Algebraic data types

Some languages offer direct support for algebraic data types and pattern matching on them. While this of course can always be simulated with manual tagging and conditionals, it allows for terse code which is easy to read, and can represent the algorithm directly.

As an example, implement insertion in a red-black-tree. A red-black-tree is a binary tree where each internal node has a color attribute red or black. Moreover, no red node can have a red child, and every path from the root to an empty node must contain the same number of black nodes. As a consequence, the tree is balanced, and must be re-balanced after an insertion.

Bracmat

<lang bracmat> ( balance

 =   a x b y c zd
   .       !arg
         : ( B
           .   ( ( R
                 .   ((R.?a,?x,?b),?y,?c)
                   | (?a,?x,(R.?b,?y,?c))
                 )
               , ?zd
               )
             | ( ?a
               , ?x
               , ( R
                 .   ((R.?b,?y,?c),?zd)
                   | (?b,?y,(R.?c,?zd))
                 )
               )
           )
       & (R.(B.!a,!x,!b),!y,(B.!c,!zd))
     | !arg
 )

& ( ins

 =   X tree a m z
   .     !arg:(?X.?tree)
       & !tree:(?C.?a,?m,?z)
       & (   !X:<!m
           & balance$(!C.ins$(!X.!a),!m,!z)
         |   !X:>!m
           & balance$(!C.!a,!m,ins$(!X.!z))
         | !tree
         )
     | (R.,!X,)
 )

& ( insert

 =   X tree
   .   !arg:(?X.?tree)
     & ins$(!X.!tree):(?.?X)
     & (B.!X)
 )

& ( insertMany

 =   L R tree
   .     !arg:(%?L_%?R.?tree)
       & insertMany$(!L.!tree):?tree
       & insertMany$(!R.!tree)
     | insert$!arg
 )

</lang>

Test: <lang bracmat> ( it allows for terse code which is easy to read

     , and can represent the algorithm directly
   . 
   )
 : ?values

& insertMany$(!values.):?tree & out$!tree & done;</lang>

Output: <lang bracmat> B . ( B

   .   (R.(B.,,),algorithm,(B.,allows,))
     , and
     , (B.,can,)
   )
 , code
 , ( R
   .   ( B
       .   (B.(R.,directly,),easy,)
         , for
         , (B.(R.,is,),it,)
       )
     , read
     , ( B
       .   (B.,represent,)
         , terse
         , (R.(B.,the,),to,(B.,which,))
       )
   )</lang>

C

[1] Link contains (2012-FEB) RB-Tree tutorial, and the site groups the code together in a library section.

Common Lisp

Common Lisp doesn't come with any pattern-matching solutions on its own, but with the help of its macro facility, it can incorporate features from other languages such as pattern matching. Macros expand into efficient code during compilation time and there isn't much difference if it's included in the core language or not. As has been said, Lisp is a ball of mud and remains one no matter what one throws at it.

This is a straighforward translation of the TCL solution. I don't know red-black-trees myself but I tried mirroring the original program as closely as possible. It uses a pattern-matching library called toadstool.

Library: toadstool

<lang lisp>(mapc #'use-package '(#:toadstool #:toadstool-system)) (defstruct (red-black-tree (:constructor tree (color left val right)))

 color left val right)

(defcomponent tree (operator macro-mixin)) (defexpand tree (color left val right)

 `(class red-black-tree red-black-tree-color ,color
                        red-black-tree-left ,left
                        red-black-tree-val ,val
                        red-black-tree-right ,right))

(pushnew 'tree *used-components*)

(defun balance (color left val right)

 (toad-ecase (color left val right)
   (('black (tree 'red (tree 'red a x b) y c) z d)
    (tree 'red (tree 'black a x b) y
          (tree 'black c z d)))
   (('black (tree 'red a x (tree 'red b y c)) z d)
    (tree 'red (tree 'black a x b) y (tree 'black c z d)))
   (('black a x (tree 'red (tree 'red b y c) z d))
    (tree 'red (tree 'black a x b) y (tree 'black c z d)))
   (('black a x (tree 'red b y (tree 'red c z d)))
    (tree 'red (tree 'black a x b) y (tree 'black c z d)))
   ((color a x b)
    (tree color a x b))))

(defun %insert (x s)

 (toad-ecase1 s
   (nil (tree 'red nil x nil))
   ((tree color a y b)
    (cond ((< x y)
           (balance color (%insert x a) y b))
          ((> x y)
           (balance color a y (%insert x b)))
          (t s)))))

(defun insert (x s)

 (toad-ecase1 (%insert x s)
   ((tree t a y b) (tree 'black a y b))))</lang>

E

Translation of: Haskell

In E, a pattern can be used almost anywhere a variable name can. Additionally, there are two operators used for pattern matching idioms: =~ (returns success as a boolean, somewhat like Perl's =~), and switch (matches multiple patterns, like Haskell's case).

Both of those operators are defined in terms of the basic bind/match operation: def pattern exit failure_handler := specimen

def balance(tree) {
  return if (
    tree =~ term`tree(black, tree(red, tree(red, @a, @x, @b), @y, @c), @z, @d)` ||
    tree =~ term`tree(black, tree(red, @a, @x, tree(red, @b, @y, @c)), @z, @d)` ||
    tree =~ term`tree(black, @a, @x, tree(red, tree(red, @b, @y, @c), @z, @d))` ||
    tree =~ term`tree(black, @a, @x, tree(red, @b, @y, tree(red, @c, @z, @d)))`
  ) {
    term`tree(red, tree(black, $a, $x, $b), $y, tree(black, $c, $z, $d))`
  } else { tree }
}
def insert(elem, tree) {
  def ins(tree) {
    return switch (tree) {
      match term`empty` { term`tree(red, empty, $elem, empty)` }
      match term`tree(@color, @a, @y, @b)` {
        if (elem < y) {
          balance(term`tree($color, ${ins(a)}, $y, $b)`)
        } else if (elem > y) {
          balance(term`tree($color, $a, $y, ${ins(b)})`)
        } else {
          tree
        }
      }
    }
  }
  def term`tree(@_, @a, @y, @b)` := ins(tree)
  return term`tree(black, $a, $y, $b)`
}

This code was tested by filling a tree with random values; you can try this at the E REPL:

? var tree := term`empty`
> for _ in 1..20 {
>   tree := insert(entropy.nextInt(100), tree)
> }
> tree

Haskell

<lang haskell>data Color = R | B data Tree a = E | T Color (Tree a) a (Tree a)

balance :: Color -> Tree a -> a -> Tree a -> Tree a balance B (T R (T R a x b) y c ) z d = T R (T B a x b) y (T B c z d) balance B (T R a x (T R b y c)) z d = T R (T B a x b) y (T B c z d) balance B a x (T R (T R b y c) z d ) = T R (T B a x b) y (T B c z d) balance B a x (T R b y (T R c z d)) = T R (T B a x b) y (T B c z d) balance col a x b = T col a x b

insert :: Ord a => a -> Tree a -> Tree a insert x s = T B a y b where

 ins E          =  T R E x E
 ins s@(T col a y b) 
   | x < y      =  balance col (ins a) y b
   | x > y      =  balance col a y (ins b)
   | otherwise  =  s
 T _ a y b = ins s</lang>

J

J incorporates a symbol data type which, in versions 6.02 and 7.01, J implements directly as a red-black tree. s: entry in the J dictionary begins Symbols are a data type and are created by the verb s:. Symbols provide a mechanism for searching, sorting, and comparisons more efficient than alternative mechanisms such as boxed strings. Structural, selection, and relational verbs work on symbols. Arithmetic verbs do not work on symbols.

The following code provides dictionary functionality using a red-black tree written in J without symbols.

<lang J> help=: noun define red-black tree Store dictionary in red-black tree. The keys can be any noun.

Reference: Left-leaning Red-Black Trees Robert Sedgewick Department of Computer Science Princeton University

verbs: insert key;value Inserts item into tree delete key Deletes item with key from tree

                 Deletion via the Sedgewick method is fairly simple.
                 However, I elected to remove the KEY;VALUE pair
                 rather than change the tree.

find key Returns the associated definition or EMPTY items any_noun Returns all the items as a rank 1 array of KEY;VALUE pairs keys any_noun Returns all the keys as a rank 1 array of boxes values any_noun Returns all the values as a rank 1 array of boxes

J stores all data as arrays. I chose to use array indexes to implement pointers. An "index" is a rank 0 length 1 array.

Internal data structure:

T This rank 2 array stores indexes of left and right at each branch point. C rank 1 array of node color. H rank 1 array of the hash value of each key. R rank 0 array stores the root index. D rank 1 array of boxes. In each box is a rank 2 array of key value

 pairs associated with the hash value.  Hash collision invokes direct
 lookup by key among the keys having same hash.

Additional test idea (done):

 Changing the hash to 0: or 2&| rapidly tests
 hash collision code for integer keys.

)

bitand=: (#. 1 0 0 0 1)b. bitxor=: (#. 1 0 1 1 0)b. hash=: [: ((4294967295) bitand (bitxor 1201&*))/ 846661 ,~ ,@:(a.&i.)@:": NB. hash=: ] [ 1&bitand NB. can choose simple hash functions for tests

setup=: 3 : 0 T=: i. 0 2 NB. Tree H=: D=: C=: i. 0 NB. Hashes, Data, Color R=: _ NB. Root 'BLACK RED'=: i. 2 EMPTY )

setup

flipColors=: monad def 'C=: -.@:{`[`]}&C (, {&T) y'

3 : 0 'test flipColors' DD=.D=: ,/<@:(;3j1&":)"0 i.3 TT=.T=: _ _,0 2,:_ _ CC=.C=: 1 0 1 RR=.R=: 1 HH=.H=: i.3 flipColors R assert C -: -. CC assert HH -: H assert TT -: T assert DD -: D assert RR -: R )

getColor=: monad def 'C ({~ :: (BLACK"_))"_ 0 y' NB. y the node

rotateTree=: dyad define NB. x left or right, y node I=. x <@:(, -.)~ y X=. I { T NB. x = root.otherside J=. X <@:, x T=: (J { T) I} T T=: y J} T C=: y (RED ,~ {)`(X , [)`]} C X )

3 : 0 'test rotateTree' DD=.D=:,/<@:(;3j1&":)"0 i.5 TT=.T=:_ _,0 2,_ _,1 4,:_ _ CC=.C=:0 1 0 0 0 R=:3 HH=.H=:i.5 assert R = rotateTree/0 1 , R assert DD -: D assert CC -: C assert HH -: H assert TT -: T )

setup

insert_privately=: adverb define

ROOT=. m HASH=. x ITEM=. y if. _ -: ROOT do. NB. new key

ROOT=. # H
H=: H , HASH
T=: T , _ _
D=: D , < ,: , ITEM
C=: C , RED

elseif. HASH = ROOT { H do. NB. change a value or hash collision

STACK=. ROOT >@:{ D
I=. STACK i.&:({."1) ITEM
STACK=. ITEM <@:(I}`,@.(I = #@])) STACK
D=: STACK ROOT } D

elseif. do. NB. Follow tree

NB. if both children are red then flipColors ROOT
flipColors^:((,~ RED) -: getColor@:({&T)) ROOT
I=. <@:(, HASH > {&H) ROOT
TEMP=. HASH (I { T) insert_privately y
T=:  TEMP I } T
NB.if (isRed(h.right) && !isRed(h.left)) h = rotateLeft(h)
ROOT=. 0&rotateTree^:((BLACK,RED) -: getColor@:({&T)) ROOT
NB.if (isRed(h.left) && isRed(h.left.left)) h = rotateRight(h)
if. RED -: getColor {. ROOT { T do.
 if. (RED -: (getColor@:(([: {&T <@:,&0)^:2) :: (BLACK"_))) ROOT do.
  ROOT=. 1 rotateTree ROOT
 end.
end.

end. ROOT )

insert=: monad define"1 assert 'boxed' -: datatype y R=: (R insert_privately~ hash@:(0&{::)) y C=: BLACK R } C y )

find_hash_index=: monad define NB. y is the hash if. 0 = # T do. return. end. NB. follow the tree I=. R NB. instead of while. y ~: I { H do. NB. direct search

J=. <@:(, y > {&H) I
if. _ > II=. J { T do. I=. II else.  return. end.

end. )

find=: monad define if. -: I=. find_hash_index hash y do. EMPTY return. end. LIST=. I {:: D K=. {. |: LIST LIST {::~ ::empty 1 ,~ K i. < y )

delete=: 3 : 0 if. -: I=. find_hash_index hash y do. EMPTY return. end. LIST=. I {:: D K=. {. |: LIST J=. K i. < y RESULT=. J ({::~ ,&1)~ LIST STACK=. J <@:({. , (}.~ >:)~) LIST D=. LIST I } D RESULT )

getPathsToLeaves=: a:&$: : (4 : 0) NB. PATH getPathsToLeaves ROOT use: getPathsToLeaves R if. 0 = # y do. getPathsToLeaves R return. end. PATH=. x ,&.> y if. _ -: y do. return. end. PATH getPathsToLeaves"0 y { T )

check=: 3 : 0 COLORS=. getColor"0&.> a: -.~ ~. , getPathsToLeaves result=. EMPTY if. 0&e.@:(= {.) +/@:(BLACK&=)@>COLORS do. result=. result,<'mismatched black count' end. if. 1 e. 1&e.@:(*. (= 1&|.))@:(RED&=)@>COLORS do. result=. result,<'successive reds' end. >result )

getPath=: 3 : 0 NB. get path to y, the key if. 0 = # H do. EMPTY return. end. HASH=. hash y PATH=. , I=. R while. HASH ~: I { H do.

J=. <@:(, HASH > {&H) I
PATH=. PATH , II=. J { T
if. _ > II do. I=. II else. EMPTY return. end.

end. PATH )

items=: 3 :';D' keys=: 3 :'0{"1 items y' values=: 3 :'1{"1 items y' </lang> With use: <lang J>

  load'rb.ijs'
  NB. populate dictionary in random order with 999 key value pairs
  insert@:(; 6j1&":)"0@:?~ 999
  find 'the' NB. 'the' has no entry.
  find 239   NB. entry 239 has the anticipated formatted string value.
239.0
  find 823823 NB. also no such entry
  NB.
  NB. tree passes the "no consecutive red" and "same number of black"
  NB. nodes to and including NULL leaves.
  check

</lang>

OCaml

<lang ocaml> type color = R | B type 'a tree = E | T of color * 'a tree * 'a * 'a tree

(** val balance : color * 'a tree * 'a * 'a tree -> 'a tree *) let balance = function

 | B, T (R, T (R,a,x,b), y, c), z, d
 | B, T (R, a, x, T (R,b,y,c)), z, d
 | B, a, x, T (R, T (R,b,y,c), z, d)
 | B, a, x, T (R, b, y, T (R,c,z,d)) -> T (R, T (B,a,x,b), y, T (B,c,z,d))
 | col, a, x, b                      -> T (col, a, x, b)

(** val insert : 'a -> 'a tree -> 'a tree *) let insert x s =

 let rec ins = function
   | E                  -> T (R,E,x,E)
   | T (col,a,y,b) as s ->

if x < y then balance (col, ins a, y, b) else if x > y then balance (col, a, y, ins b) else s

 in let T (_,a,y,b) = ins s 
 in T (B,a,y,b)

</lang>

Oz

Translation of: Haskell

Unlike Haskell, Oz does not support multiple equations per function. So we use an explicit case-statement. To match multiple variables at once, we create temporary tuples with "#".

<lang oz>fun {Balance Col A X B}

  case Col#A#X#B
  of b#t(r t(r A X B) Y C         )#Z#D                            then t(r t(b A X B) Y t(b C Z D))
  [] b#t(r A          X t(r B Y C))#Z#D                            then t(r t(b A X B) Y t(b C Z D))
  [] b#A                           #X#t(r t(r B Y C) Z D)          then t(r t(b A X B) Y t(b C Z D))
  [] b#A                           #X#t(r B          Y t(r C Z D)) then t(r t(b A X B) Y t(b C Z D))
  else t(Col A X B)
  end

end

fun {Insert X S}

  fun {Ins S}
     case S of e then t(r e X e)
     [] t(Col A Y B) then

if X < Y then {Balance Col {Ins A} Y B} elseif X > Y then {Balance Col A Y {Ins B}} else S end

     end
  end
  t(_ A Y B) = {Ins S}

in

  t(b A Y B)

end</lang>

Perl 6

Works with: Rakudo version 2012.05

Perl 6 doesn't have algebraic data types (yet), but it does have pretty good pattern matching in multi signatures. <lang perl6>enum RedBlack <R B>;

multi balance(B,[R,[R,$a,$x,$b],$y,$c],$z,$d) { [R,[B,$a,$x,$b],$y,[B,$c,$z,$d]] } multi balance(B,[R,$a,$x,[R,$b,$y,$c]],$z,$d) { [R,[B,$a,$x,$b],$y,[B,$c,$z,$d]] } multi balance(B,$a,$x,[R,[R,$b,$y,$c],$z,$d]) { [R,[B,$a,$x,$b],$y,[B,$c,$z,$d]] } multi balance(B,$a,$x,[R,$b,$y,[R,$c,$z,$d]]) { [R,[B,$a,$x,$b],$y,[B,$c,$z,$d]] }

multi balance($col, $a, $x, $b) { [$col, $a, $x, $b] }

multi ins( $x, @s [$col, $a, $y, $b] ) {

   when $x before $y     { balance $col, ins($x, $a), $y, $b }
   when $x after $y      { balance $col, $a, $y, ins($x, $b) }
   default               { @s }

} multi ins( $x, Any:U ) { [R, Any, $x, Any] }

multi insert( $x, $s ) {

   [B, ins($x,$s)[1..3]];

}

sub MAIN {

   my $t = Any;
   $t = insert($_, $t) for (1..10).pick(*);
   say $t.perl;

}</lang> This code uses generic comparison operators before and after, so it should work on any ordered type.

Output:

[RedBlack::B, [RedBlack::R, [RedBlack::B, [RedBlack::R, Any, 1, Any], 2, Any], 3, [RedBlack::B, Any, 4, [RedBlack::R, Any, 5, Any]]], 6, [RedBlack::R, [RedBlack::B, Any, 7, Any], 8, [RedBlack::B, Any, 9, [RedBlack::R, Any, 10, Any]]]]

PicoLisp

Translation of: Prolog

<lang PicoLisp>(be color (R)) (be color (B))

(be tree (@ E)) (be tree (@P (T @C @L @X @R))

  (color @C)
  (tree @P @L)
  (call @P @X)
  (tree @P @R) )

(be bal (B (T R (T R @A @X @B) @Y @C) @Z @D (T R (T B @A @X @B) @Y (T B @C @Z @D)))) (be bal (B (T R @A @X (T R @B @Y @C)) @Z @D (T R (T B @A @X @B) @Y (T B @C @Z @D)))) (be bal (B @A @X (T R (T R @B @Y @C) @Z @D) (T R (T B @A @X @B) @Y (T B @C @Z @D)))) (be bal (B @A @X (T R @B @Y (T R @C @Z @D)) (T R (T B @A @X @B) @Y (T B @C @Z @D))))

(be balance (@C @A @X @B @S)

  (bal @C @A @X @B @S)
  T )

(be balance (@C @A @X @B (T @C @A @X @B)))

(be ins (@X E (T R E @X E))) (be ins (@X (T @C @A @Y @B) @R)

  (@ < (-> @X) (-> @Y))
  (ins @X @A @Ao)
  (balance @C @Ao @Y @B @R)
  T )

(be ins (@X (T @C @A @Y @B) @R)

  (@ > (-> @X) (-> @Y))
  (ins @X @B @Bo)
  (balance @C @A @Y @Bo @R)
  T )

(be ins (@X (T @C @A @Y @B) (T @C @A @Y @B)))

(be insert (@X @S (T B @A @Y @B))

  (ins @X @S (T @ @A @Y @B)) )</lang>

Test: <lang PicoLisp>: (? (insert 2 E @A) (insert 1 @A @B) (insert 3 @B @C))

@A=(T B E 2 E) @B=(T B (T R E 1 E) 2 E) @C=(T B (T R E 1 E) 2 (T R E 3 E))

-> NIL</lang>

Prolog

color(r).
color(b).

tree(_,e).
tree(P,t(C,L,X,R)) :- color(C), tree(P,L), call(P,X), tree(P,R).

bal(b, t(r,t(r,A,X,B),Y,C), Z, D, t(r,t(b,A,X,B),Y,t(b,C,Z,D))).
bal(b, t(r,A,X,t(r,B,Y,C)), Z, D, t(r,t(b,A,X,B),Y,t(b,C,Z,D))).
bal(b, A, X, t(r,t(r,B,Y,C),Z,D), t(r,t(b,A,X,B),Y,t(b,C,Z,D))).
bal(b, A, X, t(r,B,Y,t(r,C,Z,D)), t(r,t(b,A,X,B),Y,t(b,C,Z,D))).

balance(C,A,X,B,S) :- ( bal(C,A,X,B,T) -> S = T ; S = t(C,A,X,B) ).

ins(X,e,t(r,e,X,e)).
ins(X,t(C,A,Y,B),R) :- ( X < Y -> ins(X,A,Ao), balance(C,Ao,Y,B,R)
                       ; X > Y -> ins(X,B,Bo), balance(C,A,Y,Bo,R)
                       ; X = Y -> R = t(C,A,Y,B)
                       ).

insert(X,S,t(b,A,Y,B)) :- ins(X,S,t(_,A,Y,B)).

Racket

Translation of: OCaml

<lang scheme>#lang racket

(struct t-node (color t-left value t-right))

(define (balance t)

 (match t
   [(t-node 'black (t-node 'red (t-node 'red a x b) y c) z d)
    (t-node 'red (t-node 'black a x b) y (t-node 'black c z d))]
   [(t-node 'black (t-node 'red a x (t-node 'red b y c)) z d)
    (t-node 'red (t-node 'black a x b) y (t-node 'black c z d))]
   [(t-node 'black a x (t-node 'red (t-node 'red b y c) z d))
    (t-node 'red (t-node 'black a x b) y (t-node 'black c z d))]
   [(t-node 'black a x (t-node 'red b y (t-node 'red c z d)))
    (t-node 'red (t-node 'black a x b) y (t-node 'black c z d))]
   [else t]))

(define (insert x s)

 (define (ins t)
   (match t
     ['empty (t-node 'red 'empty x 'empty)]
     [(t-node c a y b)
      (cond [(< x y)
             (balance (t-node c (ins a) y b))]
            [(> x y)
             (balance (t-node c a y (ins b)))]
            [else t])]))
 (match (ins s)
   [(t-node _ a y b) (t-node 'black a y b)]))</lang>

Rascal

Rascal offers many options for pattern matching. In essence, there are four sorts of patterns: Abstract, Concrete, PatternWithAction and classic Regular Expressions. These patterns can be used in several cases, for example switch or visit statements, on the right of the Match operator (:=), or in TryCatch statements for thrown exceptions. Each pattern binds variables in a conditional scope.

Abstract

An abstract pattern is recursively defined and may contain, among others, the following elements: Literal, VariableDeclaration, MultiVariable, Variable, List, Set, Tuple, Node, Descendant, Labelled, TypedLabelled, TypeConstrained. More explanation can be found in the Documentation. Some examples: <lang rascal> // Literal rascal>123 := 123 bool: true

// VariableDeclaration rascal>if(str S := "abc") >>>>>>> println("Match succeeds, S == \"~~\""); Match succeeds, S == "abc" ok~~

~~// MultiVariable rascal>if([10, N*, 50] := [10, 20, 30, 40, 50]) >>>>>>> println("Match succeeds, N == <N>"); Match succeeds, N == [20,30,40] ok~~

~~// Variable rascal>N = 10; int: 10 rascal>N := 10; bool: true rascal>N := 20; bool: false~~

~~// Set and List rascal>if({10, set[int] S, 50} := {50, 40, 30, 20, 10}) >>>>>>> println("Match succeeded, S = ~~"); Match succeeded, S = {30,40,20} ok~~~~

rascal>for([L1*, L2*] := [10, 20, 30, 40, 50]) >>>>>>> println("<L1> and <L2>"); [] and [10,20,30,40,50] [10] and [20,30,40,50] [10,20] and [30,40,50] [10,20,30] and [40,50] [10,20,30,40] and [50] [10,20,30,40,50] and [] list[void]: []

// Descendant rascal>T = red(red(black(leaf(1), leaf(2)), black(leaf(3), leaf(4))), black(leaf(5), leaf(4))); rascal>for(/black(_,leaf(4)) := T) >>>>>>> println("Match!"); Match! Match! list[void]: []

~~rascal>for(/black(_,leaf(int N)) := T) >>>>>>> println("Match <N>"); Match 2 Match 4 Match 4 list[void]: []~~

~~rascal>for(/int N := T) >>>>>>> append N; list[int]: [1,2,3,4,5,4]~~

~~// Labelled rascal>for(/M:black(_,leaf(4)) := T) >>>>>>> println("Match <M>"); Match black(leaf(3),leaf(4)) Match black(leaf(5),leaf(4)) list[void]: []</lang>~~

~~Concrete~~
Suppose we want to manipulate text written in some hypothetical language LANG. Then first the concrete syntax of LANG has to be defined by importing a module that declares the non-terminals and syntax rules for LANG. Next LANG programs have to be parsed. LANG programs made come from text files or they may be included in the Rascal program as literals. In both cases the text is parsed according to the defined syntax and the result is a parse tree in the form of a value of type Tree. Concrete patterns operate on these trees.
A concrete pattern is a quoted concrete syntax fragment that may contain variables. The syntax that is used to parse the concrete pattern may come from any module that has been imported in the module in which the concrete pattern occurs. Some examples of concrete patterns: <lang rascal>// Quoted pattern ` Token1 Token2 ... Tokenn ` // A typed quoted pattern (Symbol) ` Token1 Token2 ... TokenN ` // A typed variable pattern <Type Var> // A variable pattern </lang>
A full example of concrete patterns can be found in the Rascal Recipes.
~~PatternWithAction~~
There are two variants of the PatternsWitchAction. When the subject matches Pattern, the expression Exp is evaluated and the subject is replaced with the result. Secondly, when the subject matches Pattern, the (block of) Statement(s) is executed. See below for some ColoredTree examples:
<lang rascal>// Define ColoredTrees with red and black nodes and integer leaves data ColoredTree = leaf(int N)
~~| red(ColoredTree left, ColoredTree right) | black(ColoredTree left, ColoredTree right);~~
~~// Count the number of black nodes public int cntBlack(ColoredTree t){~~
~~int c = 0; visit(t) { case black(_,_): c += 1; }; return c;~~
}
// Returns if a tree is balanced public bool balance(ColoredTree t){
~~visit(t){ case black(a,b): if (cntBlack(a) == cntBlack(b)) true; else return false; case red(a,b): if (cntBlack(a) == cntBlack(b)) true; else return false; } return true;~~
~~} // Compute the sum of all integer leaves public int addLeaves(ColoredTree t){~~
~~int c = 0; visit(t) { case leaf(int N): c += N; }; return c;~~
}
// Add green nodes to ColoredTree data ColoredTree = green(ColoredTree left, ColoredTree right);
// Transform red nodes into green nodes public ColoredTree makeGreen(ColoredTree t){
~~return visit(t) { case red(l, r) => green(l, r) };~~
~~}</lang>~~
~~Regular Expressions~~
Regular expressions are noated between two slashes. Most normal regular expressions patterns are available, such as ., \n, \d, etc. Additionally, flags can be used to create case intensiveness.
<lang rascal>rascal>/XX/i := "some xx"; bool: true rascal>/a.c/ := "abc"; bool: true</lang>
~~Scala~~
Translation of: Haskell
Algebraic data types are implemented in Scala through the combination of a number of different features, to ensure principles of Object Oriented Programming.
The main type is usually defined as a sealed abstract class, which ensures it can't be instantiated, and guarantees that it can't be expanded outside the file it was defined at. This last feature is used so the compiler can verify that the pattern matching is complete, or warn when there are missing cases. It can be ommitted if preferred.
Each subtype is defined either as a case object, for non-paremeterized types, or case class, for parameterized types, all extending the main type. The case keyword is not required, but, when used, it provides a number of default methods which ensure they can be used without any further definitions.
The most important of those default methods for the purpose of algebraic data types is the extractor, a method called either unapply or unapplySeq, and which returns an Option containing the deconstructed parameters, or None if the passed object can't be deconstructed by this method. Scala uses the extractors to implement pattern matching without exposing the internal representation of the data.
This specific task is made much harder than necessary because Scala doesn't have a variant ordering class. Given that limitation, one has to either give up on a singleton object representing the empty tree, or give up on parameterizing the tree itself.
The solution below, uses the latter approach. The algebraic data types are members of a RedBlackTree class, which, itself, receives a type parameter for the keys of the tree, and an implicit parameter for an Ordering for that type. To use the tree it is thus necessary to instantiate an object of type RedBlackTree, and then reference the members of that object.
<lang scala>class RedBlackTree[A](implicit ord: Ordering[A]) {
sealed abstract class Color case object R extends Color case object B extends Color sealed abstract class Tree { def insert(x: A): Tree = ins(x) match { case T(_, a, y, b) => T(B, a, y, b) case E => E } def ins(x: A): Tree } case object E extends Tree { override def ins(x: A): Tree = T(R, E, x, E) } case class T(c: Color, left: Tree, a: A, right: Tree) extends Tree { private def balance: Tree = (c, left, a, right) match { case (B, T(R, T(R, a, x, b), y, c), z, d ) => T(R, T(B, a, x, b), y, T(B, c, z, d)) case (B, T(R, a, x, T(R, b, y, c)), z, d ) => T(R, T(B, a, x, b), y, T(B, c, z, d)) case (B, a, x, T(R, T(R, b, y, c), z, d )) => T(R, T(B, a, x, b), y, T(B, c, z, d)) case (B, a, x, T(R, b, y, T(R, c, z, d))) => T(R, T(B, a, x, b), y, T(B, c, z, d)) case _ => this } override def ins(x: A): Tree = ord.compare(x, a) match { case -1 => T(c, left ins x, a, right ).balance case 1 => T(c, left, a, right ins x).balance case 0 => this } }
}</lang>
Usage example:
scala> val rbt = new RedBlackTree[Int] rbt: RedBlackTree[Int] = RedBlackTree@17dfcf1 scala> import rbt._ import rbt._ scala> List.range(1, 17).foldLeft(E: Tree)(_ insert _) res5: rbt.Tree = T(B,T(B,T(B,T(B,E,1,E),2,T(B,E,3,E)),4,T(B,T(B,E,5,E),6,T(B,E,7,E))),8,T(B,T(B,T(B,E,9,E),10,T(B,E,11,E )),12,T(B,T(B,E,13,E),14,T(B,E,15,T(R,E,16,E)))))
~~Standard ML~~
<lang sml> datatype color = R | B datatype 'a tree = E | T of color * 'a tree * 'a * 'a tree
(** val balance = fn : color * 'a tree * 'a * 'a tree -> 'a tree *) fun balance (B, T (R, T (R,a,x,b), y, c), z, d) = T (R, T (B,a,x,b), y, T (B,c,z,d))
| balance (B, T (R, a, x, T (R,b,y,c)), z, d) = T (R, T (B,a,x,b), y, T (B,c,z,d)) | balance (B, a, x, T (R, T (R,b,y,c), z, d)) = T (R, T (B,a,x,b), y, T (B,c,z,d)) | balance (B, a, x, T (R, b, y, T (R,c,z,d))) = T (R, T (B,a,x,b), y, T (B,c,z,d)) | balance (col, a, x, b) = T (col, a, x, b)
~~(** val insert = fn : int -> int tree -> int tree *) fun insert x s = let~~
~~fun ins E = T (R,E,x,E) | ins (s as T (col,a,y,b)) =~~
~~if x < y then balance (col, ins a, y, b) else if x > y then balance (col, a, y, ins b) else s~~
~~val T (_,a,y,b) = ins s~~
in
~~T (B,a,y,b)~~
~~end </lang>~~
~~Tcl~~
Translation of: Haskell
Tcl doesn't have algebraic types built-in, but they can be simulated using tagged lists, and a custom pattern matching control structure can be built: <lang tcl># From http://wiki.tcl.tk/9547 package require Tcl 8.5 package provide datatype 0.1
namespace eval ::datatype {
~~namespace export define match matches namespace ensemble create~~
# Datatype definitions proc define {type = args} { set ns [uplevel 1 { namespace current }] foreach cons [split [join $args] |] { set name [lindex $cons 0] set args [lrange $cons 1 end] proc $ns\::$name $args [format { lreplace [info level 0] 0 0 %s } [list $name]] } return $type }
# Pattern matching # matches pattern value envVar -- # Returns 1 if value matches pattern, else 0 # Binds match variables in envVar proc matches {pattern value envVar} { upvar 1 $envVar env if {[var? $pattern]} { return [bind env $pattern $value] } if {[llength $pattern] != [llength $value]} { return 0 } if {[lindex $pattern 0] ne [lindex $value 0]} { return 0 } foreach pat [lrange $pattern 1 end] val [lrange $value 1 end] { if {![matches $pat $val env]} { return 0 } } return 1 } # A variable starts with lower-case letter or _. _ is a wildcard. proc var? term { string match {[a-z_]*} $term } proc bind {envVar var value} { upvar 1 $envVar env if {![info exists env]} { set env [dict create] } if {$var eq "_"} { return 1 } dict set env $var $value return 1 } proc match args { #puts "MATCH: $args" set values [lrange $args 0 end-1] set choices [lindex $args end] append choices \n [list return -code error -level 2 "no match for $values"] set f [list values $choices [namespace current]] lassign [apply $f $values] env body #puts "RESULT: $env -> $body" dict for {k v} $env { upvar 1 $k var; set var $v } catch { uplevel 1 $body } msg opts dict incr opts -level return -options $opts $msg } proc case args { upvar 1 values values set patterns [lrange $args 0 end-2] set body [lindex $args end] set env [dict create] if {[llength $patterns] != [llength $values]} { return } foreach pattern $patterns value $values { if {![matches $pattern $value env]} { return } } return -code return [list $env $body] } proc default body { return -code return [list {} $body] }
} </lang> We can then code our solution similar to Haskell:
<lang tcl>datatype define Color = R | B datatype define Tree = E | T color left val right
~~balance :: Color -> Tree a -> a -> Tree a -> Tree a~~
~~proc balance {color left val right} {~~
datatype match $color $left $val $right { case B [T R [T R a x b] y c] z d -> { T R [T B $a $x $b] $y [T B $c $z $d] } case B [T R a x [T R b y c]] z d -> { T R [T B $a $x $b] $y [T B $c $z $d] } case B a x [T R [T R b y c] z d] -> { T R [T B $a $x $b] $y [T B $c $z $d] } case B a x [T R b y [T R c z d]] -> { T R [T B $a $x $b] $y [T B $c $z $d] } case col a x b -> { T $col $a $x $b } }
}
~~insert :: Ord a => a -> Tree a -> Tree a~~
~~proc insert {x s} {~~
~~datatype match [ins $x $s] { case [T _ a y b] -> { T B $a $y $b } }~~
}
~~ins :: Ord a => a -> Tree a -> Tree a~~
~~proc ins {x s} {~~
datatype match $s { case E -> { T R E $x E } case [T col a y b] -> { if {$x < $y} { return [balance $col [ins $x $a] $y $b] } if {$x > $y} { return [balance $col $a $y [ins $x $b]] } return $s } }
~~}</lang>~~