Difference between revisions of "A brief introduction to Haskell"
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−  
[[Category:Tutorials]] 
[[Category:Tutorials]] 

−  
−  Inspired by the [http://www.cs.jhu.edu/~scott/pl/lectures/camlintro.html Introduction to OCaml]. 

−  
−  == Background == 

[http://haskell.org Haskell] is: 
[http://haskell.org Haskell] is: 

Line 12:  Line 7:  
* Born as an open source vehicle for programming language research 
* Born as an open source vehicle for programming language research 

* One of the youngest children of ML and Lisp 
* One of the youngest children of ML and Lisp 

−  * Particularly useful for 
+  * Particularly useful for programs that manipulate data structures (such as [http://haskell.org/ghc compilers] and [http://pugscode.org interpreters]), and for concurrent/parallel programming 
+  
+  Inspired by the article [http://www.cs.jhu.edu/~scott/pl/lectures/camlintro.html Introduction to OCaml], and translated from the OCaml by Don Stewart. 

+  
+  == Background == 

[http://haskell.org/haskellwiki/Old_news#Archives_by_year History]: 
[http://haskell.org/haskellwiki/Old_news#Archives_by_year History]: 

Line 23:  Line 18:  
* ''20002006''. Period of rapid language and community growth 
* ''20002006''. Period of rapid language and community growth 

* ''~2007''. Haskell Prime 
* ''~2007''. Haskell Prime 

+  * ''2009''. [[Haskell 2010]] 

[http://haskell.org/haskellwiki/Implementations Implementations]: 
[http://haskell.org/haskellwiki/Implementations Implementations]: 

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* [http://haskell.org/hugs Hugs] 
* [http://haskell.org/hugs Hugs] 

* [http://haskell.org/haskellwiki/Yhc YHC] 
* [http://haskell.org/haskellwiki/Yhc YHC] 

−  * [http://repetae.net/computer/jhc 
+  * [http://repetae.net/computer/jhc JHC] 
* [http://haskell.org/nhc98 nhc98] 
* [http://haskell.org/nhc98 nhc98] 

−  * [http://www.cs.uu.nl/ 
+  * [http://www.cs.uu.nl/wiki/Helium Helium] 
−  * 
+  * HBC 
== Haskell features == 
== Haskell features == 

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* Lazy evaluation: results are only computed if they're required (strictness optional) 
* Lazy evaluation: results are only computed if they're required (strictness optional) 

* Everything is an expression 
* Everything is an expression 

−  * 
+  * Firstclass functions: functions can be defined anywhere, passed as arguments, and returned as values. 
* Both compiled and interpreted implementations available 
* Both compiled and interpreted implementations available 

* Full type inference  type declarations optional 
* Full type inference  type declarations optional 

Line 50:  Line 46:  
These are all conceptually more [http://haskell.org/haskellwiki/Research_papers/Type_systems advanced ideas]. 
These are all conceptually more [http://haskell.org/haskellwiki/Research_papers/Type_systems advanced ideas]. 

−  Compared to similar functional languages, Haskell differs in 
+  Compared to similar functional languages, Haskell differs in that it 
+  has support for: 

* Lazy evaluation 
* Lazy evaluation 

−  * Pure by default 
+  * Pure functions by default 
−  * Monadic effects 
+  * Monadic side effects 
* Type classes 
* Type classes 

* Syntax based on layout 
* Syntax based on layout 

Line 68:  Line 64:  
== The Basics == 
== The Basics == 

−  Read the [http://haskell.org/onlinereport/ language definition] 
+  Read the [http://www.haskell.org/onlinereport/haskell2010/ language definition] 
−  +  to supplement these notes. For more depth and examples 

−  [http://haskell.org/haskellwiki/Books_and_tutorials see 
+  [http://haskell.org/haskellwiki/Books_and_tutorials see the Haskell wiki]. 
=== Interacting with the language === 
=== Interacting with the language === 

−  [http://haskell.org/haskellwiki/Haskell_in_5_steps Interacting with Haskell] via the GHCi interpreter: 

+  Haskell is both compiled and interpreted. For exploration purposes, 

+  we'll consider [http://haskell.org/haskellwiki/Haskell_in_5_steps interacting with Haskell] via the GHCi interpreter: 

* expressions are entered at the prompt 
* expressions are entered at the prompt 

* newline signals end of input 
* newline signals end of input 

−  Here is a GHCi 
+  Here is a GHCi session, starting from a UNIX prompt. 
$ ghci 
$ ghci 

Line 90:  Line 87:  
Loading package base1.0 ... linking ... done. 
Loading package base1.0 ... linking ... done. 

−  <haskell> 

+  Here the incredibly simple Haskell program <hask>let x = 3+4</hask> is 

−  Prelude> let x = 3 + 4 

+  compiled and loaded, and available via the variable <hask>x</hask>. 

−  </haskell> 

−  * Here the incredibly simple Haskell program <hask>let x = 3+4</hask> is 

−  compiled, loaded, and bound to the variable <hask>x</hask>. 

<haskell> 
<haskell> 

−  Prelude> 
+  Prelude> let x = 3 + 4 
−  x :: Integer 

</haskell> 
</haskell> 

Line 104:  Line 98:  
<hask>x</hask> "has type" <hask>Integer</hask>, the type of unbounded 
<hask>x</hask> "has type" <hask>Integer</hask>, the type of unbounded 

integer values. 
integer values. 

+  
+  <haskell> 

+  Prelude> :t x 

+  x :: Integer 

+  </haskell> 

+  
+  A variable evaluates to its value. 

<haskell> 
<haskell> 

Line 110:  Line 111:  
</haskell> 
</haskell> 

−  A variable evaluates to its value. 

+  The variable <hask>x</hask> is in scope, so we can reuse it in later 

+  expressions. 

<haskell> 
<haskell> 

Line 117:  Line 119:  
</haskell> 
</haskell> 

−  The variable <hask>x</hask> is in scope, so we can reuse it in later 

+  Local variables may be bound using <hask>let</hask>, which declares a 

−  expressions. 

+  new binding for a variable with local scope. 

<haskell> 
<haskell> 

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</haskell> 
</haskell> 

−  Local variables may be bound using <hask>let</hask>, which declares a 

+  Alternatively, declarations 

−  new binding for a variable with local scope. 

+  typed in at the top level are like an openended let: 

−  
−  Alternatively, declarations typed in at the top level are like an openended let: 

<haskell> 
<haskell> 

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== Basic types == 
== Basic types == 

−  There is a range of basic types, defined in the language [http://www.cse.unsw.edu.au/~dons/data/Prelude.html Prelude] 
+  There is a range of basic types, defined in the language [http://www.cse.unsw.edu.au/~dons/data/Prelude.html Prelude]. 
<haskell> 
<haskell> 

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[http://haskell.org/ghc/docs/latest/html/libraries/ standard libraries]. 
[http://haskell.org/ghc/docs/latest/html/libraries/ standard libraries]. 

−  == Functions (including Patterns and HigherOrder Functions) == 

+  === Libraries === 

+  
+  * The [http://haskell.org/ghc/docs/latest/html/libraries/base/Prelude.html Prelude] contains the core operations on basic types. It is imported by default into every Haskell module. For example; 

+  
+  <haskell> 

+  +  div mod &&  not < > == /= 

+  </haskell> 

+  
+  Learn the Prelude well. Less basic functions are found in the [http://haskell.org/ghc/docs/latest/html/libraries/ standard libraries]. For data structures such as [http://haskell.org/ghc/docs/latest/html/libraries/base/DataList.html List], [http://haskell.org/ghc/docs/latest/html/libraries/array/DataArray.html Array] and [http://haskell.org/ghc/docs/latest/html/libraries/containers/DataMap.html finite maps]. 

+  
+  To use functions from these modules you have to import them, or in GHCi, 

+  refer to the qualified name, for example to use the toUpper function on 

+  Chars: 

+  <haskell> 

+  Prelude> Char.toUpper 'x' 

+  'X' 

+  
+  Prelude> :m + Char 

+  Prelude Char> toUpper 'y' 

+  'Y' 

+  </haskell> 

+  
+  In a source file, you have to import the module explicitly: 

+  <haskell> 

+  import Char 

+  </haskell> 

+  
+  === Overloading === 

+  
+  Haskell uses ''typeclasses'' to methodically allow overloading. A 

+  typeclass describes a set of functions, and any type which provides 

+  those functions can be made an ''instance'' of that type. This avoids 

+  the syntactic redundancy of languages like OCaml. 

+  
+  For example, the function <hask>*</hask> is a method of the typeclass 

+  <hask>Num</hask>, as we can see from its type: 

+  
+  <haskell> 

+  Prelude> :t (*) 

+  (*) :: (Num a) => a > a > a 

+  </haskell> 

+  
+  Which can be read as "* is a polymorphic function, taking two values of some 

+  type 'a', and returning a result of the same type, where the type 'a' is a 

+  member of the class Num". 

+  
+  This means that it will operate on any type in the <hask>Num</hask> 

+  class, of which the following types are members: <hask>Double, Float, 

+  Int, Integer</hask>. Thus: 

+  
+  <haskell> 

+  Prelude> 2.3 * 5.7 

+  13.11 

+  </haskell> 

+  
+  or on integers: 

+  
+  <haskell> 

+  Prelude> 2 * 5 

+  10 

+  </haskell> 

+  
+  The type of the arguments determines which instance of <hask>*</hask> is 

+  used. Haskell also never performs implicit coercions, all coercions must 

+  be explicit. For example, if we try to multiply two different types, 

+  then the type check against <hask>* :: Num a => a > a > a</hask> will 

+  fail. 

+  
+  <haskell> 

+  Prelude> (2.3 :: Double) * (5 :: Int) 

+  
+  <interactive>:1:19: 

+  Couldn't match `Double' against `Int' 

+  Expected type: Double 

+  Inferred type: Int 

+  In the expression: 5 :: Int 

+  In the second argument of `(*)', namely `(5 :: Int)' 

+  </haskell> 

+  
+  To convert 5 to a <hask>Double</hask> we'd write: 

+  
+  <haskell> 

+  Prelude> (2.3 :: Double) * fromIntegral (5 :: Int) 

+  11.5 

+  </haskell> 

+  
+  Why bother  why not allow the system to implicitly coerce types? 

+  Implicit type conversions by the system are the source of innumerable 

+  hard to find bugs in languages that support them, and makes reasoning 

+  about a program harder, since you must apply not just the language's 

+  semantics, but an extra set of coercion rules. 

+  
+  Note that If we leave off the type signatures however, Haskell will 

+  helpfully infer the most general type: 

+  
+  <haskell> 

+  Prelude> 2.3 * 5 

+  11.5 

+  </haskell> 

+  
+  === Expressions === 

+  
+  In Haskell, expressions are everything. There are no pure "statements" 

+  like in Java/C++. For instance, in Haskell, <hask>ifthenelse</hask> is a kind of expression, and results in a value based on the condition part. 

+  
+  <haskell> 

+  Prelude> (if 2 == 3 then 5 else 6) + 1 

+  7 

+  </haskell> 

+  
+  <haskell> 

+  Prelude> (if 2 == 3 then 5 else 6.5) + 1 

+  7.5 

+  </haskell> 

+  
+  === Local bindings === 

+  
+  In Haskell <hask>let</hask> allows local declarations to be made in the context of a single expression. 

+  
+  <haskell> 

+  let x = 1 + 2 in x + 3 

+  </haskell> 

+  
+  is analogous to: 

+  
+  { 

+  int x = 1 + 2; 

+  ... x + 3 ... ; 

+  } 

+  
+  in C, but the Haskell variable x is given a value that is immutable (can 

+  never change). 

+  
+  === Allocation === 

+  
+  When you declare a new variable, Haskell automatically allocates that 

+  value for you  no need to explicitly manage memory. The garbage 

+  collector will then collect any unreachable values once they go out of 

+  scope. 

+  
+  Advanced users can also manage memory by hand using the foreign function 

+  interface. 

+  
+  === Lists === 

+  
+  Lists are ... lists of Haskell values. Defining a new list is trivial, 

+  easier than in Java. 

+  
+  <haskell> 

+  Prelude> [2, 1+2, 4] 

+  [2,3,4] 

+  
+  Prelude> :t [2, 1+2, 4] 

+  [2, 1+2, 4] :: (Num a) => [a] 

+  </haskell> 

+  
+  This automatically allocates space for the list and puts in the elements. 

+  Haskell is garbagecollected like Java so no explicit deallocation is 

+  needed. The type of the list is inferred automatically. All elements of 

+  a list must be of the same type. 

+  
+  <haskell> 

+  Prelude> ["e", concat ["f", "g"], "h"] 

+  ["e","fg","h"] 

+  </haskell> 

+  
+  Notice how the function call <hask>concat ["f","g"]</hask> does not require 

+  parenthesis to delimit the function's arguments. Haskell 

+  uses whitespace, and not commas, and: 

+  
+  * You don't need parentheses for function application in Haskell: <hask>sin 0.3</hask> 

+  * Multiple arguments can be passed in one at a time (curried) which means they can be separated by spaces: <hask>max 3 4</hask>. 

+  
+  Lists must be uniform in their type ("homogeneous"). 

+  
+  <haskell> 

+  Prelude> ['x', True] 

+  
+  Couldn't match `Char' against `Bool' 

+  </haskell> 

+  
+  Here we tried to build a list containing a Char and a Boolean, but the 

+  list ''constructor'', <hask>[]</hask>, has type: 

+  
+  <haskell> 

+  Prelude> :t [] 

+  [] :: [a] 

+  </haskell> 

+  
+  meaning that all elements must be of the same type, <hask>a</hask>. 

+  (For those wondering how to build a list of heterogeneous values, you 

+  would use a ''sum type''): 

+  
+  <haskell> 

+  Prelude> [Left 'x', Right True] 

+  [Left 'x',Right True] 

+  
+  Prelude> :t [Left 'x', Right True] 

+  [Left 'x', Right True] :: [Either Char Bool] 

+  </haskell> 

+  
+  List operations are numerous, as can be seen in the [http://haskell.org/ghc/docs/latest/html/libraries/base/DataList.html Data.List library]. 

+  
+  <haskell> 

+  Prelude> let x = [2,3] 

+  Prelude> let y = 1 : x  'cons' the value 1 onto the list 

+  Prelude> x  the list is immutable 

+  [2,3] 

+  Prelude> y 

+  [1,2,3] 

+  Prelude> x ++ y  joining lists 

+  [2,3,1,2,3] 

+  Prelude> head y  first element of the list is the 'head' 

+  1 

+  Prelude> tail y  the rest of the list is the 'tail' 

+  [2,3] 

+  </haskell> 

+  
+  === Pattern matching === 

+  
+  Haskell supports pattern matching on data structures. This is a powerful 

+  language feature, making code that manipulates data structures 

+  incredibly simple. The core language feature for pattern matching is the 

+  <hask>case</hask> expression: 

+  
+  <haskell> 

+  Prelude> case x of h:t > h 

+  2 

+  </haskell> 

+  
+  The <hask>case</hask> forces <hask>x</hask> (the scrutinee) to match 

+  pattern <hask>h:t</hask>, a list with head and tail, and then we extract 

+  the head, <hask>h</hask>. Tail is similar, and we can use a ''wildcard 

+  pattern'' to ignore the part of the pattern we don't care about: 

+  
+  <haskell> 

+  Prelude> case x of _:t > t 

+  [3] 

+  </haskell> 

+  
+  === Tuples === 

+  
+  Tuples are fixed length structures, whose fields may be of differing 

+  types ("heterogeneous"). They are known as ''product types'' in 

+  programming language theory. 

+  
+  <haskell> 

+  Prelude> let x = (2, "hi") 

+  Prelude> case x of (y,_) > y 

+  2 

+  </haskell> 

+  
+  Unlike the ML family of languages, Haskell uses the same syntax for the 

+  value level as on the type level. So the type of the above tuple is: 

+  
+  <haskell> 

+  Prelude> :t x 

+  x :: (Integer, [Char]) 

+  </haskell> 

+  
+  All the data mentioned so far are immutable  it is impossible to change 

+  an entry in an existing list, tuple, or record without implicitly 

+  copying the data! Also, all variables are immutable. 

+  By default Haskell is a ''pure'' language. Side effects, such as 

+  mutation, are discussed later. 

+  
+  == Functions == 

+  
+  Here is a simple recursive factorial function definition. 

+  
+  <haskell> 

+  Prelude> let fac n = if n == 0 then 1 else n * fac (n1) 

+  
+  Prelude> :t fac 

+  fac :: (Num a) => a > a 

+  
+  Prelude> fac 42 

+  1405006117752879898543142606244511569936384000000000 

+  </haskell> 

+  
+  The function name is <hask>fac</hask>, and the argument is 

+  <hask>n</hask>. This function is recursive (and there is no need to 

+  specially tag it as such, as you would in the ML family of languages). 

+  
+  When you ''apply'' (or invoke) the fac function, you don't need any 

+  special parenthesis around the code. Note that there is no return 

+  statement; instead, the value of the whole bodyexpression is implicitly 

+  what gets returned. 

+  
+  Functions of more than one argument may be defined: 

+  
+  <haskell> 

+  Prelude> let max a b = if a > b then a else b 

+  Prelude> max 3 7 

+  7 

+  </haskell> 

+  
+  Other important aspects of Haskell functions: 

+  
+  * Functions can be defined anywhere in the code via ''lambda abstractions'': 

+  
+  <haskell> 

+  Prelude> ((\x > x + 1) 4) + 7 

+  12 

+  </haskell> 

+  
+  Or, identical to <hask>let f x = x + 1</hask>: 

+  <haskell> 

+  Prelude> let f = \x > x + 1 

+  Prelude> :t f 

+  f :: Integer > Integer 

+  </haskell> 

+  
+  Anonymous functions like this can be very useful. Also, functions can 

+  be passed to and returned from functions. For example, the ''higher order'' 

+  function <hask>map</hask>, applies its function argument to each 

+  element of a list (like a forloop): 

+  
+  <haskell> 

+  Prelude> map (\x > x ^ 2) [1..10] 

+  [1,4,9,16,25,36,49,64,81,100] 

+  </haskell> 

+  
+  In Haskell, we can use ''section'' syntax for more concise anonymous 

+  functions: 

+  
+  <haskell> 

+  Prelude> map (^ 2) [1..10] 

+  [1,4,9,16,25,36,49,64,81,100] 

+  </haskell> 

+  
+  Here <hask>map</hask> takes two arguments, the function 

+  <hask>(^2) :: Integer > Integer</hask>, and a list of numbers. 

+  
+  === Currying === 

+  
+  Currying is a method by which function arguments may be passed one at a 

+  time to a function, rather than passing all arguments in one go in a 

+  structure: 

+  
+  <haskell> 

+  Prelude> let comb n m = if m == 0  m == n then 1 else comb (n1) m + comb (n1) (m1) 

+  
+  Prelude> comb 10 4 

+  210 

+  </haskell> 

+  
+  The type of comb, <hask>Num a => a > a > a</hask>, can be rewritten as 

+  <hask>Num a => a > (a > a)</hask>. That is, it takes a single argument 

+  of some numeric type <hask>a</hask>, and returns a function that takes 

+  another argument of that type! 

+  
+  Indeed, we can give comb only one argument, in which case it returns a 

+  function that we can later use: 

+  
+  <haskell> 

+  Prelude> let comb10 = comb 10 

+  Prelude> comb10 4 

+  210 

+  Prelude> comb10 3 

+  120 

+  </haskell> 

+  
+  Mutually recursive functions may be defined in the same way as normal 

+  functions: 

+  
+  <haskell> 

+  let take [] = [] 

+  (x:xs) = x : skip xs 

+  skip [] = [] 

+  (_:ys) = take ys 

+  
+  Prelude> :t take 

+  take :: [a] > [a] 

+  
+  Prelude> :t skip 

+  skip :: [a] > [a] 

+  
+  Prelude> take [1..10] 

+  [1,3,5,7,9] 

+  
+  Prelude> skip [1..10] 

+  [2,4,6,8,10] 

+  </haskell> 

+  
+  This example also shows a pattern match with multiple cases, either 

+  empty list or nonempty list. More on patterns now. 

+  
+  === Patterns === 

+  
+  Patterns make function definitions much more succinct, as we just saw. 

+  
+  <haskell> 

+  let rev [] = [] 

+  rev (x:xs) = rev xs ++ [x] 

+  </haskell> 

+  
+  In this function definition, <hask>[]</hask> and <hask>(x:xs)</hask> 

+  are patterns against which the value passed to the function is matched. 

+  The match occurs on the structure of the data  that is, on its 

+  ''constructors''. 

+  
+  Lists are defined as: 

+  
+  <haskell> 

+  data [] a = []  a : [a] 

+  </haskell> 

+  
+  That is, a list of some type a has type <hask>[a]</hask>, and it can be 

+  built two ways: 

+  * either the empty list, <hask>[]</hask> 

+  * or an element ''consed'' onto a list, such as <hask>1 : []</hask> or <hask>1 : 2 : 3 : []</hask>. 

+  * For the special case of lists, Haskell provides the syntax sugar: <hask>[1,2,3]</hask> to build the same data. 

+  
+  Thus, <hask>[]</hask> matches against the empty list constructor, and 

+  <hask>(x:xs)</hask>, match against the cons constructor, binding 

+  variables x and xs to the head and tail components of the list. 

+  
+  Remember that <hask>case</hask> is the syntactic primitive for 

+  performing pattern matching (pattern matching in let bindings is sugar 

+  for <hask>case</hask>). Also, the first successful match is taken if 

+  more than one pattern matches: 

+  
+  <haskell> 

+  case [1,2,3] of 

+  (x:y) > True 

+  (x:y:z) > False 

+  [] > True 

+  
+  Warning: Pattern match(es) are overlapped 

+  In a case alternative: (x : y : z) > ... 

+  
+  True 

+  </haskell> 

+  
+  Warnings will be generated at compile time if patterns don't cover all 

+  possibilities, or contain redundant branches. 

+  
+  <haskell> 

+  Prelude> :set Wall 

+  Prelude> case [1,2,3] of (x:_) > x 

+  
+  Warning: Pattern match(es) are nonexhaustive 

+  In a case alternative: Patterns not matched: [] 

+  
+  1 

+  </haskell> 

+  
+  An exception will be thrown at runtime if a pattern match fails: 

+  
+  <haskell> 

+  Prelude> let myhead (x:_) = x 

+  
+  Warning: Pattern match(es) are nonexhaustive 

+  In a case alternative: Patterns not matched: [] 

+  
+  Prelude> myhead [] 

+  *** Exception: <interactive>:1:1636: Nonexhaustive patterns in case 

+  </haskell> 

+  
+  As we have seen, patterns may be used in function definitions. For 

+  example, this looks like a function of two arguments, but its a function 

+  of one argument which matches a pair pattern. 

+  
+  <haskell> 

+  Prelude> let add (x,y) = x + y 

+  Prelude> add (2,3) 

+  5 

+  </haskell> 

+  
+  === Immutable declarations === 

+  
+  Immutable Declarations 

+  
+  * Important feature of letdefined variable values in Haskell (and some other functional languages): they cannot change their value later. 

+  * Greatly helps in reasoning about programswe know the variable's value is fixed. 

+  * Smalltalk also forces method arguments to be immutable; C++'s const and Java's final on fields has a similar effect. 

+  
+  <haskell> 

+  let x = 5 in 

+  let f y = x + 1 in 

+  let x = 7 in f 0  f uses the particular x in lexical scope where it is defined 

+  6 

+  </haskell> 

+  
+  Here's the one that will mess with your mind: the same thing as above 

+  but with the declarations typed into GHCi. (The GHCi environment 

+  conceptually an openended series of lets which never close). 

+  
+  <haskell> 

+  Prelude> let x = 5 

+  Prelude> let f y = x + 1 

+  Prelude> f 0 

+  6 

+  Prelude> let x = 7  not an assignment, a new declaration 

+  Prelude> f 0 

+  6 

+  </haskell> 

+  
+  === Higher order functions === 

+  
+  Haskell, like ML, makes wide use of higherorder functions: functions 

+  that either take other functions as argument or return functions as 

+  results, or both. Higherorder functions are an important component of a 

+  programmer's toolkit. 

+  
+  * They allow "pluggable" programming by passing in and out chunks of code. 

+  * Many new programming design patterns are possible. 

+  * It greatly increases the reusability of code. 

+  * [http://www.cs.kent.ac.uk/pubs/1997/224/index.html Higherorder + Polymorphic = Reusable] 

+  
+  The classic example of a function that takes another function as 

+  argument is the <hask>map</hask> function on lists. It takes a list and 

+  a function and applies the function to every element of the list. 

+  
+  <haskell> 

+  map :: (a > b) > [a] > [b] 

+  map _ [] = [] 

+  map f (x:xs) = f x : map f xs 

+  </haskell> 

+  
+  The lower case variables in the type declaration of map are ''type 

+  variables'', meaning that the function is ''polymorphic'' in that 

+  argument (can take any type). 

+  
+  <haskell> 

+  Prelude> map (*10) [4,2,7] 

+  [40,20,70] 

+  </haskell> 

+  
+  Perhaps the simplest higherorder function is the composer, in 

+  mathematics expressed as ''g o f''. it takes two functions and returns a 

+  new function which is their composition: 

+  
+  <haskell> 

+  (.) :: (b > c) > (a > b) > a > c 

+  (.) f g x = f (g x) 

+  </haskell> 

+  
+  This function takes three arguments: two functions, ''f'' and ''g'', and 

+  a value, ''x''. It then applies ''g'' to ''x'', before applying ''f'' to 

+  the result. For example: 

+  
+  <haskell> 

+  Prelude> let plus3times2 = (*2) . (+3) 

+  Prelude> plus3times2 10 

+  26 

+  </haskell> 

+  
+  As we have seen before, functions are just expressions so can also be 

+  immediately applied after being defined: 

+  
+  <haskell> 

+  Prelude> ((*2) . (+3)) 10 

+  26 

+  Prelude> (.) (*2) (+3) 10 

+  26 

+  </haskell> 

+  
+  Note how Haskell allows the infix function ''.'' to be used in prefix 

+  form, when wrapped in parenthesis. 

+  
+  === Currying === 

+  
+  Currying is an important concept of functional programming; it is named 

+  after logician [http://haskell.org/haskellwiki/Haskell_Brooks_Curry Haskell Curry], 

+  after which the languages Haskell and Curry are named! Multiargument 

+  functions as defined thus far are curried, lets look at what is really 

+  happening. 

+  
+  Here is a twoargument function defined in our usual manner. 

+  
+  <haskell> 

+  Prelude> let myadd x y = x + y 

+  Prelude> myadd 3 4 

+  7 

+  </haskell> 

+  
+  Here is another completely equivalent way to define the same function: 

+  
+  <haskell> 

+  Prelude> let myadd x = \y > x + y 

+  Prelude> :t myadd 

+  myadd :: (Num a) => a > a > a 

+  
+  Prelude> myadd 3 4 

+  7 

+  Prelude> let inc3 = myadd 3 

+  Prelude> inc3 4 

+  7 

+  </haskell> 

+  
+  The main observation is myadd is a function returning a function, so the 

+  way we supply two arguments is 

+  * Invoke the function, get a function back 

+  * Then invoke the returned function passing the second argument. 

+  * Our final value is returned, victory. 

+  * <hask>(myadd 3) 4</hask> is an inlined version of this where the function returned by <hask>myadd 3</hask> is not put in any variable 

+  
+  Here is a third equivalent way to define myadd, as an anonymous function 

+  returning another anonymous function. 

+  
+  <haskell> 

+  Prelude> let myadd = \x > \y > x + y 

+  Prelude> :t myadd 

+  myadd :: Integer > Integer > Integer 

+  </haskell> 

+  
+  With currying, all functions "really" take exactly one argument. 

+  Currying also naturally arises when functions return functions, as in 

+  the map application above showed. Multipleargument functions should 

+  always be written in curried form; all the library functions are 

+  curried. 

+  
+  Note thus far we have curried only twoargument functions; in general, 

+  nargument currying is possible. Functions can also take pairs as 

+  arguments to achieve the effect of a twoargument function: 

+  
+  <haskell> 

+  Prelude> let mypairadd (x,y) = x + y 

+  Prelude> mypairadd (2,3) 

+  5 

+  </haskell> 

+  
+  So, either we can curry or we can pass a pair. We can also write 

+  higherorder functions to switch back and forth between the two forms. 

+  
+  <haskell> 

+  fst :: (a,b) > a 

+  fst (x,_) = x 

+  
+  snd :: (a,b) > b 

+  snd (_,y) = y 

+  
+  curry :: ((a, b) > c) > a > b > c 

+  curry f x y = f (x, y) 

+  
+  uncurry :: (a > b > c) > ((a, b) > c) 

+  uncurry f p = f (fst p) (snd p) 

+  
+  Prelude> :t uncurry myadd 

+  uncurry myadd :: (Integer, Integer) > Integer 

+  
+  Prelude> :t curry mypairadd 

+  curry mypairadd :: Integer > Integer > Integer 

+  
+  Prelude> :t uncurry map 

+  uncurry map :: (a > b, [a]) > [b] 

+  
+  Prelude> :t curry (uncurry myadd)  a noop 

+  curry (uncurry myadd) :: Integer > Integer > Integer 

+  </haskell> 

+  
+  Look at the types: these mappings in both directions in some sense 

+  "implement" the wellknown isomorphism on sets: <hask>A * B > C = A > B > C</hask> 

+  
+  === A bigger example === 

+  
+  Here is a more highpowered example of the use of currying. 

+  
+  <haskell> 

+  foldr f [] y = y 

+  foldr f (x:xs) y = f x (foldr f xs y) 

+  
+  *Main> :t foldr 

+  foldr :: (a > t > t) > [a] > t > t 

+  
+  *Main> let prod = foldr (\a x > a * x) 

+  *Main> :t prod 

+  prod :: [Integer] > Integer > Integer 

+  
+  *Main> let prod0 = prod [1,2,3,4] 

+  *Main> :t prod0 

+  prod0 :: Integer > Integer 

+  
+  *Main> (prod0 1, prod0 2) 

+  (24,48) 

+  </haskell> 

+  
+  Here is an analysis of this recursive function, for the arbitrary 

+  2element list [x1,x2], the call 

+  
+  <haskell> 

+  foldr f [x1, x2] y 

+  </haskell> 

+  
+  ''reduces'' to (by inlining the body of fold): 

+  
+  <haskell> 

+  f x1 (foldr f [x2] y) 

+  </haskell> 

+  
+  which in turn reduces to: 

+  
+  <haskell> 

+  f x1 (f x2 (foldr f [] y))) 

+  </haskell> 

+  
+  and then: 

+  
+  <haskell> 

+  f x1 (f x2 y) 

+  </haskell> 

+  
+  From this we can assert that the general result returned from 

+  <hask>foldr f [x1,x2,...,xn] y</hask> is <hask>f x1 (f x2 f ...(f xn 

+  y)...))))</hask>. Currying allows us to specialize foldr to a 

+  particular function f, as with prod above. 

+  
+  === Proving program properties by induction === 

+  
+  We should in fact be able to prove this property by induction. Its 

+  easier if we reverse the numbering of the list. 

+  
+  '''Lemma'''. <hask>foldr f [xn,...,x1] y</hask> evaluates to 

+  <hask>f xn (f xn1 f ...(f x1 y)...)))</hask> for n greater than 0. 

+  
+  '''Proof'''. Proceed by induction on the length of the list 

+  <hask>[xn,..,x1]</hask>. 

+  
+  ''Base Case'' n=1, i.e. the list is [x1]. The function reduces to 

+  <hask>f x1 (foldr f [] y)</hask> which reduces to <hask>f x1 y</hask> as 

+  hypothesized. 

+  
+  ''Induction Step''. Assume 

+  
+  <haskell> 

+  foldr f [xn,...,x1] y 

+  </haskell> 

+  
+  reduces to 

+  
+  <haskell> 

+  f xn (f xn1 f ...(f x1 y)...))) 

+  </haskell> 

+  
+  and show 

+  
+  <haskell> 

+  foldr f [xn+1, xn,...,x1] y 

+  </haskell> 

+  
+  reduces to 

+  
+  <haskell> 

+  f xn+1 (f xn f ...(f x1 y)...)))) 

+  </haskell> 

+  
+  Computing 

+  
+  <haskell> 

+  foldr f [x1,x2,...,xn,xn+1] y 

+  </haskell> 

+  
+  it matches the pattern with x being xn+1 and xs being 

+  <hask>[xn,...,x1]</hask>. Thus the recursive call is 

+  
+  <haskell> 

+  foldr f [xn,...,x1] y 

+  </haskell> 

+  
+  which by our inductive assumption reduces to 

+  
+  <haskell> 

+  f xn (f xn1 f ...(f x1 y)...))) 

+  </haskell> 

+  
+  And, given this result for the recursive call, the whole function then 

+  returns 

+  
+  <haskell> 

+  f xn+1 (...result of recursive call...) 

+  </haskell> 

+  
+  which is 

+  
+  <haskell> 

+  f xn+1 (f xn (f xn1 f ...(f x1 y)...))) 

+  </haskell> 

+  
+  which is what we needed to show. '''QED''' 

+  
+  The above implementation is inefficient in that ''f'' is explicitly passed to 

+  every recursive call. Here is a more efficient version with identical 

+  functionality. 

+  
+  <haskell> 

+  foldr :: (a > b > b) > b > [a] > b 

+  foldr k z xs = go xs 

+  where 

+  go [] = z 

+  go (y:ys) = y `k` go ys 

+  </haskell> 

+  
+  This function also illustrates how functions may be defined in a local scope, 

+  using <hask>where</hask>. Observe 'go' is defined locally but then exported 

+  since it is the return value of f. 

+  
+  Question: How does the return value 'go' know where to look for k when its called?? 

+  
+  <haskell> 

+  Prelude> let summate = foldr (+) 

+  Prelude> summate [1,2,3,4] 0 

+  10 

+  </haskell> 

+  
+  <hask>summate</hask> is just <hask>go</hask> but somehow it "knows" that k is 

+  <hask>(+)</hask>, even though k is undefined at the top level: 

+  
+  <haskell> 

+  Prelude> k 

+  <interactive>:1:0: Not in scope: `k' 

+  </haskell> 

+  
+  <hask>go</hask> in fact knew the right k to call, so it must have been kept 

+  somewhere: in a ''closure''. At function definition point, the current values 

+  of variables not local to the function definition are remembered in a structure 

+  called a closure. Function values in Haskell are thus really a pair consisting 

+  of the function (pointer) and the local environment, in a closure. 

+  
+  Without making a closure, higherorder functions will do awkward things (such 

+  as binding to whatever 'k' happens to be in scope). Java, C++, C can pass and 

+  return function (pointers), but all functions are defined at the top level so 

+  they have no closures. 

+  
+  === Loading source from a file === 

+  
+  You should never type large amounts of code directly into GHCi! Its impossible 

+  to fix errors. Instead, you should edit in a file. Usingg any editor, save each 

+  group of interlinked functions in a separate file, for example "A.hs". Then, 

+  from GHCi type: 

+  
+  <haskell> 

+  Prelude> :l A.hs 

+  *Main> 

+  </haskell> 

+  
+  This will compile everything in the file. 

+  
+  === Show === 

+  
+  Haskell has the show function. 

+  
+  <haskell> 

+  *Main> show 1 

+  "1" 

+  *Main> show (1,2,'x') 

+  "(1,2,'x')" 

+  *Main> show "haskell" 

+  "\"haskell\"" 

+  *Main> show (Just ()) 

+  "Just ()" 

+  </haskell> 

+  
+  It simply returns a string representation for its arguments. 

+  
+  == Types == 

+  
+  We have generally been ignoring the type system of Haskell up to now. Its time 

+  to focus on typing in more detail. 

+  
+  === Type Declarations === 

+  
+  Haskell infers types for you, but you can add explicit type declarations if you like. 

+  
+  <haskell> 

+  let myadd :: Int > Int > Int 

+  myadd x y = x + y 

+  
+  *Main> :t myadd 

+  myadd :: Int > Int > Int 

+  
+  *Main> :set fglasgowexts 

+  let myadd (x :: Int) (y :: Int) = x + y 

+  
+  *Main> :t myadd 

+  myadd :: Int > Int > Int 

+  </haskell> 

+  
+  You can in fact put type assertions on any variable in an expression to clarify 

+  what type the variable has: 

+  
+  <haskell> 

+  *Main> let myadd (x :: Int) (y :: Int) = (x :: Int) + y 

+  *Main> :t myadd 

+  myadd :: Int > Int > Int 

+  </haskell> 

+  
+  === Type synonyms === 

+  
+  You can also make up your own name for any type. To do this, you must work in a 

+  separate file and load it into GHCi using the ":load A.hs" command. 

+  
+  <haskell> 

+  type IntPair = (Int,Int) 

+  
+  f :: IntPair > Int 

+  f (l,r) = l + r 

+  </haskell> 

+  
+  Working from GHCi: 

+  
+  <haskell> 

+  f :: IntPair > Int 

+  *Main> :l A.hs 

+  *Main> :t f 

+  f :: IntPair > Int 

+  *Main> f (2,3) 

+  5 

+  </haskell> 

+  
+  === Polymorphic Types and Type Inference === 

+  
+  <haskell> 

+  *Main> let id x = x 

+  *Main> :t id 

+  id :: forall t. t > t 

+  *Main> id 3 

+  3 

+  *Main> id True 

+  True 

+  </haskell> 

+  
+  Since <hask>id</hask> was not used as any type in particular, the type of the 

+  function is polymorphic ("many forms"). 

+  * <hask>t</hask> is a type variable, meaning it stands for some arbitrary type. 

+  * Polymorphism is really needed with type inference  inferring <hask>Int > Int</hask> would not be completely general. 

+  
+  === Parametric polymorphism === 

+  
+  The form of polymorphism in Haskell is to be precise, parametric polymorphism. The type above is parametric in ''t'': what comes out is the same type as what came in. ''Generics'' is another term for parametric polymorphism used in some communities. 

+  
+  * Java has no parametric polymorphism, but does have object polymorphism (unfortunately this is often just called polymorphism by some writers) in that a subclass object can fit into a superclassdeclared variable. 

+  * When you want parametric polymorphism in Java you declare the variable to be of type Object, but you have to cast when you get it out which requires a runtime check. 

+  * The Java JDK version 1.5 will have parametrically polymorphic types in it. 

+  
+  The general intuition to have about the type inference algorithm is everything 

+  starts out as having arbitrary types, t, u, etc, but then the use of functions 

+  on values generates constraints that "this thing has the same type as that 

+  thing". 

+  
+  Use of typespecific operators obviously restricts polymorphism: 

+  
+  <haskell> 

+  *Main> let doublenegate x = not (not x) 

+  *Main> :t doublenegate 

+  doublenegate :: Bool > Bool 

+  </haskell> 

+  
+  When a function is defined via let to have polymorphic type, every use can be at a different type: 

+  
+  <haskell> 

+  let id x = x 

+  in case id True of 

+  True > id 3 

+  False > id 4 

+  3 

+  </haskell> 

+  
+  === Algebraic Data Types === 

+  
+  Algebraic data types in Haskell are the analogue of union/variant types in 

+  C/Pascal. Following in the Haskell tradition of lists and tuples, they are not 

+  mutable. Haskell data types must be declared. Here is a really simple algebraic 

+  data type declaration to get warmed up, remember to write this in a separate 

+  file, and load it in to HHCi: 

+  
+  <haskell> 

+  data Height = Tall  Medium  Short 

+  </haskell> 

+  
+  Three constructors have been defined. These are now official constants. 

+  Constructors must be capitalized, and variables must be lowercase in Haskell. 

+  
+  <haskell> 

+  *Main> :l A.hs 

+  *Main> :t Tall 

+  Tall :: Height 

+  *Main> Tall 

+  
+  Top level: 

+  No instance for (Show Height) 

+  </haskell> 

+  
+  So we can type check them, but can't show them yet. Let's ''derive'' the 

+  typeclass <hask>Show</hask> for our data type, which generates a 'show' 

+  function for our data type, which GHCi can then use to display the value. 

+  
+  <haskell> 

+  data Height = Tall  Medium  Short 

+  deriving Show 

+  
+  *Main> :reload 

+  *Main> Tall 

+  Tall 

+  </haskell> 

+  
+  The previous type is only an enumerated type. Much more interesting data types 

+  can be defined. Remember the (recursive) list type: 

+  
+  <haskell> 

+  data [] a = []  a : [a] 

+  
+  *Main> (:) 3 ((:) (3+1) []) 

+  [3,4] 

+  *Main> 3 : 3+1 : [] 

+  [3,4] 

+  </haskell> 

+  
+  This form of type has several new features: 

+  
+  * As in C/Pascal, the data types can have values and they can be recursively defined, plus, 

+  * Polymorphic data types can be defined; ''a'' here is a type argument. 

+  * Note how there is no need to use pointers in defining recursive variant types. The compiler does all that mucking around for you. 

+  * Also note how <hask>(:)</hask>, the constructor, can be used as a function. 

+  
+  We can define trees rather simply: 

+  
+  <haskell> 

+  data Tree a = Empty  Node a (Tree a) (Tree a) 

+  </haskell> 

+  
+  Patterns automatically work for new data types. 

+  
+  === Record Declarations === 

+  
+  Records are data types with labels on fields. They are very similar to structs 

+  of C/C++. Their types are declared just like normal data types, and can be used 

+  in pattern matches. 

+  
+  <haskell> 

+  data OneTwo = OneTwo { one :: Int, two :: String } 

+  deriving Show 

+  
+  *Main> let x = OneTwo { one = 2 , two = "ni" } 

+  *Main> one x 

+  2 

+  *Main> case x of OneTwo { one = x, two = s } > x 

+  2 

+  </haskell> 

+  
+  == Imperative Programming == 

+  
+  Haskell and OCaml differ on imperative programming: OCaml mixes pure 

+  and impure code, while Haskell separates them statically. 

+  
+  The expressions and functions for I/O, mutable states, and other side 

+  effects have a special type. They enjoy a distinguished status: 

+  they are I/O instructions, and the entry point of each complete program 

+  must be one of them. The following honours this distinction by using 

+  the word ''command'' for them (another popular name is ''action''), 

+  though they are also expressions, values, functions. 

+  
+  Commands have types of the form <hask>IO a</hask>, which says it takes 

+  no parameter and it gives an answer of type <hask>a</hask>. (I will 

+  carefully avoid saying it “returns” type <hask>a</hask>, since 

+  “return” is too overloaded.) There are also functions of type 

+  <hask>b > IO a</hask>, and I will abuse terminology and call this a 

+  command needing a parameter of type <hask>b</hask>, even though the 

+  correct description should be: a function from <hask>b</hask> to 

+  commands. 

+  
+  === I/O === 

+  
+  The command for writing a line to Standard Output is 

+  
+  <haskell>putStrLn :: String > IO ()</haskell> 

+  
+  It outputs the string parameter, plus linebreak. And since there is no 

+  answer to give, the answer type is the most boring <hask>()</hask>. 

+  
+  At first, using output commands at the prompt is as easy as 

+  using expressions. 

+  
+  <haskell> 

+  Prelude> putStrLn "Hello world" 

+  Hello world 

+  </haskell> 

+  
+  You can also write a compound command with the <hask>>></hask> operator. 

+  
+  <haskell> 

+  Prelude> putStrLn "Hello" >> putStrLn "world" 

+  Hello 

+  world 

+  </haskell> 

+  
+  The fun begins when you also take input. The command for reading 

+  one line from Standard Input is: 

+  
+  <haskell>getLine :: IO String</haskell> 

+  
+  Note that the type is not <hask>String</hask> or <hask>()>String</hask>. 

+  In a purely functional language, such types promise that all calls 

+  using the same parameter yield the exactly same string. 

+  This is of course what an input command cannot promise. If you read 

+  two lines, they are very likely to be different. The type 

+  <hask>IO String</hask> does not promise to give the same string all 

+  the time. (It only promises to be the same command all the time—a 

+  rather "duh" one.) But this poses a question: how do we get at 

+  the line it reads? 

+  
+  A nonsolution is to expect an operation 

+  <hask>stripIO :: IO a > a</hask>. 

+  What's wrong with this stripthatIO mentality is that it asks to 

+  convert a command, which gives different answers at different calls, 

+  into a pure function, which gives the same answer at different calls. 

+  Contradiction! 

+  
+  But you can ask for a weaker operation: how to pass the answer on to 

+  subsequent commands (e.g., output commands) so they can use it. 

+  A moment of thought reveals that this is all you ever need. 

+  The operator sought is 

+  
+  <haskell>(>>=) :: IO a > (a > IO b) > IO b</haskell> 

+  
+  It builds a compound command from two commands, the first one of which 

+  takes no parameter and gives an answer of type <hask>a</hask>, and the 

+  second of which needs a parameter of type <hask>a</hask>. 

+  You guessed it: this operator extracts the answer from the first command 

+  and passes it to the second. Now you have some way to use the answer! 

+  
+  Here is the first example. Why don't we read a line and immediately 

+  display it? <hask>getLine</hask> answers a string, and 

+  <hask>putStrLn</hask> wants a string. Perfect match! 

+  
+  <haskell> 

+  Prelude> getLine >>= putStrLn 

+  Good morning 

+  Good morning 

+  </haskell> 

+  
+  But more often you want to output something derived from the input, 

+  rather than the input itself verbatim. To do this, you customize the 

+  second command to smuggle in the derivation. The trick of anonymous 

+  functions is very useful for this: 

+  
+  <haskell> 

+  Prelude> getLine >>= \s > putStrLn ("You entered: " ++ s) 

+  Good morning 

+  You entered: Good morning 

+  </haskell> 

+  
+  You will also want to give derived answers, especially if you write 

+  subroutines to be called from other code. This is facilitated by the 

+  command that takes a parameter and simply gives it as the answer 

+  (it is curiously named <hask>return</hask>): 

+  
+  <haskell> 

+  return :: a > IO a 

+  </haskell> 

+  
+  For example here is a routine that reads a line and answers a derived 

+  string, with a sample usage: 

+  
+  <haskell> 

+  Prelude> let mycmd = getLine >>= \s > return ("You entered: " ++ s) 

+  Prelude> mycmd >>= putStrLn 

+  Good morning 

+  You entered: Good morning 

+  </haskell> 

+  
+  Some programmers never use Standard Input. Reading files is more common. 

+  One command for this is: 

+  
+  <haskell>readFile :: String > IO String</haskell> 

+  
+  The parameter specifies the file path. Let us read a file and print 

+  out its first 10 characters (wherever available). Of course please 

+  change the filename to refer to some file you actually possess. 

+  
+  <haskell> 

+  Prelude> readFile "file.txt" >>= \s > putStrLn (take 10 s) 

+  abcdefghij 

+  </haskell> 

+  
+  Do not worry about slurping up the whole file into memory; 

+  <hask>readFile</hask> performs a magic of payasyougo reading. 

+  
+  A while ago I showed the <hask>>></hask> operator for compound commands 

+  without elaboration. I can now elaborate it: it merely uses 

+  <hask>>>=</hask> in a way that throws away the first answer: 

+  
+  <haskell> 

+  putStrLn "x" >> putStrLn "y" 

+  = putStrLn "x" >>= \_ > putStrLn "y" 

+  </haskell> 

+  
+  === doNotation === 

+  
+  To bring imperative code closer to imperative look, Haskell 

+  provides the ''donotation'', which hides the <hask>>>=</hask> 

+  operator and the anonymous functions. An example illustrates 

+  this notation well, and it should be easy to generalize: 

+  
+  <haskell> 

+  cmd0 >>= \x > cmd1 >>= \_ > cmd2 >>= \z > cmd3 

+  = do { x < cmd0; cmd1; z < cmd2; cmd3 } 

+  </haskell> 

+  
+  (<hask>cmd1</hask>, <hask>cmd2</hask>, and <hask>cmd3</hask> may 

+  use <hask>x</hask> as a parameter; similarly, <hask>cmd3</hask> 

+  may use <hask>z</hask> as a parameter. At the end, between 

+  <hask>cmd3</hask> and <hask>}</hask>, you may choose to insert 

+  or omit semicolons; similarly right after <hask>{</hask> at the 

+  beginning.) 

+  
+  Below we reexpress examples in the previous section in the donotation. 

+  
+  <haskell> 

+  Prelude> do { putStrLn "Hello"; putStrLn "world" } 

+  Hello 

+  world 

+  
+  Prelude> do { s < getLine; putStrLn s } 

+  Good morning 

+  Good morning 

+  
+  Prelude> do { s < getLine; putStrLn ("You entered: " ++ s) } 

+  Good morning 

+  You entered: Good morning 

+  
+  Prelude> let mycmd = do { s < getLine; return ("You entered: " ++ s) } 

+  Prelude> do { s < mycmd; putStrLn s } 

+  Good morning 

+  You entered: Good morning 

+  
+  Prelude> do { s < readFile "file.txt"; putStrLn (take 10 s) } 

+  abcdefghij 

+  </haskell> 

+  
+  At the prompt it is necessary to write oneliners. In a source code 

+  file it is more common to use multiple lines, one line for one 

+  command, as per tradition. In this case, ''layout rules'' allow 

+  omitting <hask>{;}</hask> in favour of indentation. Thus, here are 

+  two valid ways of writing the same doblock in a source code file, 

+  one with <hask>{;}</hask> and the other with layout. 

+  
+  <haskell> 

+  
+  foo = do { name < getLine; foo = do name < getLine 

+  s < readFile name; s < readFile name 

+  putStrLn (take 10 s) putStrLn (take 10 s) 

+  } 

+  </haskell> 

+  
+  === Mutable variables === 

+  
+  Data.IORef 

+  Data.Array.MArray 

+  Data.Array.IO 

+  
+  === Exceptions === 

+  
+  Control.Exception 

+  
+  === Concurrency === 

+  
+  Control.Concurrent 

+  Control.Concurrent.MVar 

+  
+  == Monads == 

+  
+  === Mutable variables === 

+  
+  Data.STRef 

+  Data.Array.MArray 

+  Data.Array.ST 

+  
+  === State === 

+  
+  The State monad 

+  
+  === Monad transformers === 

+  
+  == Compilation == 

+  
+  You can easily compile your Haskell modules to standalone executables. For example, write this in a file "A.hs": 

+  
+  <haskell> 

+  main = putStrLn "Hello, World!" 

+  </haskell> 

+  
+  In general, <hask>main</hask> is the entry point, and you must define it to be whatever you want run. (TODO: once the monad/IO section is done, this place should also say more about main and IO.) 

+  
+  The compiler, on unix systems, is '''ghc'''. For example "A.hs" can be compiled and run as: 

+  
+  $ ghc A.hs 

+  $ ./a.out 

+  Hello, World! 

+  
+  For multiple modules, use the ''make'' flag to GHC. Example: write these two modules: 

+  
+  B.hs: 

+  
+  <haskell> 

+  import M1 

+  main = putStrLn s 

+  </haskell> 

+  
+  M1.hs: 

+  
+  <haskell> 

+  module M1 where 

+  s = "Hi, Everyone!" 

+  </haskell> 

−  == Declaring our own types == 

+  To compile and run (this will automatically look for M1.hs): 

−  == Imperative features: monadic IO, references, mutable arrays, exceptions == 

+  $ ghc make B.hs 

+  $ ./a.out 

+  Hi, Everyone! 

−  == Type classes == 

+  In general, one and only one file must define <hask>main</hask>. In general, for all other files, the filename must match the module name. 
Latest revision as of 21:23, 29 October 2011
Haskell is:
 A language developed by the programming languages research community.
 Is a lazy, purely functional language (that also has imperative features such as side effects and mutable state, along with strict evaluation)
 Born as an open source vehicle for programming language research
 One of the youngest children of ML and Lisp
 Particularly useful for programs that manipulate data structures (such as compilers and interpreters), and for concurrent/parallel programming
Inspired by the article Introduction to OCaml, and translated from the OCaml by Don Stewart.
Contents
Background
 1990. Haskell 1.0
 1991. Haskell 1.1
 1993. Haskell 1.2
 1996. Haskell 1.3
 1997. Haskell 1.4
 1998. Haskell 98
 20002006. Period of rapid language and community growth
 ~2007. Haskell Prime
 2009. Haskell 2010
Haskell features
Has some novel features relative to Java (and C++).
 Immutable variables by default (mutable state programmed via monads)
 Pure by default (side effects are programmed via monads)
 Lazy evaluation: results are only computed if they're required (strictness optional)
 Everything is an expression
 Firstclass functions: functions can be defined anywhere, passed as arguments, and returned as values.
 Both compiled and interpreted implementations available
 Full type inference  type declarations optional
 Pattern matching on data structures  data structures are first class!
 Parametric polymorphism
 Bounded parametric polymorphism
These are all conceptually more advanced ideas.
Compared to similar functional languages, Haskell differs in that it has support for:
 Lazy evaluation
 Pure functions by default
 Monadic side effects
 Type classes
 Syntax based on layout
The GHC Haskell compiler, in particular, provides some interesting extensions:
 Generalised algebraic data types
 Impredicative types system
 Software transactional memory
 Parallel, SMP runtime system
The Basics
Read the language definition to supplement these notes. For more depth and examples see the Haskell wiki.
Interacting with the language
Haskell is both compiled and interpreted. For exploration purposes, we'll consider interacting with Haskell via the GHCi interpreter:
 expressions are entered at the prompt
 newline signals end of input
Here is a GHCi session, starting from a UNIX prompt.
$ ghci ___ ___ _ / _ \ /\ /\/ __(_) / /_\// /_/ / /   GHC Interactive, version 6.4.2, for Haskell 98. / /_\\/ __ / /___  http://www.haskell.org/ghc/ \____/\/ /_/\____/_ Type :? for help.
Loading package base1.0 ... linking ... done.
Here the incredibly simple Haskell program let x = 3+4
is
compiled and loaded, and available via the variable x
.
Prelude> let x = 3 + 4
We can ask the system what type it automaticaly inferred for our
variable. x :: Integer
means that the variable
x
"has type" Integer
, the type of unbounded
integer values.
Prelude> :t x
x :: Integer
A variable evaluates to its value.
Prelude> x
7
The variable x
is in scope, so we can reuse it in later
expressions.
Prelude> x + 4
11
Local variables may be bound using let
, which declares a
new binding for a variable with local scope.
Prelude> let x = 4 in x + 3
7
Alternatively, declarations typed in at the top level are like an openended let:
Prelude> let x = 4
Prelude> let y = x + 3
Prelude> x * x
16
Prelude> :t x
x :: Integer
Prelude> :t y
y :: Integer
Prelude> :t x * x
x * x :: Integer
Notice that type inference infers the correct type for all the expressions, without us having to ever specify the type explicitly.
Basic types
There is a range of basic types, defined in the language Prelude.
Int  bounded, wordsized integers
Integer  unbounded integers
Double  floating point values
Char  characters
String  strings
()  the unit type
Bool  booleans
[a]  lists
(a,b)  tuples / product types
Either a b  sum types
Maybe a  optional values
For example:
7
12312412412412321
3.1415
'x'
"haskell"
()
True, False
[1,2,3,4,5]
('x', 42)
Left True, Right "string"
Nothing, Just True
These types have all the usual operations on them, in the standard libraries.
Libraries
 The Prelude contains the core operations on basic types. It is imported by default into every Haskell module. For example;
+  div mod &&  not < > == /=
Learn the Prelude well. Less basic functions are found in the standard libraries. For data structures such as List, Array and finite maps.
To use functions from these modules you have to import them, or in GHCi, refer to the qualified name, for example to use the toUpper function on Chars:
Prelude> Char.toUpper 'x'
'X'
Prelude> :m + Char
Prelude Char> toUpper 'y'
'Y'
In a source file, you have to import the module explicitly:
import Char
Overloading
Haskell uses typeclasses to methodically allow overloading. A typeclass describes a set of functions, and any type which provides those functions can be made an instance of that type. This avoids the syntactic redundancy of languages like OCaml.
For example, the function *
is a method of the typeclass
Num
, as we can see from its type:
Prelude> :t (*)
(*) :: (Num a) => a > a > a
Which can be read as "* is a polymorphic function, taking two values of some type 'a', and returning a result of the same type, where the type 'a' is a member of the class Num".
This means that it will operate on any type in the Num
class, of which the following types are members: Double, Float, Int, Integer
. Thus:
Prelude> 2.3 * 5.7
13.11
or on integers:
Prelude> 2 * 5
10
The type of the arguments determines which instance of *
is
used. Haskell also never performs implicit coercions, all coercions must
be explicit. For example, if we try to multiply two different types,
then the type check against * :: Num a => a > a > a
will
fail.
Prelude> (2.3 :: Double) * (5 :: Int)
<interactive>:1:19:
Couldn't match `Double' against `Int'
Expected type: Double
Inferred type: Int
In the expression: 5 :: Int
In the second argument of `(*)', namely `(5 :: Int)'
To convert 5 to a Double
we'd write:
Prelude> (2.3 :: Double) * fromIntegral (5 :: Int)
11.5
Why bother  why not allow the system to implicitly coerce types? Implicit type conversions by the system are the source of innumerable hard to find bugs in languages that support them, and makes reasoning about a program harder, since you must apply not just the language's semantics, but an extra set of coercion rules.
Note that If we leave off the type signatures however, Haskell will helpfully infer the most general type:
Prelude> 2.3 * 5
11.5
Expressions
In Haskell, expressions are everything. There are no pure "statements"
like in Java/C++. For instance, in Haskell, ifthenelse
is a kind of expression, and results in a value based on the condition part.
Prelude> (if 2 == 3 then 5 else 6) + 1
7
Prelude> (if 2 == 3 then 5 else 6.5) + 1
7.5
Local bindings
In Haskell let
allows local declarations to be made in the context of a single expression.
let x = 1 + 2 in x + 3
is analogous to:
{ int x = 1 + 2; ... x + 3 ... ; }
in C, but the Haskell variable x is given a value that is immutable (can never change).
Allocation
When you declare a new variable, Haskell automatically allocates that value for you  no need to explicitly manage memory. The garbage collector will then collect any unreachable values once they go out of scope.
Advanced users can also manage memory by hand using the foreign function interface.
Lists
Lists are ... lists of Haskell values. Defining a new list is trivial, easier than in Java.
Prelude> [2, 1+2, 4]
[2,3,4]
Prelude> :t [2, 1+2, 4]
[2, 1+2, 4] :: (Num a) => [a]
This automatically allocates space for the list and puts in the elements. Haskell is garbagecollected like Java so no explicit deallocation is needed. The type of the list is inferred automatically. All elements of a list must be of the same type.
Prelude> ["e", concat ["f", "g"], "h"]
["e","fg","h"]
Notice how the function call concat ["f","g"]
does not require
parenthesis to delimit the function's arguments. Haskell
uses whitespace, and not commas, and:
 You don't need parentheses for function application in Haskell:
sin 0.3
 Multiple arguments can be passed in one at a time (curried) which means they can be separated by spaces:
max 3 4
.
Lists must be uniform in their type ("homogeneous").
Prelude> ['x', True]
Couldn't match `Char' against `Bool'
Here we tried to build a list containing a Char and a Boolean, but the
list constructor, []
, has type:
Prelude> :t []
[] :: [a]
meaning that all elements must be of the same type, a
.
(For those wondering how to build a list of heterogeneous values, you
would use a sum type):
Prelude> [Left 'x', Right True]
[Left 'x',Right True]
Prelude> :t [Left 'x', Right True]
[Left 'x', Right True] :: [Either Char Bool]
List operations are numerous, as can be seen in the Data.List library.
Prelude> let x = [2,3]
Prelude> let y = 1 : x  'cons' the value 1 onto the list
Prelude> x  the list is immutable
[2,3]
Prelude> y
[1,2,3]
Prelude> x ++ y  joining lists
[2,3,1,2,3]
Prelude> head y  first element of the list is the 'head'
1
Prelude> tail y  the rest of the list is the 'tail'
[2,3]
Pattern matching
Haskell supports pattern matching on data structures. This is a powerful
language feature, making code that manipulates data structures
incredibly simple. The core language feature for pattern matching is the
case
expression:
Prelude> case x of h:t > h
2
The case
forces x
(the scrutinee) to match
pattern h:t
, a list with head and tail, and then we extract
the head, h
. Tail is similar, and we can use a wildcard
pattern to ignore the part of the pattern we don't care about:
Prelude> case x of _:t > t
[3]
Tuples
Tuples are fixed length structures, whose fields may be of differing types ("heterogeneous"). They are known as product types in programming language theory.
Prelude> let x = (2, "hi")
Prelude> case x of (y,_) > y
2
Unlike the ML family of languages, Haskell uses the same syntax for the value level as on the type level. So the type of the above tuple is:
Prelude> :t x
x :: (Integer, [Char])
All the data mentioned so far are immutable  it is impossible to change an entry in an existing list, tuple, or record without implicitly copying the data! Also, all variables are immutable. By default Haskell is a pure language. Side effects, such as mutation, are discussed later.
Functions
Here is a simple recursive factorial function definition.
Prelude> let fac n = if n == 0 then 1 else n * fac (n1)
Prelude> :t fac
fac :: (Num a) => a > a
Prelude> fac 42
1405006117752879898543142606244511569936384000000000
The function name is fac
, and the argument is
n
. This function is recursive (and there is no need to
specially tag it as such, as you would in the ML family of languages).
When you apply (or invoke) the fac function, you don't need any special parenthesis around the code. Note that there is no return statement; instead, the value of the whole bodyexpression is implicitly what gets returned.
Functions of more than one argument may be defined:
Prelude> let max a b = if a > b then a else b
Prelude> max 3 7
7
Other important aspects of Haskell functions:
 Functions can be defined anywhere in the code via lambda abstractions:
Prelude> ((\x > x + 1) 4) + 7
12
Or, identical to let f x = x + 1
:
Prelude> let f = \x > x + 1
Prelude> :t f
f :: Integer > Integer
Anonymous functions like this can be very useful. Also, functions can
be passed to and returned from functions. For example, the higher order
function map
, applies its function argument to each
element of a list (like a forloop):
Prelude> map (\x > x ^ 2) [1..10]
[1,4,9,16,25,36,49,64,81,100]
In Haskell, we can use section syntax for more concise anonymous functions:
Prelude> map (^ 2) [1..10]
[1,4,9,16,25,36,49,64,81,100]
Here map
takes two arguments, the function
(^2) :: Integer > Integer
, and a list of numbers.
Currying
Currying is a method by which function arguments may be passed one at a time to a function, rather than passing all arguments in one go in a structure:
Prelude> let comb n m = if m == 0  m == n then 1 else comb (n1) m + comb (n1) (m1)
Prelude> comb 10 4
210
The type of comb, Num a => a > a > a
, can be rewritten as
Num a => a > (a > a)
. That is, it takes a single argument
of some numeric type a
, and returns a function that takes
another argument of that type!
Indeed, we can give comb only one argument, in which case it returns a function that we can later use:
Prelude> let comb10 = comb 10
Prelude> comb10 4
210
Prelude> comb10 3
120
Mutually recursive functions may be defined in the same way as normal functions:
let take [] = []
(x:xs) = x : skip xs
skip [] = []
(_:ys) = take ys
Prelude> :t take
take :: [a] > [a]
Prelude> :t skip
skip :: [a] > [a]
Prelude> take [1..10]
[1,3,5,7,9]
Prelude> skip [1..10]
[2,4,6,8,10]
This example also shows a pattern match with multiple cases, either empty list or nonempty list. More on patterns now.
Patterns
Patterns make function definitions much more succinct, as we just saw.
let rev [] = []
rev (x:xs) = rev xs ++ [x]
In this function definition, []
and (x:xs)
are patterns against which the value passed to the function is matched.
The match occurs on the structure of the data  that is, on its
constructors.
Lists are defined as:
data [] a = []  a : [a]
That is, a list of some type a has type [a]
, and it can be
built two ways:
 either the empty list,
[]
 or an element consed onto a list, such as
1 : []
or1 : 2 : 3 : []
.  For the special case of lists, Haskell provides the syntax sugar:
[1,2,3]
to build the same data.
Thus, []
matches against the empty list constructor, and
(x:xs)
, match against the cons constructor, binding
variables x and xs to the head and tail components of the list.
Remember that case
is the syntactic primitive for
performing pattern matching (pattern matching in let bindings is sugar
for case
). Also, the first successful match is taken if
more than one pattern matches:
case [1,2,3] of
(x:y) > True
(x:y:z) > False
[] > True
Warning: Pattern match(es) are overlapped
In a case alternative: (x : y : z) > ...
True
Warnings will be generated at compile time if patterns don't cover all possibilities, or contain redundant branches.
Prelude> :set Wall
Prelude> case [1,2,3] of (x:_) > x
Warning: Pattern match(es) are nonexhaustive
In a case alternative: Patterns not matched: []
1
An exception will be thrown at runtime if a pattern match fails:
Prelude> let myhead (x:_) = x
Warning: Pattern match(es) are nonexhaustive
In a case alternative: Patterns not matched: []
Prelude> myhead []
*** Exception: <interactive>:1:1636: Nonexhaustive patterns in case
As we have seen, patterns may be used in function definitions. For example, this looks like a function of two arguments, but its a function of one argument which matches a pair pattern.
Prelude> let add (x,y) = x + y
Prelude> add (2,3)
5
Immutable declarations
Immutable Declarations
 Important feature of letdefined variable values in Haskell (and some other functional languages): they cannot change their value later.
 Greatly helps in reasoning about programswe know the variable's value is fixed.
 Smalltalk also forces method arguments to be immutable; C++'s const and Java's final on fields has a similar effect.
let x = 5 in
let f y = x + 1 in
let x = 7 in f 0  f uses the particular x in lexical scope where it is defined
6
Here's the one that will mess with your mind: the same thing as above but with the declarations typed into GHCi. (The GHCi environment conceptually an openended series of lets which never close).
Prelude> let x = 5
Prelude> let f y = x + 1
Prelude> f 0
6
Prelude> let x = 7  not an assignment, a new declaration
Prelude> f 0
6
Higher order functions
Haskell, like ML, makes wide use of higherorder functions: functions that either take other functions as argument or return functions as results, or both. Higherorder functions are an important component of a programmer's toolkit.
 They allow "pluggable" programming by passing in and out chunks of code.
 Many new programming design patterns are possible.
 It greatly increases the reusability of code.
 Higherorder + Polymorphic = Reusable
The classic example of a function that takes another function as
argument is the map
function on lists. It takes a list and
a function and applies the function to every element of the list.
map :: (a > b) > [a] > [b]
map _ [] = []
map f (x:xs) = f x : map f xs
The lower case variables in the type declaration of map are type variables, meaning that the function is polymorphic in that argument (can take any type).
Prelude> map (*10) [4,2,7]
[40,20,70]
Perhaps the simplest higherorder function is the composer, in mathematics expressed as g o f. it takes two functions and returns a new function which is their composition:
(.) :: (b > c) > (a > b) > a > c
(.) f g x = f (g x)
This function takes three arguments: two functions, f and g, and a value, x. It then applies g to x, before applying f to the result. For example:
Prelude> let plus3times2 = (*2) . (+3)
Prelude> plus3times2 10
26
As we have seen before, functions are just expressions so can also be immediately applied after being defined:
Prelude> ((*2) . (+3)) 10
26
Prelude> (.) (*2) (+3) 10
26
Note how Haskell allows the infix function . to be used in prefix form, when wrapped in parenthesis.
Currying
Currying is an important concept of functional programming; it is named after logician Haskell Curry, after which the languages Haskell and Curry are named! Multiargument functions as defined thus far are curried, lets look at what is really happening.
Here is a twoargument function defined in our usual manner.
Prelude> let myadd x y = x + y
Prelude> myadd 3 4
7
Here is another completely equivalent way to define the same function:
Prelude> let myadd x = \y > x + y
Prelude> :t myadd
myadd :: (Num a) => a > a > a
Prelude> myadd 3 4
7
Prelude> let inc3 = myadd 3
Prelude> inc3 4
7
The main observation is myadd is a function returning a function, so the way we supply two arguments is
 Invoke the function, get a function back
 Then invoke the returned function passing the second argument.
 Our final value is returned, victory.

(myadd 3) 4
is an inlined version of this where the function returned bymyadd 3
is not put in any variable
Here is a third equivalent way to define myadd, as an anonymous function returning another anonymous function.
Prelude> let myadd = \x > \y > x + y
Prelude> :t myadd
myadd :: Integer > Integer > Integer
With currying, all functions "really" take exactly one argument. Currying also naturally arises when functions return functions, as in the map application above showed. Multipleargument functions should always be written in curried form; all the library functions are curried.
Note thus far we have curried only twoargument functions; in general, nargument currying is possible. Functions can also take pairs as arguments to achieve the effect of a twoargument function:
Prelude> let mypairadd (x,y) = x + y
Prelude> mypairadd (2,3)
5
So, either we can curry or we can pass a pair. We can also write higherorder functions to switch back and forth between the two forms.
fst :: (a,b) > a
fst (x,_) = x
snd :: (a,b) > b
snd (_,y) = y
curry :: ((a, b) > c) > a > b > c
curry f x y = f (x, y)
uncurry :: (a > b > c) > ((a, b) > c)
uncurry f p = f (fst p) (snd p)
Prelude> :t uncurry myadd
uncurry myadd :: (Integer, Integer) > Integer
Prelude> :t curry mypairadd
curry mypairadd :: Integer > Integer > Integer
Prelude> :t uncurry map
uncurry map :: (a > b, [a]) > [b]
Prelude> :t curry (uncurry myadd)  a noop
curry (uncurry myadd) :: Integer > Integer > Integer
Look at the types: these mappings in both directions in some sense
"implement" the wellknown isomorphism on sets: A * B > C = A > B > C
A bigger example
Here is a more highpowered example of the use of currying.
foldr f [] y = y
foldr f (x:xs) y = f x (foldr f xs y)
*Main> :t foldr
foldr :: (a > t > t) > [a] > t > t
*Main> let prod = foldr (\a x > a * x)
*Main> :t prod
prod :: [Integer] > Integer > Integer
*Main> let prod0 = prod [1,2,3,4]
*Main> :t prod0
prod0 :: Integer > Integer
*Main> (prod0 1, prod0 2)
(24,48)
Here is an analysis of this recursive function, for the arbitrary 2element list [x1,x2], the call
foldr f [x1, x2] y
reduces to (by inlining the body of fold):
f x1 (foldr f [x2] y)
which in turn reduces to:
f x1 (f x2 (foldr f [] y)))
and then:
f x1 (f x2 y)
From this we can assert that the general result returned from
foldr f [x1,x2,...,xn] y
is f x1 (f x2 f ...(f xn y)...))))
. Currying allows us to specialize foldr to a
particular function f, as with prod above.
Proving program properties by induction
We should in fact be able to prove this property by induction. Its easier if we reverse the numbering of the list.
Lemma. foldr f [xn,...,x1] y
evaluates to
f xn (f xn1 f ...(f x1 y)...)))
for n greater than 0.
Proof. Proceed by induction on the length of the list
[xn,..,x1]
.
Base Case n=1, i.e. the list is [x1]. The function reduces to
f x1 (foldr f [] y)
which reduces to f x1 y
as
hypothesized.
Induction Step. Assume
foldr f [xn,...,x1] y
reduces to
f xn (f xn1 f ...(f x1 y)...)))
and show
foldr f [xn+1, xn,...,x1] y
reduces to
f xn+1 (f xn f ...(f x1 y)...))))
Computing
foldr f [x1,x2,...,xn,xn+1] y
it matches the pattern with x being xn+1 and xs being
[xn,...,x1]
. Thus the recursive call is
foldr f [xn,...,x1] y
which by our inductive assumption reduces to
f xn (f xn1 f ...(f x1 y)...)))
And, given this result for the recursive call, the whole function then returns
f xn+1 (...result of recursive call...)
which is
f xn+1 (f xn (f xn1 f ...(f x1 y)...)))
which is what we needed to show. QED
The above implementation is inefficient in that f is explicitly passed to every recursive call. Here is a more efficient version with identical functionality.
foldr :: (a > b > b) > b > [a] > b
foldr k z xs = go xs
where
go [] = z
go (y:ys) = y `k` go ys
This function also illustrates how functions may be defined in a local scope,
using where
. Observe 'go' is defined locally but then exported
since it is the return value of f.
Question: How does the return value 'go' know where to look for k when its called??
Prelude> let summate = foldr (+)
Prelude> summate [1,2,3,4] 0
10
summate
is just go
but somehow it "knows" that k is
(+)
, even though k is undefined at the top level:
Prelude> k
<interactive>:1:0: Not in scope: `k'
go
in fact knew the right k to call, so it must have been kept
somewhere: in a closure. At function definition point, the current values
of variables not local to the function definition are remembered in a structure
called a closure. Function values in Haskell are thus really a pair consisting
of the function (pointer) and the local environment, in a closure.
Without making a closure, higherorder functions will do awkward things (such as binding to whatever 'k' happens to be in scope). Java, C++, C can pass and return function (pointers), but all functions are defined at the top level so they have no closures.
Loading source from a file
You should never type large amounts of code directly into GHCi! Its impossible to fix errors. Instead, you should edit in a file. Usingg any editor, save each group of interlinked functions in a separate file, for example "A.hs". Then, from GHCi type:
Prelude> :l A.hs
*Main>
This will compile everything in the file.
Show
Haskell has the show function.
*Main> show 1
"1"
*Main> show (1,2,'x')
"(1,2,'x')"
*Main> show "haskell"
"\"haskell\""
*Main> show (Just ())
"Just ()"
It simply returns a string representation for its arguments.
Types
We have generally been ignoring the type system of Haskell up to now. Its time to focus on typing in more detail.
Type Declarations
Haskell infers types for you, but you can add explicit type declarations if you like.
let myadd :: Int > Int > Int
myadd x y = x + y
*Main> :t myadd
myadd :: Int > Int > Int
*Main> :set fglasgowexts
let myadd (x :: Int) (y :: Int) = x + y
*Main> :t myadd
myadd :: Int > Int > Int
You can in fact put type assertions on any variable in an expression to clarify what type the variable has:
*Main> let myadd (x :: Int) (y :: Int) = (x :: Int) + y
*Main> :t myadd
myadd :: Int > Int > Int
Type synonyms
You can also make up your own name for any type. To do this, you must work in a separate file and load it into GHCi using the ":load A.hs" command.
type IntPair = (Int,Int)
f :: IntPair > Int
f (l,r) = l + r
Working from GHCi:
f :: IntPair > Int
*Main> :l A.hs
*Main> :t f
f :: IntPair > Int
*Main> f (2,3)
5
Polymorphic Types and Type Inference
*Main> let id x = x
*Main> :t id
id :: forall t. t > t
*Main> id 3
3
*Main> id True
True
Since id
was not used as any type in particular, the type of the
function is polymorphic ("many forms").

t
is a type variable, meaning it stands for some arbitrary type.  Polymorphism is really needed with type inference  inferring
Int > Int
would not be completely general.
Parametric polymorphism
The form of polymorphism in Haskell is to be precise, parametric polymorphism. The type above is parametric in t: what comes out is the same type as what came in. Generics is another term for parametric polymorphism used in some communities.
 Java has no parametric polymorphism, but does have object polymorphism (unfortunately this is often just called polymorphism by some writers) in that a subclass object can fit into a superclassdeclared variable.
 When you want parametric polymorphism in Java you declare the variable to be of type Object, but you have to cast when you get it out which requires a runtime check.
 The Java JDK version 1.5 will have parametrically polymorphic types in it.
The general intuition to have about the type inference algorithm is everything starts out as having arbitrary types, t, u, etc, but then the use of functions on values generates constraints that "this thing has the same type as that thing".
Use of typespecific operators obviously restricts polymorphism:
*Main> let doublenegate x = not (not x)
*Main> :t doublenegate
doublenegate :: Bool > Bool
When a function is defined via let to have polymorphic type, every use can be at a different type:
let id x = x
in case id True of
True > id 3
False > id 4
3
Algebraic Data Types
Algebraic data types in Haskell are the analogue of union/variant types in C/Pascal. Following in the Haskell tradition of lists and tuples, they are not mutable. Haskell data types must be declared. Here is a really simple algebraic data type declaration to get warmed up, remember to write this in a separate file, and load it in to HHCi:
data Height = Tall  Medium  Short
Three constructors have been defined. These are now official constants. Constructors must be capitalized, and variables must be lowercase in Haskell.
*Main> :l A.hs
*Main> :t Tall
Tall :: Height
*Main> Tall
Top level:
No instance for (Show Height)
So we can type check them, but can't show them yet. Let's derive the
typeclass Show
for our data type, which generates a 'show'
function for our data type, which GHCi can then use to display the value.
data Height = Tall  Medium  Short
deriving Show
*Main> :reload
*Main> Tall
Tall
The previous type is only an enumerated type. Much more interesting data types can be defined. Remember the (recursive) list type:
data [] a = []  a : [a]
*Main> (:) 3 ((:) (3+1) [])
[3,4]
*Main> 3 : 3+1 : []
[3,4]
This form of type has several new features:
 As in C/Pascal, the data types can have values and they can be recursively defined, plus,
 Polymorphic data types can be defined; a here is a type argument.
 Note how there is no need to use pointers in defining recursive variant types. The compiler does all that mucking around for you.
 Also note how
(:)
, the constructor, can be used as a function.
We can define trees rather simply:
data Tree a = Empty  Node a (Tree a) (Tree a)
Patterns automatically work for new data types.
Record Declarations
Records are data types with labels on fields. They are very similar to structs of C/C++. Their types are declared just like normal data types, and can be used in pattern matches.
data OneTwo = OneTwo { one :: Int, two :: String }
deriving Show
*Main> let x = OneTwo { one = 2 , two = "ni" }
*Main> one x
2
*Main> case x of OneTwo { one = x, two = s } > x
2
Imperative Programming
Haskell and OCaml differ on imperative programming: OCaml mixes pure and impure code, while Haskell separates them statically.
The expressions and functions for I/O, mutable states, and other side effects have a special type. They enjoy a distinguished status: they are I/O instructions, and the entry point of each complete program must be one of them. The following honours this distinction by using the word command for them (another popular name is action), though they are also expressions, values, functions.
Commands have types of the form IO a
, which says it takes
no parameter and it gives an answer of type a
. (I will
carefully avoid saying it “returns” type a
, since
“return” is too overloaded.) There are also functions of type
b > IO a
, and I will abuse terminology and call this a
command needing a parameter of type b
, even though the
correct description should be: a function from b
to
commands.
I/O
The command for writing a line to Standard Output is
putStrLn :: String > IO ()
It outputs the string parameter, plus linebreak. And since there is no
answer to give, the answer type is the most boring ()
.
At first, using output commands at the prompt is as easy as using expressions.
Prelude> putStrLn "Hello world"
Hello world
You can also write a compound command with the >>
operator.
Prelude> putStrLn "Hello" >> putStrLn "world"
Hello
world
The fun begins when you also take input. The command for reading one line from Standard Input is:
getLine :: IO String
Note that the type is not String
or ()>String
.
In a purely functional language, such types promise that all calls
using the same parameter yield the exactly same string.
This is of course what an input command cannot promise. If you read
two lines, they are very likely to be different. The type
IO String
does not promise to give the same string all
the time. (It only promises to be the same command all the time—a
rather "duh" one.) But this poses a question: how do we get at
the line it reads?
A nonsolution is to expect an operation
stripIO :: IO a > a
.
What's wrong with this stripthatIO mentality is that it asks to
convert a command, which gives different answers at different calls,
into a pure function, which gives the same answer at different calls.
Contradiction!
But you can ask for a weaker operation: how to pass the answer on to subsequent commands (e.g., output commands) so they can use it. A moment of thought reveals that this is all you ever need. The operator sought is
(>>=) :: IO a > (a > IO b) > IO b
It builds a compound command from two commands, the first one of which
takes no parameter and gives an answer of type a
, and the
second of which needs a parameter of type a
.
You guessed it: this operator extracts the answer from the first command
and passes it to the second. Now you have some way to use the answer!
Here is the first example. Why don't we read a line and immediately
display it? getLine
answers a string, and
putStrLn
wants a string. Perfect match!
Prelude> getLine >>= putStrLn
Good morning
Good morning
But more often you want to output something derived from the input, rather than the input itself verbatim. To do this, you customize the second command to smuggle in the derivation. The trick of anonymous functions is very useful for this:
Prelude> getLine >>= \s > putStrLn ("You entered: " ++ s)
Good morning
You entered: Good morning
You will also want to give derived answers, especially if you write
subroutines to be called from other code. This is facilitated by the
command that takes a parameter and simply gives it as the answer
(it is curiously named return
):
return :: a > IO a
For example here is a routine that reads a line and answers a derived string, with a sample usage:
Prelude> let mycmd = getLine >>= \s > return ("You entered: " ++ s)
Prelude> mycmd >>= putStrLn
Good morning
You entered: Good morning
Some programmers never use Standard Input. Reading files is more common. One command for this is:
readFile :: String > IO String
The parameter specifies the file path. Let us read a file and print out its first 10 characters (wherever available). Of course please change the filename to refer to some file you actually possess.
Prelude> readFile "file.txt" >>= \s > putStrLn (take 10 s)
abcdefghij
Do not worry about slurping up the whole file into memory;
readFile
performs a magic of payasyougo reading.
A while ago I showed the >>
operator for compound commands
without elaboration. I can now elaborate it: it merely uses
>>=
in a way that throws away the first answer:
putStrLn "x" >> putStrLn "y"
= putStrLn "x" >>= \_ > putStrLn "y"
doNotation
To bring imperative code closer to imperative look, Haskell
provides the donotation, which hides the >>=
operator and the anonymous functions. An example illustrates
this notation well, and it should be easy to generalize:
cmd0 >>= \x > cmd1 >>= \_ > cmd2 >>= \z > cmd3
= do { x < cmd0; cmd1; z < cmd2; cmd3 }
(cmd1
, cmd2
, and cmd3
may
use x
as a parameter; similarly, cmd3
may use z
as a parameter. At the end, between
cmd3
and }
, you may choose to insert
or omit semicolons; similarly right after {
at the
beginning.)
Below we reexpress examples in the previous section in the donotation.
Prelude> do { putStrLn "Hello"; putStrLn "world" }
Hello
world
Prelude> do { s < getLine; putStrLn s }
Good morning
Good morning
Prelude> do { s < getLine; putStrLn ("You entered: " ++ s) }
Good morning
You entered: Good morning
Prelude> let mycmd = do { s < getLine; return ("You entered: " ++ s) }
Prelude> do { s < mycmd; putStrLn s }
Good morning
You entered: Good morning
Prelude> do { s < readFile "file.txt"; putStrLn (take 10 s) }
abcdefghij
At the prompt it is necessary to write oneliners. In a source code
file it is more common to use multiple lines, one line for one
command, as per tradition. In this case, layout rules allow
omitting {;}
in favour of indentation. Thus, here are
two valid ways of writing the same doblock in a source code file,
one with {;}
and the other with layout.
foo = do { name < getLine; foo = do name < getLine
s < readFile name; s < readFile name
putStrLn (take 10 s) putStrLn (take 10 s)
}
Mutable variables
Data.IORef Data.Array.MArray Data.Array.IO
Exceptions
Control.Exception
Concurrency
Control.Concurrent Control.Concurrent.MVar
Monads
Mutable variables
Data.STRef Data.Array.MArray Data.Array.ST
State
The State monad
Monad transformers
Compilation
You can easily compile your Haskell modules to standalone executables. For example, write this in a file "A.hs":
main = putStrLn "Hello, World!"
In general, main
is the entry point, and you must define it to be whatever you want run. (TODO: once the monad/IO section is done, this place should also say more about main and IO.)
The compiler, on unix systems, is ghc. For example "A.hs" can be compiled and run as:
$ ghc A.hs $ ./a.out Hello, World!
For multiple modules, use the make flag to GHC. Example: write these two modules:
B.hs:
import M1
main = putStrLn s
M1.hs:
module M1 where
s = "Hi, Everyone!"
To compile and run (this will automatically look for M1.hs):
$ ghc make B.hs $ ./a.out Hi, Everyone!
In general, one and only one file must define main
. In general, for all other files, the filename must match the module name.