write <hask>let fn = f . g . h</hask> than to
 
write <hask>let fn x = f (g (h x))</hask>.
 
   
 
This style is particularly useful when deriving efficient programs by
 
This style is particularly useful when deriving efficient programs by
calculation, but it is good discipline in general. It helps the writer
+
calculation and, in general, constitutes good discipline. It helps the writer
 
(and reader) think about composing functions (high level), rather than
 
(and reader) think about composing functions (high level), rather than
 
shuffling data (low level).
 
shuffling data (low level).
Line 53: Line 53:
 
== But pointfree has more points! ==
 
== But pointfree has more points! ==
   
A common misconception is that the 'points' of pointfree style are the
+
A common misconception is that the 'points' of pointfree style are the <hask>(.)</hask> operator (function composition, as an ASCII symbol), which uses the same identifier as the decimal point. This is wrong. The term originated in topology, a branch of mathematics which works with spaces composed of points, and functions between those spaces. So a 'points-free' definition of a function is one which does not explicitly mention the points (values) of the space on which the function acts. In Haskell, our 'space' is some type, and 'points' are values. In the declaration
<hask>(.)</hask> operator (function composition, as an ASCII symbol),
 
which uses the same identifier as the decimal point. This is wrong. The
 
term originated in topology, a branch of mathematics which works with
 
spaces composed of points, and functions between those spaces. So a
 
'points-free' definition of a function is one which does not explicitly
 
mention the points (values) of the space on which the function acts. In
 
Haskell, our 'space' is some type, and 'points' are values. In the
 
declaration:
 
 
<haskell>
 
<haskell>
 
f x = x + 1
 
f x = x + 1
 
</haskell>
 
</haskell>
we define the function <hask>f</hask> in terms of its action on an
+
we define the function <hask>f</hask> in terms of its action on an arbitrary point <hask>x</hask>. Contrast this with the points-free version:
arbitrary point <hask>x</hask>. Contrast this with the points-free
 
version:
 
 
<haskell>
 
<haskell>
 
f = (+ 1)
 
f = (+ 1)
Line 65: Line 65:
 
== Background ==
 
== Background ==
   
To find out more about this style, search for Squiggol and the
+
To find out more about this style, search for Squiggol and the Bird-Meertens Formalism, a style of functional programming by calculation that was developed by [http://web.comlab.ox.ac.uk/oucl/work/richard.bird/publications.html Richard Bird], [http://www.kestrel.edu/home/people/meertens/ Lambert Meertens], and others at Oxford University. [http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/ Jeremy Gibbons] has also written a number of papers about the topic, which are cited below.
Bird-Meertens Formalism, a style of functional programming by
 
calculation that was developed by [http://web.comlab.ox.ac.uk/oucl/work/richard.bird/publications.html Richard Bird], [http://www.kestrel.edu/home/people/meertens/ Lambert Meertens], and
 
others at Oxford University. [http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/ Jeremy Gibbons] has also written a number of
 
papers about the topic, which are cited below.
 
   
== Tool Support ==
+
== Tool support ==
   
 
Thomas Yaeger has
 
Thomas Yaeger has
 
[http://www.cse.unsw.edu.au/~dons/code/lambdabot/Plugins/Pl/ written] a
 
[http://www.cse.unsw.edu.au/~dons/code/lambdabot/Plugins/Pl/ written] a
[http://www.cse.unsw.edu.au/~dons/lambdabot.html Lambdabot]
+
[http://haskell.org/haskellwiki/Lambdabot Lambdabot]
 
plugin to automatically convert a large subset of Haskell expressions to
 
plugin to automatically convert a large subset of Haskell expressions to
 
pointfree form. This tool has made it easier to use the more abstract
 
pointfree form. This tool has made it easier to use the more abstract
 
pointfree encodings (as it saves some mental gymnastics on the part of
 
pointfree encodings (as it saves some mental gymnastics on the part of
the programmer). You can experiment with this in the [[IRC channel|Haskell IRC channel]].
+
the programmer). You can experiment with this in the [[IRC channel|Haskell IRC channel]]. A stand-alone command-line version is available at [http://hackage.haskell.org/package/pointfree HackageDB] (package pointfree).
   
The @pl (point-less) plugin is rather infamous for using the <hask>(-> a)</hask>
+
The @pl (point-less) plugin is rather infamous for using the <hask>(->) a</hask> [[Monad|monad]] to obtain concise code. It also makes use of [[Arrow|Arrows]]. It also sometimes produces (amusing) code blow ups with the
[[Monad|monad]] to obtain concise code. It also makes use of [[Arrow|Arrows]].
+
<hask>(.)</hask> operator.
It also sometimes produces (amusing) code blow ups with the
+
<hask>(.)</hask> operator.
+
Recently, @unpl has been written, which (attempts) to unscramble @pl-ified code. It also has a [http://hackage.haskell.org/package/pointful stand-alone command-line version] (package pointful).
   
 
A transcript:
 
A transcript:
   
 
<haskell>
 
<haskell>
> @pl \x y -> x y
+
> pl \x y -> x y
 
id
 
id
   
> @pl \x y -> x + 1
+
> unpl id
  +
(\ a -> a)
  +
  +
> pl \x y -> x + 1
 
const . (1 +)
 
const . (1 +)
   
> @pl \v1 v2 -> sum (zipWith (*) v1 v2)
+
> unpl const . (1 +)
  +
(\ e _ -> 1 + e)
  +
  +
> pl \v1 v2 -> sum (zipWith (*) v1 v2)
 
(sum .) . zipWith (*)
 
(sum .) . zipWith (*)
   
> @pl \x y z -> f (g x y z)
+
> unpl (sum .) . zipWith (*)
  +
(\ d g -> sum (zipWith (*) d g))
  +
  +
> pl \x y z -> f (g x y z)
 
((f .) .) . g
 
((f .) .) . g
   
> @pl \x y z -> f (g x y) z
+
> unpl ((f .) .) . g
  +
(\ e j m -> f (g e j m))
  +
  +
> pl \x y z -> f (g x y) z
 
(f .) . g
 
(f .) . g
   
> @pl \x y z -> f z (g x y)
+
> unpl (f .) . g
  +
(\ d i -> f (g d i))
  +
  +
> pl \x y z -> f z (g x y)
 
(flip f .) . g
 
(flip f .) . g
   
> @pl \(a,b) -> (b,a)
+
> unpl (flip f .) . g
  +
(\ i l c -> f c (g i l))
  +
  +
> pl \(a,b) -> (b,a)
 
uncurry (flip (,))
 
uncurry (flip (,))
   
> @pl f a b = b a
+
> pl f a b = b a
 
f = flip id
 
f = flip id
   
> @pl \ x -> x * x
+
> pl \ x -> x * x
 
join (*)
 
join (*)
 
 
> @pl \a b -> a:b:[]
+
> pl \a b -> a:b:[]
 
(. return) . (:)
 
(. return) . (:)
   
> @pl \x -> x+x+x
+
> pl \x -> x+x+x
 
(+) =<< join (+)
 
(+) =<< join (+)
   
> @pl \a b -> Nothing
+
> pl \a b -> Nothing
 
const (const Nothing)
 
const (const Nothing)
   
> @pl \(a,b) -> (f a, g b)
+
> pl \(a,b) -> (f a, g b)
 
f *** g
 
f *** g
   
> @pl \f g h x -> f x `h` g x
+
> pl \f g h x -> f x `h` g x
 
flip . (ap .) . flip (.)
 
flip . (ap .) . flip (.)
   
> \x y -> x . f . y
+
> pl \x y -> x . f . y
 
(. (f .)) . (.)
 
(. (f .)) . (.)
   
> @pl \f xs -> xs >>= return . f
+
> pl \f xs -> xs >>= return . f
 
fmap
 
fmap
   
> @pl \h f g x -> f x `h` g x
+
> pl \h f g x -> f x `h` g x
 
liftM2
 
liftM2
   
> @pl \f a b c d -> f b c d a
+
> pl \f a b c d -> f b c d a
 
flip . ((flip . (flip .)) .)
 
flip . ((flip . (flip .)) .)
   
> @pl \a (b,c) -> a c b
+
> pl \a (b,c) -> a c b
 
(`ap` snd) . (. fst) . flip
 
(`ap` snd) . (. fst) . flip
   
> @pl \x y -> compare (f x) (f y)
+
> pl \x y -> compare (f x) (f y)
 
((. f) . compare .)
 
((. f) . compare .)
 
</haskell>
 
</haskell>
   
For many many more examples, google for the results of '@pl' in the
+
For many many more examples, google for the results of '@pl' in the [[IRC_channel|#haskell]] logs. (Or join #haskell on FreeNode and try it yourself!) It can, of course, get out of hand:
[[IRC_channel|#haskell] logs. (Or join #haskell on FreeNode and try it
 
yourself!) It can, of course, get out of hand:
 
   
 
<haskell>
 
<haskell>
> @pl \(a,b) -> a:b:[]
+
> pl \(a,b) -> a:b:[]
 
uncurry ((. return) . (:))
 
uncurry ((. return) . (:))
   
> @pl \a b c -> a*b+2+c
+
> pl \a b c -> a*b+2+c
 
((+) .) . flip flip 2 . ((+) .) . (*)
 
((+) .) . flip flip 2 . ((+) .) . (*)
   
> @pl \f (a,b) -> (f a, f b)
+
> pl \f (a,b) -> (f a, f b)
 
(`ap` snd) . (. fst) . (flip =<< (((.) . (,)) .))
 
(`ap` snd) . (. fst) . (flip =<< (((.) . (,)) .))
   
> @pl \f g (a,b) -> (f a, g b)
+
> pl \f g (a,b) -> (f a, g b)
 
flip flip snd . (ap .) . flip flip fst . ((.) .) . flip . (((.) . (,)) .)
 
flip flip snd . (ap .) . flip flip fst . ((.) .) . flip . (((.) . (,)) .)
  +
  +
> unpl flip flip snd . (ap .) . flip flip fst . ((.) .) . flip . (((.) . (,)) .)
  +
(\ aa f ->
  +
(\ p w -> ((,)) (aa (fst p)) (f w)) >>=
  +
\ ao -> snd >>= \ an -> return (ao an))
 
</haskell>
 
</haskell>
   
 
== Combinator discoveries ==
 
== Combinator discoveries ==
   
Some fun combinators have been found via @pl. Here we list some of the
+
Some fun combinators have been found via @pl. Here we list some of the best:
best:
 
   
 
=== The owl ===
 
=== The owl ===
   
 
<haskell>
 
<haskell>
(.)$(.))
+
((.)$(.))
 
</haskell>
 
</haskell>
   
The owl has type <hask>(a -> b -> c) -> a -> (a1 -> b) -> a1 ->
+
The owl has type <hask>(a -> b -> c) -> a -> (a1 -> b) -> a1 -> c</hask>, and in pointful style can be written as <hask> f a b c d = a b (c d)</hask>.
c</hask>, and in pointful style can be written as <hask> f a b c d = a b
 
(c d)</hask>.
 
   
 
Example
 
Example
Line 178: Line 183:
 
> ((.)$(.)) (==) 1 (1+) 0
 
> ((.)$(.)) (==) 1 (1+) 0
 
True
 
True
  +
</haskell>
  +
  +
=== Dot ===
  +
  +
<haskell>
  +
dot = ((.).(.))
  +
  +
dot :: (b -> c) -> (a -> a1 -> b) -> a -> a1 -> c
  +
</haskell>
  +
  +
Example:
  +
  +
<haskell>
  +
sequence `dot` replicate ==
  +
(sequence .) . replicate ==
  +
replicateM
  +
  +
(=<<) == join `dot` fmap
 
</haskell>
 
</haskell>
   
 
=== Swing ===
 
=== Swing ===
  +
  +
-- Note: @pl had nothing to do with the invention of this combinator. I constructed it by hand after noticing a common pattern. -- Cale
   
 
<haskell>
 
<haskell>
 
swing :: (((a -> b) -> b) -> c -> d) -> c -> a -> d
 
swing :: (((a -> b) -> b) -> c -> d) -> c -> a -> d
 
swing = flip . (. flip id)
 
swing = flip . (. flip id)
swing f c a = f ($ c) a
+
swing f = flip (f . runCont . return)
  +
swing f c a = f ($ a) c
  +
</haskell>
  +
  +
Some examples of use:
  +
  +
<haskell>
  +
swing map :: forall a b. [a -> b] -> a -> [b]
  +
swing any :: forall a. [a -> Bool] -> a -> Bool
  +
swing foldr :: forall a b. b -> a -> [a -> b -> b] -> b
  +
swing zipWith :: forall a b c. [a -> b -> c] -> a -> [b] -> [c]
  +
swing find :: forall a. [a -> Bool] -> a -> Maybe (a -> Bool)
  +
-- applies each of the predicates to the given value, returning the first predicate which succeeds, if any
  +
swing partition :: forall a. [a -> Bool] -> a -> ([a -> Bool], [a -> Bool])
 
</haskell>
 
</haskell>
   
Line 200: Line 225:
 
</haskell>
 
</haskell>
   
== Obfuscation ==
+
[[/Combine|An actually useful example]], numbering lines of a file.
  +
  +
== Problems with pointfree ==
   
 
Point-free style can (clearly) lead to [[Obfuscation]] when used unwisely.
 
Point-free style can (clearly) lead to [[Obfuscation]] when used unwisely.
Line 208: Line 233:
 
arguments) go missing.
 
arguments) go missing.
   
Perhaps this is why pointfree style is sometimes (often?) referred to as
+
Point-free style often times leads to code which is difficult to modify. A function written in a pointfree style may have to be radically changed to make minor changes in functionality. This is because the function becomes more complicated than a composition of lambdas and other functions, and compositions must be changed to application for a pointful function.
  +
  +
Perhaps these are why pointfree style is sometimes (often?) referred to as
 
''pointless style''.
 
''pointless style''.
   
Line 222: Line 247:
 
* http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/index.html#radix
 
* http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/index.html#radix
   
This style underlies a lot of expert Haskeller's intuitions.
+
This style underlies a lot of expert Haskeller's intuitions. A rather infamous paper (for all the cute symbols) is Erik Meijer et. al's:
A rather infamous paper (for all the cute symbols) is Erik Meijer et. al's:
 
   
 
* Functional Programming with Bananas, Lenses, and Barbed Wire, http://wwwhome.cs.utwente.nl/~fokkinga/mmf91m.ps.
 
* Functional Programming with Bananas, Lenses, and Barbed Wire, http://wwwhome.cs.utwente.nl/~fokkinga/mmf91m.ps.
Line 231: Line 256:
 
* The Squiggolist, ed Johan Jeuring, published irregularly by CWI Amsterdam.
 
* The Squiggolist, ed Johan Jeuring, published irregularly by CWI Amsterdam.
   
[http://wiki.di.uminho.pt/wiki/bin/view/Alcino/PointlessHaskell Pointless Haskell] is a library for point-free programming with recursion patterns defined as hylomorphisms. It also allows the visualization of the intermediate data structure of the hylomorphisms with GHood. This feature together with the DrHylo tool allows us to easily visualize recursion trees of Haskell functions.
+
[http://wiki.di.uminho.pt/twiki/bin/view/Personal/Alcino/PointlessHaskell Pointless Haskell] is a library for point-free programming with recursion patterns defined as hylomorphisms. It also allows the visualization of the intermediate data structure of the hylomorphisms with GHood. This feature together with the DrHylo tool allows us to easily visualize recursion trees of Haskell functions. [http://wiki.di.uminho.pt/wiki/pub/Ze/Bic/report.pdf Haskell Manipulation] by Jose Miguel Paiva Proenca discusses this tool based approach to re-factoring.
   
 
This project is written by [http://www.di.uminho.pt/~mac/ Manuel Alcino Cunha], see his homepage for more related materials on the topic.
 
This project is written by [http://www.di.uminho.pt/~mac/ Manuel Alcino Cunha], see his homepage for more related materials on the topic.
Line 241: Line 266:
   
 
Are there pointfree approaches to [[relational algebra]]?
 
Are there pointfree approaches to [[relational algebra]]?
See [http://www.di.uminho.pt/~jno/ps/_.pdf First Steps in Pointfree Functional Dependency Theory] written by Jos Nuno Oliveira. A concise and deep approach. See also [http://www.di.uminho.pt/~jno/html/ the author's homepage] and also [http://www.di.uminho.pt/~jno/html/jnopub.html his many other papers] -- many materials related to this topic can be found there.
+
See [http://www.di.uminho.pt/~jno/ps/_.pdf First Steps in Pointfree Functional Dependency Theory] written by José Nuno Oliveira. A concise and deep approach. See also [http://www.di.uminho.pt/~jno/html/ the author's homepage] and also [http://www.di.uminho.pt/~jno/html/jnopub.html his many other papers] -- many materials related to this topic can be found there.
   
 
[[Category:Idioms]]
 
[[Category:Idioms]]

Latest revision as of 14:44, 5 June 2011

Contents


Pointfree Style

It is very common for functional programmers to write functions as a composition of other functions, never mentioning the actual arguments they will be applied to. For example, compare:

 sum = foldr (+) 0

with:

 sum' xs = foldr (+) 0 xs
These functions perform the same operation, however, the former is more compact, and is considered cleaner. This is closely related to function pipelines (and to unix shell scripting): it is clearer to write
let fn = f . g . h
than to write
let fn x = f (g (h x))
.

This style is particularly useful when deriving efficient programs by calculation and, in general, constitutes good discipline. It helps the writer (and reader) think about composing functions (high level), rather than shuffling data (low level).

It is a common experience when rewriting expressions in pointfree style to derive more compact, clearer versions of the code -- explicit points often obscure the underlying algorithm.

Point-free map fusion:

 foldr f e . map g == foldr (f . g) e

versus pointful map fusion:

 foldr f e . map g == foldr f' e
      where f' a b = f (g a) b

Some more examples:

 -- point-wise, and point-free member
 mem, mem' :: Eq a => a -> [a] -> Bool
 
 mem x lst = any (== x) lst
 mem'      = any . (==)

[edit] 1 But pointfree has more points!

A common misconception is that the 'points' of pointfree style are the
(.)
operator (function composition, as an ASCII symbol), which uses the same identifier as the decimal point. This is wrong. The term originated in topology, a branch of mathematics which works with spaces composed of points, and functions between those spaces. So a 'points-free' definition of a function is one which does not explicitly mention the points (values) of the space on which the function acts. In Haskell, our 'space' is some type, and 'points' are values. In the declaration
 f x = x + 1
we define the function
f
in terms of its action on an arbitrary point
x
. Contrast this with the points-free version:
 f = (+ 1)

where there is no mention of the value on which the function is acting.

[edit] 2 Background

To find out more about this style, search for Squiggol and the Bird-Meertens Formalism, a style of functional programming by calculation that was developed by Richard Bird, Lambert Meertens, and others at Oxford University. Jeremy Gibbons has also written a number of papers about the topic, which are cited below.

[edit] 3 Tool support

Thomas Yaeger has written a Lambdabot plugin to automatically convert a large subset of Haskell expressions to pointfree form. This tool has made it easier to use the more abstract pointfree encodings (as it saves some mental gymnastics on the part of the programmer). You can experiment with this in the Haskell IRC channel. A stand-alone command-line version is available at HackageDB (package pointfree).

The @pl (point-less) plugin is rather infamous for using the
(->) a
monad to obtain concise code. It also makes use of Arrows. It also sometimes produces (amusing) code blow ups with the
(.)
operator.

Recently, @unpl has been written, which (attempts) to unscramble @pl-ified code. It also has a stand-alone command-line version (package pointful).

A transcript:

 > pl \x y -> x y
 id
 
 > unpl id
 (\ a -> a)
 
 > pl \x y -> x + 1
 const . (1 +)
 
 > unpl const . (1 +)
 (\ e _ -> 1 + e)
 
 > pl \v1 v2 -> sum (zipWith (*) v1 v2)
 (sum .) . zipWith (*)
 
 > unpl (sum .) . zipWith (*)
 (\ d g -> sum (zipWith (*) d g))
 
 > pl \x y z -> f (g x y z)
 ((f .) .) . g
 
 > unpl ((f .) .) . g
 (\ e j m -> f (g e j m))
 
 > pl \x y z -> f (g x y) z
 (f .) . g
 
 > unpl (f .) . g
 (\ d i -> f (g d i))
 
 > pl \x y z -> f z (g x y)
 (flip f .) . g
 
 > unpl (flip f .) . g
 (\ i l c -> f c (g i l))
 
 > pl \(a,b) -> (b,a)
 uncurry (flip (,))
 
 > pl f a b = b a
 f = flip id
 
 > pl \ x -> x * x
 join (*)
 
 > pl \a b -> a:b:[]
 (. return) . (:)
 
 > pl \x -> x+x+x
 (+) =<< join (+)
 
 > pl \a b -> Nothing
 const (const Nothing)
 
 > pl \(a,b) -> (f a, g b)
 f *** g
 
 > pl \f g h x -> f x `h` g x
 flip . (ap .) . flip (.)
 
 > pl \x y -> x . f . y
 (. (f .)) . (.)
 
 > pl \f xs -> xs >>= return . f
 fmap
 
 > pl \h f g x -> f x `h` g x
 liftM2
 
 > pl \f a b c d -> f b c d a
 flip . ((flip . (flip .)) .)
 
 > pl \a (b,c) -> a c b
 (`ap` snd) . (. fst) . flip
 
 > pl \x y -> compare (f x) (f y)
 ((. f) . compare .)

For many many more examples, google for the results of '@pl' in the #haskell logs. (Or join #haskell on FreeNode and try it yourself!) It can, of course, get out of hand:

 > pl \(a,b) -> a:b:[]
 uncurry ((. return) . (:))
 
 > pl \a b c -> a*b+2+c
 ((+) .) . flip flip 2 . ((+) .) . (*)
 
 > pl \f (a,b) -> (f a, f b)
 (`ap` snd) . (. fst) . (flip =<< (((.) . (,)) .))
 
 > pl \f g (a,b) -> (f a, g b)
 flip flip snd . (ap .) . flip flip fst . ((.) .) . flip . (((.) . (,)) .)
 
 > unpl flip flip snd . (ap .) . flip flip fst . ((.) .) . flip . (((.) . (,)) .)
 (\ aa f ->
   (\ p w -> ((,)) (aa (fst p)) (f w)) >>=
      \ ao -> snd >>= \ an -> return (ao an))

[edit] 4 Combinator discoveries

Some fun combinators have been found via @pl. Here we list some of the best:

[edit] 4.1 The owl

((.)$(.))
The owl has type
(a -> b -> c) -> a -> (a1 -> b) -> a1 -> c
, and in pointful style can be written as
 f a b c d = a b (c d)
.

Example

>  ((.)$(.)) (==) 1 (1+) 0
True

[edit] 4.2 Dot

dot = ((.).(.))
 
dot :: (b -> c) -> (a -> a1 -> b) -> a -> a1 -> c

Example:

sequence `dot` replicate == 
(sequence .) . replicate ==
replicateM
 
(=<<) == join `dot` fmap

[edit] 4.3 Swing

-- Note: @pl had nothing to do with the invention of this combinator. I constructed it by hand after noticing a common pattern. -- Cale

swing :: (((a -> b) -> b) -> c -> d) -> c -> a -> d
swing = flip . (. flip id)
swing f = flip (f . runCont . return)
swing f c a = f ($ a) c

Some examples of use:

swing map :: forall a b. [a -> b] -> a -> [b]
swing any :: forall a. [a -> Bool] -> a -> Bool
swing foldr :: forall a b. b -> a -> [a -> b -> b] -> b
swing zipWith :: forall a b c. [a -> b -> c] -> a -> [b] -> [c]
swing find :: forall a. [a -> Bool] -> a -> Maybe (a -> Bool)
   -- applies each of the predicates to the given value, returning the first predicate which succeeds, if any
swing partition :: forall a. [a -> Bool] -> a -> ([a -> Bool], [a -> Bool])

[edit] 4.4 Squish

f >>= a . b . c =<< g

Example:

(readFile y >>=) . ((a . b) .) . c =<< readFile x

An actually useful example, numbering lines of a file.

[edit] 5 Problems with pointfree

Point-free style can (clearly) lead to Obfuscation when used unwisely. As higher-order functions are chained together, it can become harder to mentally infer the types of expressions. The mental cues to an expression's type (explicit function arguments, and the number of arguments) go missing.

Point-free style often times leads to code which is difficult to modify. A function written in a pointfree style may have to be radically changed to make minor changes in functionality. This is because the function becomes more complicated than a composition of lambdas and other functions, and compositions must be changed to application for a pointful function.

Perhaps these are why pointfree style is sometimes (often?) referred to as pointless style.

[edit] 6 References

One early reference is

  • Backus, J. 1978. "Can Programming Be Liberated from the von Neumann Style? A Functional Style and Its Algebra of Programs," Communications of the Association for Computing Machinery 21:613-641.

which appears to be available (as a scan) at http://www.stanford.edu/class/cs242/readings/backus.pdf

A paper specifically about point-free style:

This style underlies a lot of expert Haskeller's intuitions. A rather infamous paper (for all the cute symbols) is Erik Meijer et. al's:

Squiggol, and the Bird-Meertens Formalism:

Pointless Haskell is a library for point-free programming with recursion patterns defined as hylomorphisms. It also allows the visualization of the intermediate data structure of the hylomorphisms with GHood. This feature together with the DrHylo tool allows us to easily visualize recursion trees of Haskell functions. Haskell Manipulation by Jose Miguel Paiva Proenca discusses this tool based approach to re-factoring.

This project is written by Manuel Alcino Cunha, see his homepage for more related materials on the topic. An extended verson of his paper Point-free Programming with Hylomorphisms can be found here.

[edit] 7 Other areas

Combinatory logic and also Recursive function theory can be said in some sense pointfree.

Are there pointfree approaches to relational algebra? See First Steps in Pointfree Functional Dependency Theory written by José Nuno Oliveira. A concise and deep approach. See also the author's homepage and also his many other papers -- many materials related to this topic can be found there.