Difference between revisions of "Pointfree"
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+  __TOC__ 

+  
'''Pointfree Style''' 
'''Pointfree Style''' 

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they will be applied to. For example, compare: 
they will be applied to. For example, compare: 

+  <haskell> 

sum = foldr (+) 0 
sum = foldr (+) 0 

+  </haskell> 

with: 
with: 

+  <haskell> 

sum' xs = foldr (+) 0 xs 
sum' xs = foldr (+) 0 xs 

+  </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: 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 
This style is particularly useful when deriving efficient programs by 

−  calculation, 
+  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 27:  Line 30:  
Pointfree map fusion: 
Pointfree map fusion: 

+  <haskell> 

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

+  </haskell> 

versus pointful map fusion: 
versus pointful map fusion: 

+  <haskell> 

foldr f e . map g == foldr f' e 
foldr f e . map g == foldr f' e 

where f' a b = f (g a) b 
where f' a b = f (g a) b 

+  </haskell> 

Some more examples: 
Some more examples: 

+  <haskell> 

 pointwise, and pointfree member 
 pointwise, and pointfree member 

mem, mem' :: Eq a => a > [a] > Bool 
mem, mem' :: Eq a => a > [a] > Bool 

Line 41:  Line 49:  
mem x lst = any (== x) lst 
mem x lst = any (== x) lst 

mem' = any . (==) 
mem' = any . (==) 

+  </haskell> 

== 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 'pointsfree' 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 

−  ''(.)'' operator (function composition, as an ascii symbol), which 

+  <haskell> 

−  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 'pointsfree' 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 
f x = x + 1 

−  we define the function 'f' in terms of its action on an arbitrary point 'x'. Contrast this with the pointsfree version: 

+  </haskell> 

+  we define the function <hask>f</hask> in terms of its action on an arbitrary point <hask>x</hask>. Contrast this with the pointsfree version: 

+  <haskell> 

f = (+ 1) 
f = (+ 1) 

+  </haskell> 

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

== 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 BirdMeertens 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. 

−  BirdMeertens 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. 

−  == Tool 
+  == 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:// 
+  [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 
+  the programmer). You can experiment with this in the [[IRC channelHaskell IRC channel]]. A standalone commandline version is available at [http://hackage.haskell.org/package/pointfree HackageDB] (package pointfree). 
−  channel. 

−  The @pl (pointless) plugin is rather infamous for using the 
+  The @pl (pointless) plugin is rather infamous for using the <hask>(>) a</hask> [[Monadmonad]] to obtain concise code. It also makes use of [[ArrowArrows]]. It also sometimes produces (amusing) code blow ups with the 
−  +  <hask>(.)</hask> operator. 

−  +  
+  Recently, @unpl has been written, which (attempts) to unscramble @plified code. It also has a [http://hackage.haskell.org/package/pointful standalone commandline version] (package pointful). 

A transcript: 
A transcript: 

−  > @pl \x y > x y 

+  <haskell> 

+  > 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 (*) 

−  > 
+  > unpl (sum .) . zipWith (*) 
+  (\ d g > sum (zipWith (*) d g)) 

+  
+  > pl \x y z > f (g x y z) 

((f .) .) . g 
((f .) .) . g 

−  > 
+  > unpl ((f .) .) . g 
+  (\ e j m > f (g e j m)) 

+  
+  > pl \x y z > f (g x y) z 

(f .) . g 
(f .) . g 

−  > 
+  > unpl (f .) . g 
+  (\ d i > f (g d i)) 

+  
+  > pl \x y z > f z (g x y) 

(flip f .) . g 
(flip f .) . g 

−  > 
+  > 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 
f = flip id 
f = flip id 

−  > 
+  > pl \ x > x * x 
join (*) 
join (*) 

−  > 
+  > pl \a b > a:b:[] 
(. return) . (:) 
(. return) . (:) 

−  > 
+  > pl \x > x+x+x 
(+) =<< join (+) 
(+) =<< join (+) 

−  > 
+  > pl \a b > Nothing 
const (const Nothing) 
const (const Nothing) 

−  > 
+  > pl \(a,b) > (f a, g b) 
f *** g 
f *** g 

−  > 
+  > 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 
fmap 
fmap 

−  > 
+  > pl \h f g x > f x `h` g x 
liftM2 
liftM2 

−  > 
+  > pl \f a b c d > f b c d a 
flip . ((flip . (flip .)) .) 
flip . ((flip . (flip .)) .) 

−  > 
+  > pl \a (b,c) > a c b 
(`ap` snd) . (. fst) . flip 
(`ap` snd) . (. fst) . flip 

−  > 
+  > pl \x y > compare (f x) (f y) 
((. f) . compare .) 
((. f) . compare .) 

+  </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: 
−  #haskell logs. (Or join #haskell on FreeNode and try it yourself!) It can, of course, get out of hand: 

−  > @pl \(a,b) > a:b:[] 

+  <haskell> 

+  > pl \(a,b) > a:b:[] 

uncurry ((. return) . (:)) 
uncurry ((. return) . (:)) 

−  > 
+  > pl \a b c > a*b+2+c 
((+) .) . flip flip 2 . ((+) .) . (*) 
((+) .) . flip flip 2 . ((+) .) . (*) 

−  > 
+  > 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) 
flip flip snd . (ap .) . flip flip fst . ((.) .) . flip . (((.) . (,)) .) 
flip flip snd . (ap .) . flip flip fst . ((.) .) . flip . (((.) . (,)) .) 

−  == Obfuscation == 

+  > 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> 

+  
+  == Combinator discoveries == 

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

+  
+  === The owl === 

+  
+  <haskell> 

+  ((.)$(.)) 

+  </haskell> 

+  
+  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>. 

+  
+  Example 

+  <haskell> 

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

+  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> 

+  
+  === 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> 

+  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 

+  </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> 

+  
+  === Squish === 

+  
+  <haskell> 

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

+  </haskell> 

+  
+  Example: 

+  
+  <haskell> 

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

+  </haskell> 

+  
+  [[/CombineAn actually useful example]], numbering lines of a file. 

+  
+  == Problems with pointfree == 

Pointfree style can (clearly) lead to [[Obfuscation]] when used unwisely. 
Pointfree style can (clearly) lead to [[Obfuscation]] when used unwisely. 

As higherorder functions are chained together, it can become harder to 
As higherorder functions are chained together, it can become harder to 

mentally infer the types of expressions. The mental cues to an 
mentally infer the types of expressions. The mental cues to an 

−  expression's type ( 
+  expression's type (explicit function arguments, and the number of 
arguments) go missing. 
arguments) go missing. 

−  Perhaps this is why pointfree style is sometimes (often?) referred to as 

+  Pointfree 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 165:  Line 262:  
One early reference is 
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 21:613641. 

−  A Functional Style and Its Algebra of Programs," Communications of the 

−  Association for Computing Machinery 21:613641. 

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

A paper specifically about pointfree style: 
A paper specifically about pointfree style: 

−  +  * 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. 

[http://en.wikipedia.org/wiki/Squiggol Squiggol], and the BirdMeertens Formalism: 
[http://en.wikipedia.org/wiki/Squiggol Squiggol], and the BirdMeertens Formalism: 

−  +  * http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/index.html#squiggolintro. 

−  +  * A Calculus of Functions for Program Derivation, R.S. Bird, in Res Topics in Fnl Prog, D. Turner ed, AW 1990. 

−  +  * The Squiggolist, ed Johan Jeuring, published irregularly by CWI Amsterdam. 

+  
+  [http://wiki.di.uminho.pt/twiki/bin/view/Personal/Alcino/PointlessHaskell Pointless Haskell] is a library for pointfree 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 refactoring. 

+  
+  This project is written by [http://www.di.uminho.pt/~mac/ Manuel Alcino Cunha], see his homepage for more related materials on the topic. 

+  An extended verson of his paper ''Pointfree Programming with Hylomorphisms'' can be found [http://web.comlab.ox.ac.uk/oucl/research/pdt/ap/dgp/workshop2004/cunha.pdf here]. 

+  
+  == Other areas == 

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

+  
+  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 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.
Pointfree 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:
 pointwise, and pointfree member
mem, mem' :: Eq a => a > [a] > Bool
mem x lst = any (== x) lst
mem' = any . (==)
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 'pointsfree' 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 pointsfree version:
f = (+ 1)
where there is no mention of the value on which the function is acting.
Background
To find out more about this style, search for Squiggol and the BirdMeertens 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.
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 standalone commandline version is available at HackageDB (package pointfree).
The @pl (pointless) 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 @plified code. It also has a standalone commandline 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))
Combinator discoveries
Some fun combinators have been found via @pl. Here we list some of the best:
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
Dot
dot = ((.).(.))
dot :: (b > c) > (a > a1 > b) > a > a1 > c
Example:
sequence `dot` replicate ==
(sequence .) . replicate ==
replicateM
(=<<) == join `dot` fmap
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])
Squish
f >>= a . b . c =<< g
Example:
(readFile y >>=) . ((a . b) .) . c =<< readFile x
An actually useful example, numbering lines of a file.
Problems with pointfree
Pointfree style can (clearly) lead to Obfuscation when used unwisely. As higherorder 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.
Pointfree 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.
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 21:613641.
which appears to be available (as a scan) at http://www.stanford.edu/class/cs242/readings/backus.pdf
A paper specifically about pointfree 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:
 Functional Programming with Bananas, Lenses, and Barbed Wire, http://wwwhome.cs.utwente.nl/~fokkinga/mmf91m.ps.
Squiggol, and the BirdMeertens Formalism:
 http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/index.html#squiggolintro.
 A Calculus of Functions for Program Derivation, R.S. Bird, in Res Topics in Fnl Prog, D. Turner ed, AW 1990.
 The Squiggolist, ed Johan Jeuring, published irregularly by CWI Amsterdam.
Pointless Haskell is a library for pointfree 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 refactoring.
This project is written by Manuel Alcino Cunha, see his homepage for more related materials on the topic. An extended verson of his paper Pointfree Programming with Hylomorphisms can be found here.
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.