Changes between Version 1 and Version 2 of ViewPatternsArchive

Jul 23, 2007 8:57:26 AM (12 years ago)

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  • ViewPatternsArchive

    v1 v2  
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     2= View patterns: lightweight views for Haskell =
     4This page describes a rather lightweight proposal for adding views to
     5Haskell Prime.  I'm thinking of prototyping the idea in GHC, so I'm looking
     6for feedback.
     8This page is open to editing by anyone.  (Chase the "Wiki notes" link in the sidebar to find out how.)
     10== The problem ==
     12We are keen on abstraction, but pattern matching is so convenient that
     13we break abstractions all the time.  It's our dirty little secret.
     14Looked at this way, object-oriented folk are much more obsessive
     15about abstraction than we are: everything (including field access
     16these days) is a method.
     18Views have, in one form or another, repeatedly been proposed as a
     19solution for this problem.   (See the end for a comparison with related work.)
     22== The lightweight view proposal ==
     23=== Informally ===
     25The proposal introduces a new form of pattern, called a '''view pattern'''
     26Here are some function definitions using view patterns.
     27To read these definitions, imagine that `sing` is
     28a sort of constructor that matches singleton lists.
     30  f :: [Int] -> Int
     31  f (sing -> n) = n+1   -- Equiv to: f [n] = ...
     32  f other     = 0
     34  g :: [Bool] -> Int
     35  g (sing -> True)  = 0         -- Equiv to: g [True] = ...
     36  g (sing -> False) = 1         -- Equiv to: g [False] = ...
     37  g other           = 2
     39  h :: [[Int]] -> Int   
     40  h (sing -> x : sing -> y : _) = x+y
     41                        -- Equiv to: h ([x]:[y]:_) = ...
     42  h other = 0
     44So what is `sing`?  It is just an ordinary Haskell function that
     45returns a `Maybe` type:
     47  sing :: [a] -> Maybe a
     48  sing [x]   = Just x
     49  sing other = Nothing
     51So `sing` simply identifies singleton lists, and returns the payload (that is,
     52the singleton element; otherwise it returns `Nothing`.
     53It is very important that '''there is nothing special about `sing`'''.  It is
     54not declared to be a view; it can be called as a normal Haskell function; the author
     55of `sing` might not have intended it to be used in pattern matching. 
     57=== More formally ===
     59The only special stuff is in the pattern. 
     60The sole change is this: add a single new sort of pattern, of the
     62        (''expr'' `->` ''pat'')
     64where ''expr'' is an arbitrary Haskell expression.   I'll call a pattern
     65of this form a '''view pattern'''.
     67From a '''scoping''' point of view, the variables bound by the pattern (''expr'' `->` ''pat'')
     68are simply the variables bound by ``pat``.
     69Any variables in ``expr`` are bound occurrences.
     71The rule for '''pattern-matching''' is this:
     72To match a value ''v'' against a pattern ''(expr -> p)'',
     73  * Evaluate ''(expr v)''
     74  * If the result is ''(`Just` w)'', match ''w'' against ''p''
     75  * If the result is `Nothing`, the match fails.
     77The '''typing rule''' is similarly simple. 
     78The expression ''expr'' must have type
     79''t1 `-> Maybe` t2''. Then the pattern ''pat'' must have type ''t2'', and the
     80whole pattern (''expr'' `->` ''pat'') has type ''t1''.
     82=== Features ===
     84For the different features this proposal (and others) have, see [[ref(Features views can have)]].
     85The proposal
     86  * has the value input feature
     87  * has the implicit `Maybe` feature
     88  * doesn't have the transparent ordinary patterns feature
     89  * has the nesting feature
     91=== Possible extension 1: multi-argument view patterns ===
     93It would be quite useful to allow more than one sub-pattern in a view
     94pattern.  To do this we'd need a `Maybe` data type that returns more than
     95one result, thus:
     97  data Maybe2 a b   = Nothing2 | Just2 a b
     98  data Maybe3 a b c = Nothing3 | Just3 a b c
     99        -- ..etc..., up to 8 perhaps (sigh)
     101With this in hand we can extend the views story to have multiple sub-patterns.
     104  snoc :: [a] -> Maybe2 [a] a
     105  snoc [] = Nothing2
     106  snoc (x:xs) = case snoc xs of
     107                  Nothing2   -> Just2 [] x
     108                  Just2 ys y -> Just2 (x:ys) y
     110  last :: [Int] -> Int
     111  last (snoc -> xs x) = x
     112  last other = error "empty list"
     114It is tiresome that we need types `Maybe2`, `Maybe3` etc, but we already have
     115that in Haskell; consider `zip3`, `zip4` and so on.
     116We could always get away without it, by sticking to unary view patterns and
     117using tuples, thus:
     119  snoc :: [a] -> Maybe ([a], a)
     120  snoc [] = Nothing
     121  snoc (x:xs) = case snoc xs of
     122                  Nothing     -> Just ([], x)
     123                  Just (ys,y) -> Just (x:ys, y)
     125  last :: [Int] -> Int
     126  last (snoc -> (xs, x)) = x
     127  last other = error "empty list"
     129But the tuple looks a bit clumsy.
     131Under this proposal, the number of sub-patterns in the view pattern determines
     132which return type the view function should have.  E.g. in the pattern '(e -> p1 p2 p3)',
     133'e' should return a `Maybe3`.
     135If n=0, then we want `Maybe0`, which is called `Bool`.  Thus
     137  even :: Int -> Bool
     138  even n = n `div` 2 == 0
     140  f (even ->) = ...     -- Matches even numbers
     141  f other     = ...
     143Here `even` is used as a nullary view pattern, with no sub-patterns
     144following the `->`.
     146Another variation (call it "extension 1b"), which avoids the tiresome need to define new types, is this: supplying multiple sub-patterns in a view pattern is synonymous with tupling.  Thus `(f -> p1 p2)` would be synonymous with `(f -> (p1,p2))`.  Here the effect is purely syntactic, allowing you to omit parens and commas without confusion.  No new types.  The power-to-weight ratio is probably better for this alternative.
     148=== Possible extension 2: the implicit `Maybe`  ===
     150Thus far, the view function is required to return a `Maybe` type, with `Nothing` to indicate match
     151failure.  An alternative, presented in the Erwig paper on transformational patterns (see Related work below),
     152this implicit matching is not performed; instead, the sub-pattern is matched against
     153whatever the view function returns.  So you'd have to write:
     155f (snoc -> Just2 xs x) = ...
     157(Note the tiresome `Just2`.)
     159For more one the consequences of removing the implicit `Maybe`, see the [[ref(Implicit `Maybe` feature)]]
     161I can think of three alternatives:
     162 * The `Maybe` stuff is built-in. This is the main proposal, because I think it is often exactly what you want.
     163 * No built-in `Maybe` stuff.  Arguably this is more consistent with pattern-guards.
     164 * Both are available, with different syntax.  For example
     165    * ''(expr `->` pat)'' for the built-in `Maybe` story
     166    * ''(expr `=>` pat)'' with no bulit-in `Maybe`
     168=== Concrete syntax ===
     170A disadvantage of the arrow syntax is that it looks a bit confusing
     171when it appears in a case expression:
     173  last xs = case xs of
     174                (snoc -> x xs) -> x
     176(Also that "x xs" looks a bit like `x` applied to `xs`.)
     178Here are some other possible syntax choices I've considered:
     180  f ($snoc x xs) = ...          -- Use prefix "$"
     181  g ($(bits 3) x bs) = ...      -- Extra parens for the value input feature
     183  f (%snoc x xs) = ...          -- Or use prefix "%" instead
     184  f (.snoc x xs) = ...          -- Or use prefix "." instead
     185  f (?snoc x xs) = ...          -- Or use prefix "?" instead
     187  f (snoc? x xs) = ...          -- Postfix "?"
     188  g ((bits 3)? x bs) = ...      -- With parens
     190  f (snoc | x xs) = ..          -- Use "|" instead of "->"
     191  g (bits 3 | b bs) = ...
     193Another possibility is to use a backward arrow, more like pattern guards:
     195  f ((x,xs) <- snoc) = ...  -- More like pattern guards
     197But that messes up the left-to-right flow that is useful in some cases.
     198For example, compare these:
     200  parsePacket1 (bits 3 -> n (bits n -> val bs)) = ...
     202  parsePacket2 (n (val bs <- bits n) <- bits 3) = ...
     205I also thought about infix view patterns, where the view function
     206appears between its (pattern) arguments, but I could not think of a
     207nice syntax for it, so it is not provided by this proposal.
     209=== Remarks ===
     211The key feature of this proposal is its modesty, rather than its ambition:
     212  * There is no new form of declaration (e.g. 'view' or 'pattern synonym'). 
     213  * The functions used in view patterns are ordinary Haskell functions, and can be called from ordinary Haskell code.  They are not special view functions.
     214  * Since the view functions are ordinary Haskell functions, no changes are needed to import or export mechanisms.
     215  * Both static and dynamic semantics are extremely simple.
     216It is essentially some simple syntactic sugar for patterns.
     217However, sometimes modest syntactic sugar can have profound consequences.
     218In this case, it's possible that people would start routinely hiding
     219the data representation and exporting view functions instead, which might
     220be an excellent thing.
     222All this could be done with pattern guards.  For example `parsePacket` could be written
     224  parsePacket bs | Just (n, bs')    <- bits 3 bs
     225                 | Just (val, bs'') <- bits n bs'
     226                 = ...
     228Indeed, one might ask whether the extra syntax for view patterns is worth
     229it when they are so close to pattern guards. 
     230That's a good question.  I'm hoping that support for view patterns
     231might encourage people to export view functions (ones with `Maybe`
     232return types, and encouage their use in patten matching).  That is,
     233just lower the barrier for abstraction a bit.
     235'''Completeness'''.  It is hard to check for completeness of pattern matching; and likewise for
     236overlap.  But guards already make both of these hard; and GADTs make completness hard too.
     237So matters are not much worse than before.
     241== Features views can have ==
     243The main goal of views is to introduce computations into pattern matches thus introducing abstraction from hard wired constructors. To distinguish between the different proposals, we pick out the key features
     245=== Value input feature ===
     247This features allows to introduce additional parameters into the computation. Perhaps the most basic example are (n+k) patterns
     249  fib :: Int -> Int
     250  fib 0 = 1
     251  fib 1 = 1
     252  fib (n + 2) = fib (n + 1) + fib n
     254Here, the number 2 can be arbitrary, we are not fixed to a "finite" set of "constructors" (+1), (+2) etc.
     256Of course, the real power unfolds when the extra parameter can be given at runtime
     258   f :: Int -> Int -> ...
     259   f n (n + m) = m           -- f a b = (b - a)
     262In the proposed view pattern (''expr'' `->` ''pat''), ''expr'' is an arbitrary Haskell expression. Thus, the lightweight proposal has the '''value input feature'''. For another example, suppose you wrote a regular expression matching function:
     264  regexp :: String -> String -> Maybe (String, String)
     265  -- (regexp r s) parses a string matching regular expression r
     266  --    the front of s, returning the matched string and remainder of s
     268then you could use it in patterns thus:
     270  f :: String -> String
     271  f (regexp "[a-z]*" -> (name, rest)) = ...
     273Of course, the argument does not need to be a constant as it is here.
     275With the value input feature, in a sense, patterns become first class. For example, one could pass a pattern as an argument to a function thus:
     277  g :: (Int -> Maybe Int) -> Int -> ...
     278  g p (p -> x) = ...
     280Here the first argument `p` can be thought of as a pattern passed to `g`, which
     281is used to match the second argument of `g`.
     283Here is another rather cute example:
     285unfoldr :: (b -> Maybe (a, b)) -> b -> [a]
     286unfoldr f (f -> (a, b)) = a : unfoldr f b
     287unfoldr f other         = []
     290=== Implicit `Maybe` feature ===
     292In functional languages, pattern matching is used to inspect a sum types like `Either Int String` and to proceed with the matching alternative. We can always project a choice between multiple alternatives to choice between one alternative (`Just`) and failure (`Nothing`):
     294   maybeLeft  :: Either a b -> Maybe a
     295   maybeRight :: Either a b -> Maybe b
     298Some proposals build their patterns entirely from from such single alternative de-constructors functions of type `a -> Maybe b`, while some allow projection to multiple alternatives.
     300By restricting de-constructors to be of type `a -> Maybe b`, the Maybe can be made implicit, it doesn't show up in the pattern. Example:
     302    data Product = ....some big data type...
     303    type Size = Int
     305    smallProd, medProd, bigProd :: Product -> Maybe Size
     306    smallProd p = ...
     307    medProd   p = ...
     308    bigProd   p = ...
     310    f :: Product -> ...
     311    f (smallProd -> s) = ...
     312    f (medProd   -> s) = ...
     313    f (bigProd   -> s) = ...
     316Projection to multiple alternatives requires a new (or existing) data type for every group of alternatives introduced.
     318    data Dimensions = Small | Medium | Big      -- View type
     319    prodSize :: Product -> Dimensions
     320    prodSize = ...
     322    f :: Product -> ...
     323    f (prodSize -> Small)  = ...
     324    f (prodSize -> Medium) = ...
     325    f (prodSize -> Big)    = ...
     327Using a fixed set of multiple alternatives makes it more obvious whether the match is exhaustive or not.
     329While the implicit `Maybe a` is more compositional and nicely integrates with already existing uses of the `Maybe`-type, it cannot share expensive computations across multiple alternatives. This is a strong argument against the implicit `Maybe a`. To illustrate the problem, suppose that
     332   data Graph
     334represents a graph and that we want a function
     336   g :: Graph -> [...]
     337   g (forest -> xs) = concatMap g xs
     338   g (tree ->)      = ...
     339   g (dag  ->)      = ...
     341These three properties are expensive to calculate but all three only
     342depend on the result of a single depth first search. By projecting the
     343disjoint sum to several `Maybe`s, the depth first search has to be
     344repeated every time. More importantly, there is *no way* for the compiler to optimize this because that would mean common subexpression elimination across
     347Some would argue that implicit the 'Maybe a' is ''less'' compositional than the explicit version.  If no 'Maybe' is required, then the result of the view function can be any type at all, which can be pattern-matched in the ordinary way.  Some examples of cute programming of well-known combinators:
     349map f [] = []
     350map f (x: map f -> xs) = x:xs
     352foldr f z [] = z
     353foldr f z (x: foldr f z -> xs) =  x `f` xs
     356=== Transparent ordinary Patterns ===
     358The lightweight view proposal has different syntax for view functions and ordinary pattern matches, they are not interchangeable. To use the abstraction view functions offer, you have to anticipate whether you can stick to ordinary constructors or whether you will switch to abstract constructors at some time. For example, a stack abstraction would have to use view patterns if we want to be able to change the concrete representation of stacks later on.
     360    type Stack a = [a]
     362    f :: Stack a
     363    f (null?)     = ...
     364    f (pop? x xs) = ...
     366This certainly discourages ordinary pattern matching. In other words,
     367implementing the proposal has considerable impact on ordinary pattern
     368matching (not in semantics but in use).
     370The problem that needs to be solved is to introduce abstraction "after the fact".
     372On the other hand, view patterns can do arbitrary computation, perhaps expensive. So it's good to have a syntactically-distinct notation that reminds the programmer that some computation beyond ordinary pattern matching may be going on.
     374=== Nesting ===
     376In the lightweight proposal, view patterns are just an extra syntactic form of pattern, and they nest inside other patterns, and other patterns nest inside them.  So one can write
     378  f (sing -> x, True) = ...
     379  g (Just (sing -> x)) = ...
     380  h (Just (sing -> Just x)) = ...
     382And by the same token, view patterns nest inside each other:
     384  k :: [[a]] -> a
     385  k (sing -> sing -> x) = x
     387This convenient nesting is perhaps the biggest practical
     388difference between view patterns and pattern guards.
     390The majority of the proposals allow nesting.
     393=== Integration with type classes ===
     395A view mechanism that integrates nicely with type classes would allow
     396a single "view" to decompose multiple different data types.  For
     397example, a view might work on any type in class Num, or in class Sequence.
     399A good example is Haskell's existing (n+k) patterns.  Here is how they
     400can be expressed using the view pattern proposed in this page (with different
     401syntax, of course):
     403   np :: Num a => a -> a -> Maybe a
     404   np k n | k <= n    = Just (n-k)
     405          | otherwise = Nothing
     407   g :: Int -> Int
     408   g (np 3 -> n) = n*2
     410   h :: Integer -> Integer
     411   h (np 9 -> n) = n*2
     413   f :: Num a => a -> Int
     414   f (np 10 -> n) = n           -- Matches values >= 10, f a = (a - 10)
     415   f (np 4  -> n) = 1           -- Matches values >= 4
     416   f other        = 2
     418Here a single, overloaded view, `np`, can be used
     419in `g`, and `h` to match against values of different types and,
     420in `f`'s case, any type in class Num. (Notice too the use of the value
     421input feature.)
     423This feature falls out very nicely from view patterns, but
     424not from all other proposals.
     427== Efficiency of Views ==
     429View patterns can do arbitrary computation, perhaps expensive.
     431It's reasonable to expect the compiler to avoid repeated computation when
     432pattern line up in a column:
     434  f (snoc -> x xs) True  = ...
     435  f (snoc -> x xs) False = ...
     437In pattern-guard form, common sub-expression should achieve the same
     438effect, but it's quite a bit less obvious.  We should be able to give
     439clear rules for when the avoidance of repeat computation is
     443== Use cases and examples ==
     445Whether views are really worth it can only be decide on the base of examples. Some are situations where you programmed and thought "I wish I had a view for that". Some are those snippets of code that unexpectedly use views to good effect.
     447=== Sequences ===
     449Lists, queues, ByteStrings and 2-3-finger trees are all implementations of sequences, but only ordinary lists can be deconstructed using pattern matching. The need for list patterns on arbitrary sequence data structures is desperate. As if to ease the pain, Data.Sequence even defines the views from the left and from the right
     451   data ViewL a
     452   = EmptyL
     453   | (:<) a (Seq a)
     455   viewl :: Seq a -> ViewL a
     457   data ViewR a
     458   = EmptyR
     459   | (:>) (Seq a) a
     461   viewr :: Seq a -> ViewR a
     464Thus, the presence of views has a direct impact on existing Haskell libraries. Arguably, a view proposal that wants to be effective for abstract data types likely has to have the transparent ordinary patterns feature.
     466The observations from [ Okasaki: Breadth-First Numbering - Lessons ... ] suggest that not having abstract pattern matching (for sequences) can indeed have great impact on the abstractions functional programmers can think of.
     468=== Designing data structures ===
     470The abstractional power views offer can also be put to good use when designing data structures, as the following papers show
     472  * [ R.Hinze: A fresh look on binary search trees].
     473  * [ R.Hinze:  A Simple Implementation Technique for Priority Search Queues]
     475=== Sets and Inductive Graphs ===
     477Having the value input feature, even set like data structures come in reach for pattern matching. In fact, the key idea of [ M.Erwig: Inductive Graphs and Functional Graph Algorithms] is to introduce a suitable view of graphs. This way, graphs can be liberated from their notoriously imperative touch.
     479Here is a small module that allows to decompose sets with repsect to a given element, deleting it hereby.
     481module Set( Set, empty, insert, delete, has) where
     483  newtype Set a = S [a]
     485  has :: Eq a => a -> Set a -> Maybe (Set a)
     486  has x (S xs) | x `elem` xs = Just (xs \\ x)
     487               | otherwise   = Nothing
     489  delete :: a -> Set a -> Set a
     490  delete x (has x -> s) = s
     491  delete x s            = s
     493  insert :: a -> Set a -> Set a
     494  insert x s@(has x -> _) = s
     495  insert x (S xs)         = S (x:xs)
     497Notice that in the left-hand side `delete x (has x -> s)`, the first `x` is a binding occurrence, but the second is merely an argument to `has` and is a bound occurrence.
     499=== Erlang-style parsing ===
     501The expression to the left of the `->` can mention variables bound in earlier patterns.
     502For example, Sagonas et al describe an extension to Erlang that supports pattern-matching on bit-strings ([ "Application, implementation and performance evaluation of bit-stream programming in Erlang", PADL'07]).  Suppose we had a parsing function thus:
     504  bits :: Int -> ByteString -> Maybe2 Word ByteString
     505  -- (bits n bs) parses n bits from the front of bs, returning
     506  -- the n-bit Word, and the remainder of bs
     508Then we could write patterns like this:
     510  parsePacket :: ByteString -> ...
     511  parsePacket (bits 3 -> n (bits n -> val bs)) = ...
     513This parses 3 bits to get the value of `n`, and then parses `n` bits to get
     514the value of `val`. 
     516=== N+k patterns ===
     518You can test for values.  For example here's a view function that tests for values
     519greater than or equal to n:
     521   np :: Num a => a -> a -> Maybe a
     522   np k n | k <= n    = Just (n-k)
     523          | otherwise = Nothing
     525   f :: Num a => a -> a
     526   f (np 10 -> n) = 0           -- Matches values >= 10
     527   f (np 4  -> n) = 1           -- Matches values >= 4
     528   f other        = 2
     530You will recognise these as (n+k) patterns, albeit with slightly different syntax.
     531(Incidentally, this example shows that the view function can be overloaded, and
     532that its use in a view pattern gives rise to a type-class constraint (in this case,
     533that in turn makes `f` overloaded).
     535=== Naming constants in one place ===
     537We are always taught to write magic numbers, or constants, in one place only.
     538In C you can write
     540  #define ERRVAL 4
     542and then use `ERRVAL` in `switch` labels.  You can't do that in Haskell, which is tiresome.
     543But with view pattern you can:
     545  errVal :: Int -> Bool
     546  errVal = (== 4)
     548  f (errVal ->) = ...
     553== Related work ==
     555=== Wadler's original paper (POPL 1987) ===
     557Wadler's paper was the first concrete proposal.  It proposed a top-level view
     558declaration, together with functions ''in both directions'' between the view type
     559and the original type, which are required to be mutually inverse. 
     560This allows view constructors to be used in expressions
     561as well as patterns, which seems cool. Unfortunately this dual role proved
     562problematic for equational reasoning, and every subsequent proposal restricted
     563view constructors to appear in patterns only.
     565=== [ Burton et al views (1996)] ===
     567This proposal is substantially more complicated than the one above; in particular it
     568requires new form of top-level declaration for a view type. For example:
     570  view Backwards a of [a] = [a] `Snoc` a | Nil
     571     where
     572     backwards [] = Nil
     573     backwards (x:[]) = [] `Snoc` x
     574     backwards (x1:(xs `Snoc` xn)) = (x1:xs) `Snoc` xn
     576Furthermore, it is in some ways less expressive than the proposal here;
     577the (n+k) pattern, Erlang `bits` pattern, and `regexp` examples are not
     578definable, because all rely on the value input feature.
     580I think this proposal is substantially the same as "Pattern matching and
     581abstract data types", Burton and Cameron, JFP 3(2), Apr 1993.
     583=== [ Okasaki: views in Standard ML] ===
     585Okasaki's design is very similar to Burton et al's, apart from differences due
     586to the different host language.  Again, the value input feature is not supported.
     588=== [ Erwig: active patterns] ===
     590Erwig's proposal for active patterns renders the Set example like this:
     592data Set a = Empty | Add a (Set a)
     594pat Add' x _ =
     595  Add y s => if x==y then Add y s
     596             else let Add' x t = s
     597                  in Add x (Add y t)
     599delete x (Add' x s) = s
     600delete x s          = s
     602This requires a new top-leven declaration form `pat`; and I don't think it's nearly
     603as easy to understand as the current proposal.  Notably, in the first equation for
     604`delete` it's ahrd to see that the second `x` is a bound occurrence; this somehow
     605follows from the `pat` declaration.
     607Still the proposal does support the value input feature.
     609=== [ Palao et al: active destructors (ICFP'96)] ===
     611Active Desctructors (ADs) are defined by a new form of top-level declaration.  Our
     612singleton example would look like this:
     614  Sing x match [x]
     616Here '''match''' is the keyword, and `Sing` is the AD.  ADs are quite like view patterns:
     617the can do computation, and can fail to match.  But they are definitely not normal
     618Haskell functions, and need their own form of top-level declaration.  They even have
     619a special form of type to describe them.
     621The value-input feature is supported, but only via a sort of mode declaration (indicated by a down-arrow) on the
     622new form of type.
     624They also introduce a combining form for ADs, to make a kind of and-pattern.  For
     625example, suppose we had
     627  Head x match (x:_)
     628  Tail x match (_:xs)
     630  f :: [a] -> [a]
     631  f ((Head x)@(Tail ys)) = x:x:ys
     633Here `(Head x)@(Tail ys)` is a pattern that matches ''both'' `(Head x)` and `(Tail ys)`
     634against the argument, binding `x` and `ys` respectively.  We can model that with view patterns,
     635only a bit more clumsily:
     637  headV (x:xs) = Just x
     638  headV []     = Nothing
     640  tailV (x:xs) = Just xs
     641  tailV []     = Nothing
     643  (@) :: (a -> Maybe b) -> (a -> Maybe c) -> a -> Maybe (b,c)
     644  (f @ g) x = do { b <- f x; c <- g x; return (b,c) }
     646  f :: [a] -> [a]
     647  f (headV @ tailV -> (x,ys)) = x:x:ys
     649The clumsiness is that the "`@`" combines functions, with a kind of positional
     650binding; the pattern `(x,ys)` is separated from the combiner so that it's less clear
     651that `headV` binds `x` and `tailV` binds `y`.
     653In exchange, although view patterns are a bit less convenient here, they
     654are a much, much smaller language change than ADs.
     656=== [ Erwig/Peyton Jones: transformational patterns] ===
     658This paper describes pattern guards, but it also introduces '''transformational patterns'''.  (Although
     659it is joint-authored, the transformational-pattern idea is Martin's.)  Transformational patterns
     660are very close to what we propose here.  In particular, view functions are ordinary Haskell functions,
     661so that the only changes are to patterns themselves.
     663There are two main differences (apart from syntax).
     664First, transformational patterns didn't have the value input feature, althought it'd be easy
     665to add (indeed that's what we've done). Second, transformational patterns as described by
     666Erwig do no stripping of the `Maybe` (see "Possible extension 2" above).
     668=== [ F# Active Patterns] ===
     670Simon started this design discussion after talking to Don Syme about F#'s '''active patterns''', which serve a very similar purpose. These combine both “total” discrimination (views) and “partial” discrimination (implicit maybe) into one mechanism. It does this by embedding the names of discriminators in the names of matching functions, via “values with structured names”.  Sample uses include matching on .NET objects and XML.
     672Here is [ a full paper describing the design], by Don Syme, Gregory Neverov, and James Margetson (April 2007).
     674The feature is implemented in F# 1.9. Some code snippets are below.
     676    let (|Rect|) (x:complex) = (x.RealPart, x.ImaginaryPart)
     677    let (|Polar|) (x:complex) = (x.Magnitude , x.Phase)
     679    let mulViaRect c1 c2 =
     680        match c1,c2 with
     681        | Rect(ar,ai), Rect(br,bi) -> Complex.mkRect(ar*br - ai*bi, ai*br + bi*ar)
     683    let mulViaPolar c1 c2 =
     684        match c1,c2 with
     685        | Polar (r1,th1),Polar (r2,th2) -> Complex.mkPolar(r1*r2, th1+th2)
     687    let mulViaRect2  (Rect(ar,ai))   (Rect(br,bi))   = Complex.mkRect(ar*br - ai*bi,
     688                                                                      ai*br + bi*ar)
     689    let mulViaPolar2 (Polar(r1,th1)) (Polar(r2,th2)) = Complex.mkPolar(r1*r2, th1+th2)
     691And for views:
     693    open System
     695    let (|Named|Array|Ptr|Param|) (typ : System.Type) =
     696        if typ.IsGenericType        then Named(typ.GetGenericTypeDefinition(),
     697                                               typ.GetGenericArguments())
     698        elif not typ.HasElementType then Named(typ, [| |])
     699        elif typ.IsArray            then Array(typ.GetElementType(),
     700                                               typ.GetArrayRank())
     701        elif typ.IsByRef            then Ptr(true,typ.GetElementType())
     702        elif typ.IsPointer          then Ptr(false,typ.GetElementType())
     703        elif typ.IsGenericParameter then Param(typ.GenericParameterPosition,
     704                                               typ.GetGenericParameterConstraints())
     705        else failwith "MSDN says this can't happen"
     707    let rec freeVarsAcc typ acc =
     708        match typ with
     709        | Named (con, args) -> Array.fold_right freeVarsAcc args acc
     710        | Array (arg, rank) -> freeVarsAcc arg acc
     711        | Ptr (_,arg)       -> freeVarsAcc arg acc
     712        | Param(pos,cxs)    -> Array.fold_right freeVarsAcc cxs (typ :: acc)
     715=== [ Emir, Odersky, Williams: Matching objects with patterns] ===
     717Scala is an OO language with lots of functional features.  It has algebraic data types and
     718pattern matching.  It also has a form of view called '''extractors''', which are
     719pretty similar to view patterns, albeit in OO clothing.  Notably, by packaging a constructor
     720and an extractor in a class, they can use the same class name in both expressions and terms,
     721implicitly meaning "use the constructor in expressions, and use the extractor in patterns".
     723The paper does a comparative evaluation of various OO paradigms for matching, and
     724concludes that case expressions and extractors work pretty well.
     726=== Pattern synonyms  ===
     728[ Pattern synonyms]
     729are a requested Haskell Prime feature. John Reppy had the same idea years ago for Standard ML; see
     730[ Abstract value constructors],
     731Reppy & Aiken, TR 92-1290, Cornell, June 1992.
     733The one way in which pattern synonyms are better than view patterns is that
     734they define by-construction bi-directional maps.  Example
     736  data Term = Var String | Term String [Term]
     738  -- 'const' introduces a pattern synonym
     739  const Plus a b = Term "+" [a,b]
     741  f :: Term -> Term
     742  f (Plus a b) = Plus (f a) (f b)
     743  f ... = ...
     745With pattern views, we'd have to write two functions for the "plus" view:
     747  plus :: Term -> Term -> Term
     748  plus a b = Term "+" [a,b]
     750  isPlus :: Term -> Maybe2 Term Term
     751  isPlus (Term "+" [a,b]) = Just2 a b
     752  isPlus other            = Nothing
     754  f :: Term -> Term
     755  f (isPlus -> a b) = plus (f a) (f b)
     757But perhaps that is not so bad.  Pattern synonyms also require a new form of top level declaration;
     758and are much more limited than view patterns (by design they cannot do computation).
     760=== [ Tullsen: First Class Patterns] ===
     762First Class Patterns is an approach that attempts to
     763add the minimum of syntax to the language which---in combination with
     764pattern combinators written within the language---can achieve everything
     765and more that Haskell patterns can do.  They have the value-input feature.
     767The advantages are  1) They are simpler than Haskell's patterns;  2) Patterns are first class.
     7683) The binding mechanism (the pattern binder) is orthogonal to the the pattern combinators:
     769the hope is that one can stop changing the syntax/semantics of patterns and concentrate on writing the
     770combinators (as Haskell functions).
     772The disadvantages are as follows: 1) An extra syntactic construct that binds variables, the pattern binder, is required.
     7732) Even with pattern binders, simple patterns look clunkier than Haskell's patterns.
     7743) No attempt is made to check for exhaustiveness of patterns.
     7754) No attempt is made to integrate with Haskell's patterns, the idea is a proposal for an alternative when one needs more than simple patterns.
     777The examples at the top of this page would look like this with first class patterns:
     779  f = {%sing n} .-> n+1
     780                |>> 0
     782  g =  {%sing True}  .-> 0
     783    .| {%sing False} .-> 1
     784                     |>> 2 
     786=== First class abstractions ===
     788Several proposals suggest first class ''abstractions'' rather that first-class ''patterns''.  By a "first class abstraction" I mean a value of type
     789(''something'' `->` ''something'')
     790with a syntax something like
     791(`\` ''pattern'' `->` ''result'').
     792The abstraction includes both the pattern and the result.  In contrast, view patterns tackle only the syntax of patterns; the pattern of a first-class abstraction. 
     794Here are the ones I know of
     796 * [ Claus Reinke's lambda-match proposal]
     797 * [ Matthias Blume: Extensible programming with first-class cases] (ICFP06)
     799All these proposals are more or less orthogonal to this one. For example, Reinke
     800proposes a compositional syntax for lambda abstractions
     801`(\p -> e)` where pattern matching failure on `p` can be caught and composed
     802with a second abstraction. Thus
     804   (| Just x -> x+1 ) +++ (| Nothing -> 0 )
     806combines two abstractions, with failure from the first falling through to the seoond. 
     808None of these proposals say
     809anything about the patterns themselves, which in turn is all this
     810proposal deals with.  Hence orthgonal.
     812=== Barry Jay: First class patterns ===
     814A yet more ambitious scheme is to treat patterns themselves as first class, even though they have free (binding) variables.  This is the approach that Barry Jay has taken in his very interesting project on the ''pattern calculus''.  His [ home page] has more info.