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<?xml version="1.0" encoding="utf-8" ?>
<!DOCTYPE chapter SYSTEM "chapter.dtd">

<chapter>
  <header>
    <copyright>
      <year>2003</year><year>2013</year>
      <holder>Ericsson AB. All Rights Reserved.</holder>
    </copyright>
    <legalnotice>
      Licensed under the Apache License, Version 2.0 (the "License");
      you may not use this file except in compliance with the License.
      You may obtain a copy of the License at
 
          http://www.apache.org/licenses/LICENSE-2.0

      Unless required by applicable law or agreed to in writing, software
      distributed under the License is distributed on an "AS IS" BASIS,
      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
      See the License for the specific language governing permissions and
      limitations under the License.
    
    </legalnotice>

    <title>List Comprehensions</title>
    <prepared></prepared>
    <docno></docno>
    <date></date>
    <rev></rev>
    <file>list_comprehensions.xml</file>
  </header>

  <section>
    <title>Simple Examples</title>
    <p>This section starts with a simple example, showing a generator and a filter:</p>
    <pre>
> <input>[X || X &lt;- [1,2,a,3,4,b,5,6], X > 3].</input>
[a,4,b,5,6]</pre>
    <p>This is read as follows: The list of X such that X is taken from the list
        <c>[1,2,a,...]</c> and X is greater than 3.</p>
    <p>The notation <c><![CDATA[X <- [1,2,a,...]]]></c> is a generator and
      the expression <c>X > 3</c> is a filter.</p>
    <p>An additional filter, <c>integer(X)</c>, can be added to restrict
      the result to integers:</p>
    <pre>
> <input>[X || X &lt;- [1,2,a,3,4,b,5,6], integer(X), X > 3].</input>
[4,5,6]</pre>
    <p>Generators can be combined. For example, the Cartesian product
      of two lists can be written as follows:</p>
    <pre>
> <input>[{X, Y} || X &lt;- [1,2,3], Y &lt;- [a,b]].</input>
[{1,a},{1,b},{2,a},{2,b},{3,a},{3,b}]</pre>
  </section>

  <section>
    <title>Quick Sort</title>
    <p>The well-known quick sort routine can be written as follows:</p>
    <code type="none"><![CDATA[
sort([Pivot|T]) ->
    sort([ X || X <- T, X < Pivot]) ++
    [Pivot] ++
    sort([ X || X <- T, X >= Pivot]);
sort([]) -> [].]]></code>
    <p>The expression <c><![CDATA[[X || X <- T, X < Pivot]]]></c> is the list of
      all elements in <c>T</c> that are less than <c>Pivot</c>.</p>
    <p><c><![CDATA[[X || X <- T, X >= Pivot]]]></c> is the list of all elements in
      <c>T</c> that are greater than or equal to <c>Pivot</c>.</p>
      <p>A list sorted as follows:</p>
      <list type="bulleted">
       <item>The first element in the list is isolated
       and the list is split into two sublists.</item>
       <item>The first sublist contains
      all elements that are smaller than the first element in
      the list.</item>
      <item>The second sublist contains all elements that are greater
      than, or equal to, the first element in the list.</item>
       <item>Then the sublists are sorted and the results are combined.</item>
     </list>
  </section>

  <section>
    <title>Permutations</title>
    <p>The following example generates all permutations of
      the elements in a list:</p>
    <code type="none"><![CDATA[
perms([]) -> [[]];
perms(L)  -> [[H|T] || H <- L, T <- perms(L--[H])].]]></code>
    <p>This takes <c>H</c> from <c>L</c> in all possible ways.
      The result is the set of all lists <c>[H|T]</c>, where <c>T</c>
      is the set of all possible permutations of <c>L</c>, with
      <c>H</c> removed:</p>
    <pre>
> <input>perms([b,u,g]).</input>
[[b,u,g],[b,g,u],[u,b,g],[u,g,b],[g,b,u],[g,u,b]]</pre>
  </section>

  <section>
    <title>Pythagorean Triplets</title>
    <p>Pythagorean triplets are sets of integers <c>{A,B,C}</c> such
      that <c>A**2 + B**2 = C**2</c>.</p>
    <p>The function <c>pyth(N)</c> generates a list of all integers
      <c>{A,B,C}</c> such that <c>A**2 + B**2 = C**2</c> and where
      the sum of the sides is equal to, or less than, <c>N</c>:</p>
    <code type="none"><![CDATA[
pyth(N) ->
    [ {A,B,C} ||
        A <- lists:seq(1,N),
        B <- lists:seq(1,N),
        C <- lists:seq(1,N),
        A+B+C =< N,
        A*A+B*B == C*C 
    ].]]></code>
    <pre>
> <input>pyth(3).</input>
[].
> <input>pyth(11).</input>
[].
> <input>pyth(12).</input>
[{3,4,5},{4,3,5}]
> <input>pyth(50).</input>
[{3,4,5},
 {4,3,5},
 {5,12,13},
 {6,8,10},
 {8,6,10},
 {8,15,17},
 {9,12,15},
 {12,5,13},
 {12,9,15},
 {12,16,20},
 {15,8,17},
 {16,12,20}]</pre>
    <p>The following code reduces the search space and is more
      efficient:</p>
    <code type="none"><![CDATA[
pyth1(N) ->
   [{A,B,C} ||
       A <- lists:seq(1,N-2),
       B <- lists:seq(A+1,N-1),
       C <- lists:seq(B+1,N),
       A+B+C =< N,
       A*A+B*B == C*C ].]]></code>
  </section>

  <section>
    <title>Simplifications With List Comprehensions</title>
    <p>As an example, list comprehensions can be used to simplify some
      of the functions in <c>lists.erl</c>:</p>
    <code type="none"><![CDATA[
append(L)   ->  [X || L1 <- L, X <- L1].
map(Fun, L) -> [Fun(X) || X <- L].
filter(Pred, L) -> [X || X <- L, Pred(X)].]]></code>
  </section>

  <section>
    <title>Variable Bindings in List Comprehensions</title>
    <p>The scope rules for variables that occur in list
      comprehensions are as follows:</p>
    <list type="bulleted">
      <item>All variables that occur in a generator pattern are
       assumed to be "fresh" variables.</item>
      <item>Any variables that are defined before the list
       comprehension, and that are used in filters, have the values
       they had before the list comprehension.</item>
      <item>Variables cannot be exported from a list comprehension.</item>
    </list>
    <p>As an example of these rules, suppose you want to write
      the function <c>select</c>, which selects certain elements from
      a list of tuples. Suppose you write
      <c><![CDATA[select(X, L) ->  [Y || {X, Y} <- L].]]></c> with the intention
      of extracting all tuples from <c>L</c>, where the first item is
      <c>X</c>.</p>
    <p>Compiling this gives the following diagnostic:</p>
    <code type="none">
./FileName.erl:Line: Warning: variable 'X' shadowed in generate</code>
    <p>This diagnostic warns that the variable <c>X</c> in
      the pattern is not the same as the variable <c>X</c>
      that occurs in the function head.</p>
    <p>Evaluating <c>select</c> gives the following result:</p>
    <pre>
> <input>select(b,[{a,1},{b,2},{c,3},{b,7}]).</input>
[1,2,3,7]</pre>
    <p>This is not the wanted result. To achieve the desired
      effect, <c>select</c> must be written as follows:</p>
    <code type="none"><![CDATA[
select(X, L) ->  [Y || {X1, Y} <- L, X == X1].]]></code>
    <p>The generator now contains unbound variables and the test has
      been moved into the filter.</p>
      <p>This now works as expected:</p>
    <pre>
> <input>select(b,[{a,1},{b,2},{c,3},{b,7}]).</input>
[2,7]</pre>
    <p>A consequence of the rules for importing variables into a
      list comprehensions is that certain pattern matching operations
      must be moved into the filters and cannot be written directly
      in the generators.</p>
      <p>To illustrate this, do <em>not</em> write as follows:</p>
    <code type="none"><![CDATA[
f(...) ->
    Y = ...
    [ Expression || PatternInvolving Y  <- Expr, ...]
    ...]]></code>
    <p>Instead, write as follows:</p>
    <code type="none"><![CDATA[
f(...) ->
    Y = ...
    [ Expression || PatternInvolving Y1  <- Expr, Y == Y1, ...]
    ...]]></code>
  </section>
</chapter>