Difference between revisions of "Higher-Order Functions"

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This page talks about ''higher-order functions'' introduced with [[XQuery 3.0]]. The BaseX-specific <code>hof</code> module containing some more very usful functions can be found at [[Higher-Order Functions Module]].
+
This page present some ''higher-order functions'' of the XQuery specification. The BaseX-specific [[Higher-Order Functions Module]] contains some additional useful functions.
 
 
{{Version|7.7}}: In the upcoming version of the [http://www.w3.org/TR/xpath-functions-30 XQuery Functions and Operators] specification, some functions will be modified! Function arguments are now placed last in the function signature. Details are found below.
 
  
 
=Function Items=
 
=Function Items=
  
Probably the most important new feature in XQuery 3.0 are ''function items'', i. e. items that act as functions, but can also be passed to and from other functions and expressions, making functions ''first-class citizens'' of the language.
+
Probably the most important new feature in XQuery 3.0 are ''function items'', i. e., items that act as functions, but can also be passed to and from other functions and expressions. This feature makes functions ''first-class citizens'' of the language. The [[XQuery_3.0#Function_Items|XQuery 3.0]] page goes into details on how function items can be obtained.
 
 
The [[XQuery_3.0#Function_Items|XQuery 3.0]] page goes into details on how function items can be obtained.
 
  
 
== Function Types ==
 
== Function Types ==
  
Like every XQuery item, function items have a ''sequence type''. It can be
+
Like every XQuery item, function items have a ''sequence type''. It can be used to specify the ''arity'' (number of arguments the function takes) and the argument and result types.
used to specify the ''arity'' (number of arguments the function takes) and
 
the argument and result types.
 
  
The most general function type is <code>function(*)</code>. It's the type of all
+
The most general function type is <code>function(*)</code>. It's the type of all function items. The following query for example goes through a list of XQuery items and, if it is a function item, prints its arity:
function items. The following query for example goes through a list of XQuery
 
items and, if it is a function item, prints its arity:
 
  
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
for $item in (1, 'foo', fn:concat#3, function($a) { 42 * $a })
 
for $item in (1, 'foo', fn:concat#3, function($a) { 42 * $a })
 
where $item instance of function(*)
 
where $item instance of function(*)
 
return fn:function-arity($item)
 
return fn:function-arity($item)
</pre>
+
</syntaxhighlight>
 
''Result:'' <code>3 1</code>
 
''Result:'' <code>3 1</code>
  
The notation for specifying argument and return types is quite intuitive, as it
+
The notation for specifying argument and return types is quite intuitive, as it closely resembles the function declaration. The XQuery function
closely resembles the function declaration. The XQuery function
 
  
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
declare function local:char-at(
 
declare function local:char-at(
 
   $str as xs:string,
 
   $str as xs:string,
Line 36: Line 27:
 
   fn:substring($str, $pos, 1)
 
   fn:substring($str, $pos, 1)
 
};
 
};
</pre>
+
</syntaxhighlight>
  
for example has the type <code>function(xs:string, xs:integer) as xs:string</code>. It isn't possible to specify only the argument and not the result
+
for example has the type <code>function(xs:string, xs:integer) as xs:string</code>. It isn't possible to specify only the argument and not the result type or the other way round. A good place-holder to use when no restriction is wanted is <code>item()*</code>, as it matches any XQuery value.
type or the other way round. A good place-holder to use when no restriction
 
is wanted is <code>item()*</code>, as it matches any XQuery value.
 
  
 
Function types can also be nested. As an example we take <code>local:on-sequences</code>, which takes a function defined on single items and makes it work on sequences as well:
 
Function types can also be nested. As an example we take <code>local:on-sequences</code>, which takes a function defined on single items and makes it work on sequences as well:
  
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
declare function local:on-sequences(
 
declare function local:on-sequences(
 
   $fun as function(item()) as item()*
 
   $fun as function(item()) as item()*
Line 50: Line 39:
 
   fn:for-each($fun, ?)
 
   fn:for-each($fun, ?)
 
};
 
};
</pre>
+
</syntaxhighlight>
We'll see later how <code>fn:for-each(...)</code> works. The type of <code>local:on-sequences(...)</code> on the other hand is easily constructed, if a bit long:
+
 
 +
We willl see later how <code>fn:for-each(...)</code> works. The type of <code>local:on-sequences(...)</code> on the other hand is easily constructed, if a bit long:
  
 
<code>function(function(item()) as item()*) as function(item()*) as item()*</code>.
 
<code>function(function(item()) as item()*) as function(item()*) as item()*</code>.
Line 59: Line 49:
 
A ''higher-order function'' is a function that takes other functions as arguments and/or returns them as results. <code>fn:for-each</code> and <code>local:on-sequences</code> from the last chapter are nice examples.
 
A ''higher-order function'' is a function that takes other functions as arguments and/or returns them as results. <code>fn:for-each</code> and <code>local:on-sequences</code> from the last chapter are nice examples.
  
With the help of higher-order functions, one can extract common patterns of
+
With the help of higher-order functions, one can extract common patterns of ''behavior'' and abstract them into a library function.
''behaviour'' and abstract them into a library function.
 
  
== Higher-Order Functions on Sequences ==
+
== Sequences ==
  
Some usage patterns on sequences are so common that the higher-order functions
+
Some usage patterns on sequences are so common that the higher-order functions describing them are in the XQuery standard libraries. They are listed here, together with their possible XQuery implementation and some motivating examples.
describing them are in the XQuery standard libraries. They are listed here, together
 
with their possible XQuery implementation and some motivating examples.
 
  
 
===fn:for-each===
 
===fn:for-each===
 
{{Mark|Updated with Version 7.7:}} the function has been renamed, and the arguments have been swapped.<br/>
 
  
 
{|
 
{|
 
|-
 
|-
 
| width='120' | '''Signatures'''
 
| width='120' | '''Signatures'''
|{{Func|fn:for-each|$seq as item()*, $fun as function(item()) as item()*)|item()*}}<br/><font color='gray'>Old signature: fn:map($fun as function(item()) as item()*, $seq as item()*) as item()*</font>
+
|{{Func|fn:for-each|$seq as item()*, $function as function(item()) as item()*)|item()*}}
 
|-
 
|-
 
| '''Summary'''
 
| '''Summary'''
|Applies the function item <code>$fun</code> to every element of the sequence <code>$seq</code> and returns all of the results as a sequence.
+
|Applies the specified <code>$function</code> to every item of <code>$seq</code> and returns all results as a single sequence.
 
|-
 
|-
 
| '''Examples'''
 
| '''Examples'''
 
|
 
|
<ul><li>Squaring all numbers from 1 to 10:
+
<ul><li>Square all numbers from 1 to 10:
  <pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
fn:for-each(1 to 10, math:pow(?, 2))
 
fn:for-each(1 to 10, math:pow(?, 2))
</pre>
+
</syntaxhighlight>
 
''Result:'' <code>1 4 9 16 25 36 49 64 81 100</code>
 
''Result:'' <code>1 4 9 16 25 36 49 64 81 100</code>
 
</li>
 
</li>
<li>Applying a list of functions to a string:
+
<li>Apply a list of functions to a string:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
let $fs := (
 
let $fs := (
 
   fn:upper-case#1,
 
   fn:upper-case#1,
Line 96: Line 81:
 
)
 
)
 
return fn:for-each($fs, function($f) { $f('foobar') })
 
return fn:for-each($fs, function($f) { $f('foobar') })
</pre>
+
</syntaxhighlight>
 
''Result:'' <code>FOOBAR bar 6</code>
 
''Result:'' <code>FOOBAR bar 6</code>
 +
</li>
 +
<li>Process each item of a sequence with the arrow operator:
 +
<syntaxhighlight lang="xquery">
 +
("one", "two", "three") => fn:for-each(fn:upper-case(?))
 +
</syntaxhighlight>
 +
''Result:'' <code>ONE TWO THREE</code>
 
</li></ul>
 
</li></ul>
 
|-
 
|-
 
| '''XQuery 1.0'''
 
| '''XQuery 1.0'''
|<pre class="brush:xquery">
+
|At the core, for-each is nothing else than a simple FLWOR expression:
 +
<syntaxhighlight lang="xquery">
 
declare function local:for-each(
 
declare function local:for-each(
 
   $seq as item()*,
 
   $seq as item()*,
Line 109: Line 101:
 
   return $fun($s)
 
   return $fun($s)
 
};
 
};
</pre>
+
</syntaxhighlight>
 
|}
 
|}
  
 
===fn:filter===
 
===fn:filter===
 
{{Mark|Updated with Version 7.7:}} the arguments have been swapped.<br/>
 
  
 
{|
 
{|
 
|-
 
|-
 
| width='120' | '''Signatures'''
 
| width='120' | '''Signatures'''
|{{Func|fn:filter|$seq as item()*, $pred as function(item()) as xs:boolean)|item()*}}<br /><font color='gray'>fn:filter($pred as function(item()) as xs:boolean, $seq as item()*) as item()*</font>
+
|{{Func|fn:filter|$seq as item()*, $pred as function(item()) as xs:boolean)|item()*}}
 
|-
 
|-
 
| '''Summary'''
 
| '''Summary'''
Line 126: Line 116:
 
| '''Examples'''
 
| '''Examples'''
 
|<ul><li>All even integers until 10:
 
|<ul><li>All even integers until 10:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
fn:filter(1 to 10, function($x) { $x mod 2 eq 0 })
 
fn:filter(1 to 10, function($x) { $x mod 2 eq 0 })
</pre>
+
</syntaxhighlight>
 
''Result:'' <code>2 4 6 8 10</code>
 
''Result:'' <code>2 4 6 8 10</code>
 
</li>
 
</li>
 
<li>Strings that start with an upper-case letter:
 
<li>Strings that start with an upper-case letter:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
let $first-upper := function($str) {
 
let $first-upper := function($str) {
 
   let $first := fn:substring($str, 1, 1)
 
   let $first := fn:substring($str, 1, 1)
Line 138: Line 128:
 
}
 
}
 
return fn:filter(('FooBar', 'foo', 'BAR'), $first-upper)
 
return fn:filter(('FooBar', 'foo', 'BAR'), $first-upper)
</pre>
+
</syntaxhighlight>
 
''Result:'' <code>FooBar BAR</code>
 
''Result:'' <code>FooBar BAR</code>
 
</li>
 
</li>
 
<li>Inefficient prime number generator:
 
<li>Inefficient prime number generator:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
let $is-prime := function($x) {
 
let $is-prime := function($x) {
 
   $x gt 1 and (every $y in 2 to ($x - 1) satisfies $x mod $y ne 0)
 
   $x gt 1 and (every $y in 2 to ($x - 1) satisfies $x mod $y ne 0)
 
}
 
}
 
return filter(1 to 20, $is-prime)
 
return filter(1 to 20, $is-prime)
</pre>
+
</syntaxhighlight>
 
''Result:'' <code>2 3 5 7 11 13 17 19</code>
 
''Result:'' <code>2 3 5 7 11 13 17 19</code>
 
</li></ul>
 
</li></ul>
Line 153: Line 143:
 
| '''Note'''
 
| '''Note'''
 
|<code>fn:filter</code> can be easily implemented with <code>fn:for-each</code>:
 
|<code>fn:filter</code> can be easily implemented with <code>fn:for-each</code>:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
declare function local:filter($seq, $pred) {
 
declare function local:filter($seq, $pred) {
 
   for-each(
 
   for-each(
Line 162: Line 152:
 
   )
 
   )
 
};
 
};
</pre>
+
</syntaxhighlight>
 
|-
 
|-
 
| '''XQuery 1.0'''
 
| '''XQuery 1.0'''
|<pre class="brush:xquery">
+
|At the core, for-each is nothing else than a filter expression:
 +
<syntaxhighlight lang="xquery">
 
declare function local:filter(
 
declare function local:filter(
 
   $seq as item()*,
 
   $seq as item()*,
Line 172: Line 163:
 
   $seq[$pred(.)]
 
   $seq[$pred(.)]
 
};
 
};
</pre>
+
</syntaxhighlight>
 
|}
 
|}
  
 
===fn:for-each-pair===
 
===fn:for-each-pair===
 
{{Mark|Updated with Version 7.7:}} the function has been renamed, and the arguments have been swapped.<br/>
 
  
 
{|
 
{|
 
|-
 
|-
 
| width='120' | '''Signatures'''
 
| width='120' | '''Signatures'''
|{{Func|fn:for-each-pair|$seq1 as item()*, $seq2 as item()*, $fun as function(item(), item()) as item()*|item()*}}<br /><font color='gray'>Old signature: fn:map-pairs($fun as function(item(), item()) as item()*, $seq1 as item()*, $seq2 as item()*) as item()*</font>
+
|{{Func|fn:for-each-pair|$seq1 as item()*, $seq2 as item()*, $function as function(item(), item()) as item()*|item()*}}
 
|-
 
|-
 
| '''Summary'''
 
| '''Summary'''
|''zips'' the elements from the two sequences <code>$seq1</code> and <code>$seq2</code> together with the function <code>$f</code>. It stops after the shorter sequence ends.
+
|Applies the specified {{Code|$function}} to the successive pairs of items of <code>$seq1</code> and <code>$seq2</code>. Evaluation is stopped if one sequence yields no more items.
 
|-
 
|-
 
| '''Examples'''
 
| '''Examples'''
 
|<ul><li>Adding one to the numbers at odd positions:
 
|<ul><li>Adding one to the numbers at odd positions:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
fn:for-each-pair(
 
fn:for-each-pair(
 
   fn:for-each(1 to 10, function($x) { $x mod 2 }),
 
   fn:for-each(1 to 10, function($x) { $x mod 2 }),
Line 195: Line 184:
 
   function($a, $b) { $a + $b }
 
   function($a, $b) { $a + $b }
 
)
 
)
</pre>
+
</syntaxhighlight>
 
''Result:'' <code>2 1 2 1 2</code>
 
''Result:'' <code>2 1 2 1 2</code>
 
</li>
 
</li>
 
<li>Line numbering:
 
<li>Line numbering:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
let $number-lines := function($str) {
+
let $number-words := function($str) {
 
   fn:string-join(
 
   fn:string-join(
 
     fn:for-each-pair(
 
     fn:for-each-pair(
       1 to 1000,
+
       1 to 1000000000,
       tokenize($str, '\r?\n|\r'),
+
       tokenize($str, ' +'),
 
       concat(?, ': ', ?)
 
       concat(?, ': ', ?)
 
     ),
 
     ),
     '&amp;#xa;'
+
     '&#xa;'
 
   )
 
   )
 
}
 
}
return $number-lines('hello world,
+
return $number-words('how are you?')
how are you?')
+
</syntaxhighlight>
</pre>
 
 
''Result:''
 
''Result:''
<pre>
+
<syntaxhighlight lang="xquery">
1: hello world,
+
1: how
2: how are you?
+
2: are
</pre>
+
3: you?
 +
</syntaxhighlight>
 
</li>
 
</li>
 
<li>Checking if a sequence is sorted:
 
<li>Checking if a sequence is sorted:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
let $is-sorted := function($seq) {
 
let $is-sorted := function($seq) {
 
   every $b in
 
   every $b in
Line 234: Line 223:
 
   $is-sorted((1, 2, 42, 4, 5))
 
   $is-sorted((1, 2, 42, 4, 5))
 
)
 
)
</pre>
+
</syntaxhighlight>
 
''Result:'' <code>true false</code></li></ul>
 
''Result:'' <code>true false</code></li></ul>
 
|-
 
|-
 
| '''XQuery 1.0'''
 
| '''XQuery 1.0'''
|<pre class="brush:xquery">
+
|<syntaxhighlight lang="xquery">
 
declare function local:for-each-pair(
 
declare function local:for-each-pair(
 
   $seq1 as item()*,
 
   $seq1 as item()*,
Line 247: Line 236:
 
   return $fun($seq1[$pos], $seq2[$pos])
 
   return $fun($seq1[$pos], $seq2[$pos])
 
};
 
};
</pre>
+
</syntaxhighlight>
 
|}
 
|}
  
 
== Folds ==
 
== Folds ==
  
A ''fold'', also called ''reduce'' or ''accumulate'' in other languages, is a very
+
A ''fold'', also called ''reduce'' or ''accumulate'' in other languages, is a very basic higher-order function on sequences. It starts from a seed value and incrementally builds up a result, consuming one element from the sequence at a time and combining it with the aggregate of a user-defined function.
basic higher-order function on sequences. It starts from a seed value and incrementally
 
builds up a result, consuming one element from the sequence at a time and combining it with
 
the aggregate with a user-defined function.
 
  
Folds are one solution to the problem of not having ''state'' in functional programs.
+
Folds are one solution to the problem of not having ''state'' in functional programs. Solving a problem in ''imperative'' programming languages often means repeatedly updating the value of variables, which isn't allowed in functional languages.
Solving a problem in ''imperative'' programming languages often means repeatedly updating
 
the value of variables, which isn't allowed in functional languages.
 
  
 
Calculating the ''product'' of a sequence of integers for example is easy in <code>Java</code>:
 
Calculating the ''product'' of a sequence of integers for example is easy in <code>Java</code>:
<pre class="brush:java">
+
 
 +
<syntaxhighlight lang="java">
 
public int product(int[] seq) {
 
public int product(int[] seq) {
 
   int result = 1;
 
   int result = 1;
Line 270: Line 255:
 
   return result;
 
   return result;
 
}
 
}
</pre>
+
</syntaxhighlight>
 +
 
 
Nice and efficient implementations using folds will be given below.
 
Nice and efficient implementations using folds will be given below.
  
The ''linear'' folds on sequences come in two flavours. They differ in the direction in which they traverse the sequence:
+
The ''linear'' folds on sequences come in two flavors. They differ in the direction in which they traverse the sequence:
  
 
===fn:fold-left===
 
===fn:fold-left===
 
{{Mark|Updated with Version 7.7:}} the <code>$seq</code> and <code>$fun</code> arguments have been swapped.<br/>
 
  
 
{|
 
{|
 
|-
 
|-
 
| width='120' | '''Signatures'''
 
| width='120' | '''Signatures'''
|{{Func|fn:fold-left|$seq as item()*, $seed as item()*, $fun as function(item()*, item()) as item()*|item()*}}<br /><font color='gray'>Old signature: fn:fold-left</b>($fun as function(item()*, item()) as item()*, $seed as item()*, $seq as item()*) as item()*</font>
+
|{{Func|fn:fold-left|$seq as item()*, $seed as item()*, $function as function(item()*, item()) as item()*|item()*}}
 
|-
 
|-
 
| '''Summary'''
 
| '''Summary'''
 
|The ''left fold'' traverses the sequence from the left.
 
|The ''left fold'' traverses the sequence from the left.
 
The query <code>fn:fold-left(1 to 5, 0, $f)</code> for example would be evaluated as:
 
The query <code>fn:fold-left(1 to 5, 0, $f)</code> for example would be evaluated as:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
$f($f($f($f($f(0, 1), 2), 3), 4), 5)
 
$f($f($f($f($f(0, 1), 2), 3), 4), 5)
</pre>
+
</syntaxhighlight>
 
|-
 
|-
 
| '''Examples'''
 
| '''Examples'''
 
|<ul><li>Product of a sequence of integers:
 
|<ul><li>Product of a sequence of integers:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
let $product := fn:fold-left(?, 1,
+
fn:fold-left(1 to 5, 1,
   function($result, $i) { $result * $i }
+
   function($result, $curr) { $result * $curr }
 
)
 
)
return $product(1 to 5)
+
</syntaxhighlight>
</pre>
 
 
''Result:'' <code>120</code>
 
''Result:'' <code>120</code>
 
</li>
 
</li>
 
<li>Illustrating the evaluation order:
 
<li>Illustrating the evaluation order:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
fn:fold-left(1 to 5, '$seed',
 
fn:fold-left(1 to 5, '$seed',
 
   concat('$f(', ?, ', ', ?, ')')
 
   concat('$f(', ?, ', ', ?, ')')
 
)
 
)
</pre>
+
</syntaxhighlight>
 
''Result:'' <code>$f($f($f($f($f($seed, 1), 2), 3), 4), 5)</code>
 
''Result:'' <code>$f($f($f($f($f($seed, 1), 2), 3), 4), 5)</code>
 
</li>
 
</li>
 
<li>Building a decimal number from digits:
 
<li>Building a decimal number from digits:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
let $from-digits := fold-left(?, 0,
 
let $from-digits := fold-left(?, 0,
 
   function($n, $d) { 10 * $n + $d }
 
   function($n, $d) { 10 * $n + $d }
Line 318: Line 301:
 
   $from-digits((4, 2))
 
   $from-digits((4, 2))
 
)
 
)
</pre>
+
</syntaxhighlight>
 
''Result:'' <code>12345 42</code>
 
''Result:'' <code>12345 42</code>
 
</li></ul>
 
</li></ul>
Line 324: Line 307:
 
| '''XQuery 1.0'''
 
| '''XQuery 1.0'''
 
|As folds are more general than ''FLWOR'' expressions, the implementation isn't as concise as the former ones:
 
|As folds are more general than ''FLWOR'' expressions, the implementation isn't as concise as the former ones:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
declare function local:fold-left(
 
declare function local:fold-left(
 
   $seq as item()*,
 
   $seq as item()*,
 
   $seed as item()*,
 
   $seed as item()*,
   $fun as function(item()*, item()) as item()*
+
   $function as function(item()*, item()) as item()*
 
) as item()* {
 
) as item()* {
 
   if(empty($seq)) then $seed
 
   if(empty($seq)) then $seed
 
   else local:fold-left(
 
   else local:fold-left(
 
     fn:tail($seq),
 
     fn:tail($seq),
     $fun($seed, fn:head($seq)),
+
     $function($seed, fn:head($seq)),
     $fun
+
     $function
 
   )  
 
   )  
 
};
 
};
</pre>
+
</syntaxhighlight>
 
|}
 
|}
  
 
===fn:fold-right===
 
===fn:fold-right===
 
{{Mark|Updated with Version 7.7:}} the {{Code|$seq}} and {{Code|$fun}} arguments have been swapped.<br/>
 
  
 
{|
 
{|
 
|-
 
|-
 
| width='120' | '''Signatures'''
 
| width='120' | '''Signatures'''
|{{Func|fn:fold-right|$seq as item()*, $seed as item()*, $fun as function(item(), item()*|item()*}}<br /><font color='gray'>Old signature: fn:fold-right</b>($fun as function(item(), item()*) as item()*, $seed as item()*, $seq as item()*) as item()*</font>
+
|{{Func|fn:fold-right|$seq as item()*, $seed as item()*, $function as function(item(), item()*) as item()*|item()*}}
 
|-
 
|-
 
| '''Summary'''
 
| '''Summary'''
|The ''right fold'' <code>fn:fold-right($seq, $seed, $fun)</code> traverses the from the right.
+
|The ''right fold'' <code>fn:fold-right($seq, $seed, $fun)</code> traverses the sequence from the right.
 
The query <code>fn:fold-right(1 to 5, 0, $f)</code> for example would be evaluated as:
 
The query <code>fn:fold-right(1 to 5, 0, $f)</code> for example would be evaluated as:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
$f(1, $f(2, $f(3, $f(4, $f(5, 0)))))
 
$f(1, $f(2, $f(3, $f(4, $f(5, 0)))))
</pre>
+
</syntaxhighlight>
 
|-
 
|-
 
| '''Examples'''
 
| '''Examples'''
 
|<ul><li>Product of a sequence of integers:
 
|<ul><li>Product of a sequence of integers:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
let $product := fn:fold-right(?, 1,
+
fn:fold-right(1 to 5, 1,
   function($i, $result) { $result * $i }
+
   function($curr, $result) { $result * $curr }
 
)
 
)
return $product(1 to 5)
+
</syntaxhighlight>
</pre>
 
 
''Result:'' <code>120</code>
 
''Result:'' <code>120</code>
 
</li>
 
</li>
 
<li>Illustrating the evaluation order:
 
<li>Illustrating the evaluation order:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
fn:fold-right(1 to 5, '$seed',
 
fn:fold-right(1 to 5, '$seed',
 
   concat('$f(', ?, ', ', ?, ')')
 
   concat('$f(', ?, ', ', ?, ')')
 
)
 
)
</pre>
+
</syntaxhighlight>
 
''Result:'' <code>$f(1, $f(2, $f(3, $f(4, $f(5, $seed)))))</code>
 
''Result:'' <code>$f(1, $f(2, $f(3, $f(4, $f(5, $seed)))))</code>
 
</li>
 
</li>
 
<li>Reversing a sequence of items:
 
<li>Reversing a sequence of items:
<pre class="brush:xquery">
+
<syntaxhighlight lang="xquery">
 
let $reverse := fn:fold-right(?, (),
 
let $reverse := fn:fold-right(?, (),
 
   function($item, $rev) {
 
   function($item, $rev) {
Line 382: Line 362:
 
)
 
)
 
return $reverse(1 to 10)
 
return $reverse(1 to 10)
</pre>
+
</syntaxhighlight>
 
''Result:'' <code>10 9 8 7 6 5 4 3 2 1</code>
 
''Result:'' <code>10 9 8 7 6 5 4 3 2 1</code>
 
</li></ul>
 
</li></ul>
 
|-
 
|-
 
| '''XQuery 1.0'''
 
| '''XQuery 1.0'''
|<pre class="brush:xquery">
+
|<syntaxhighlight lang="xquery">
 
declare function local:fold-right(
 
declare function local:fold-right(
 
   $seq as item()*,
 
   $seq as item()*,
 
   $seed as item()*,
 
   $seed as item()*,
   $fun as function(item(), item()*) as item()*
+
   $function as function(item(), item()*) as item()*
 
) as item()* {
 
) as item()* {
 
   if(empty($seq)) then $seed
 
   if(empty($seq)) then $seed
   else $fun(
+
   else $function(
 
     fn:head($seq),
 
     fn:head($seq),
     local:fold-right(tail($seq), $seed, $fun)
+
     local:fold-right(tail($seq), $seed, $function)
 
   )
 
   )
 
};
 
};
</pre>
+
</syntaxhighlight>
 
Note that the order of the arguments of <code>$fun</code> are inverted compared to that in <code>fn:fold-left(...)</code>.
 
Note that the order of the arguments of <code>$fun</code> are inverted compared to that in <code>fn:fold-left(...)</code>.
 
|}
 
|}
 
[[Category:XQuery]]
 

Latest revision as of 14:17, 27 February 2020

This page present some higher-order functions of the XQuery specification. The BaseX-specific Higher-Order Functions Module contains some additional useful functions.

Function Items[edit]

Probably the most important new feature in XQuery 3.0 are function items, i. e., items that act as functions, but can also be passed to and from other functions and expressions. This feature makes functions first-class citizens of the language. The XQuery 3.0 page goes into details on how function items can be obtained.

Function Types[edit]

Like every XQuery item, function items have a sequence type. It can be used to specify the arity (number of arguments the function takes) and the argument and result types.

The most general function type is function(*). It's the type of all function items. The following query for example goes through a list of XQuery items and, if it is a function item, prints its arity:

for $item in (1, 'foo', fn:concat#3, function($a) { 42 * $a })
where $item instance of function(*)
return fn:function-arity($item)

Result: 3 1

The notation for specifying argument and return types is quite intuitive, as it closely resembles the function declaration. The XQuery function

declare function local:char-at(
  $str as xs:string,
  $pos as xs:integer
) as xs:string {
  fn:substring($str, $pos, 1)
};

for example has the type function(xs:string, xs:integer) as xs:string. It isn't possible to specify only the argument and not the result type or the other way round. A good place-holder to use when no restriction is wanted is item()*, as it matches any XQuery value.

Function types can also be nested. As an example we take local:on-sequences, which takes a function defined on single items and makes it work on sequences as well:

declare function local:on-sequences(
  $fun as function(item()) as item()*
) as function(item()*) as item()* {
  fn:for-each($fun, ?)
};

We willl see later how fn:for-each(...) works. The type of local:on-sequences(...) on the other hand is easily constructed, if a bit long:

function(function(item()) as item()*) as function(item()*) as item()*.

Higher-Order Functions[edit]

A higher-order function is a function that takes other functions as arguments and/or returns them as results. fn:for-each and local:on-sequences from the last chapter are nice examples.

With the help of higher-order functions, one can extract common patterns of behavior and abstract them into a library function.

Sequences[edit]

Some usage patterns on sequences are so common that the higher-order functions describing them are in the XQuery standard libraries. They are listed here, together with their possible XQuery implementation and some motivating examples.

fn:for-each[edit]

Signatures fn:for-each($seq as item()*, $function as function(item()) as item()*)) as item()*
Summary Applies the specified $function to every item of $seq and returns all results as a single sequence.
Examples
  • Square all numbers from 1 to 10:
    fn:for-each(1 to 10, math:pow(?, 2))
    

    Result: 1 4 9 16 25 36 49 64 81 100

  • Apply a list of functions to a string:
    let $fs := (
      fn:upper-case#1,
      fn:substring(?, 4),
      fn:string-length#1
    )
    return fn:for-each($fs, function($f) { $f('foobar') })
    

    Result: FOOBAR bar 6

  • Process each item of a sequence with the arrow operator:
    ("one", "two", "three") => fn:for-each(fn:upper-case(?))
    

    Result: ONE TWO THREE

XQuery 1.0 At the core, for-each is nothing else than a simple FLWOR expression:
declare function local:for-each(
  $seq as item()*,
  $fun as function(item()) as item()*
) as item()* {
  for $s in $seq
  return $fun($s)
};

fn:filter[edit]

Signatures fn:filter($seq as item()*, $pred as function(item()) as xs:boolean)) as item()*
Summary Applies the boolean predicate $pred to all elements of the sequence $seq, returning those for which it returns true().
Examples
  • All even integers until 10:
    fn:filter(1 to 10, function($x) { $x mod 2 eq 0 })
    

    Result: 2 4 6 8 10

  • Strings that start with an upper-case letter:
    let $first-upper := function($str) {
      let $first := fn:substring($str, 1, 1)
      return $first eq fn:upper-case($first)
    }
    return fn:filter(('FooBar', 'foo', 'BAR'), $first-upper)
    

    Result: FooBar BAR

  • Inefficient prime number generator:
    let $is-prime := function($x) {
      $x gt 1 and (every $y in 2 to ($x - 1) satisfies $x mod $y ne 0)
    }
    return filter(1 to 20, $is-prime)
    

    Result: 2 3 5 7 11 13 17 19

Note fn:filter can be easily implemented with fn:for-each:
declare function local:filter($seq, $pred) {
  for-each(
    $seq,
    function($x) {
      if($pred($x)) then $x else ()
    }
  )
};
XQuery 1.0 At the core, for-each is nothing else than a filter expression:
declare function local:filter(
  $seq as item()*,
  $pred as function(item()) as xs:boolean
) as item()* {
  $seq[$pred(.)]
};

fn:for-each-pair[edit]

Signatures fn:for-each-pair($seq1 as item()*, $seq2 as item()*, $function as function(item(), item()) as item()*) as item()*
Summary Applies the specified $function to the successive pairs of items of $seq1 and $seq2. Evaluation is stopped if one sequence yields no more items.
Examples
  • Adding one to the numbers at odd positions:
    fn:for-each-pair(
      fn:for-each(1 to 10, function($x) { $x mod 2 }),
      (1, 1, 1, 1, 1),
      function($a, $b) { $a + $b }
    )
    

    Result: 2 1 2 1 2

  • Line numbering:
    let $number-words := function($str) {
      fn:string-join(
        fn:for-each-pair(
          1 to 1000000000,
          tokenize($str, ' +'),
          concat(?, ': ', ?)
        ),
        '&#xa;'
      )
    }
    return $number-words('how are you?')
    

    Result:

    1: how
    2: are
    3: you?
    
  • Checking if a sequence is sorted:
    let $is-sorted := function($seq) {
      every $b in
        fn:for-each-pair(
          $seq,
          fn:tail($seq),
          function($a, $b) { $a le $b }
        )
      satisfies $b
    }
    return (
      $is-sorted(1 to 10),
      $is-sorted((1, 2, 42, 4, 5))
    )
    
    Result: true false
XQuery 1.0
declare function local:for-each-pair(
  $seq1 as item()*,
  $seq2 as item()*,
  $fun as function(item(), item()) as item()*
) as item()* {
  for $pos in 1 to min((count($seq1), count($seq2)))
  return $fun($seq1[$pos], $seq2[$pos])
};

Folds[edit]

A fold, also called reduce or accumulate in other languages, is a very basic higher-order function on sequences. It starts from a seed value and incrementally builds up a result, consuming one element from the sequence at a time and combining it with the aggregate of a user-defined function.

Folds are one solution to the problem of not having state in functional programs. Solving a problem in imperative programming languages often means repeatedly updating the value of variables, which isn't allowed in functional languages.

Calculating the product of a sequence of integers for example is easy in Java:

public int product(int[] seq) {
  int result = 1;
  for(int i : seq) {
    result = result * i;
  }
  return result;
}

Nice and efficient implementations using folds will be given below.

The linear folds on sequences come in two flavors. They differ in the direction in which they traverse the sequence:

fn:fold-left[edit]

Signatures fn:fold-left($seq as item()*, $seed as item()*, $function as function(item()*, item()) as item()*) as item()*
Summary The left fold traverses the sequence from the left.

The query fn:fold-left(1 to 5, 0, $f) for example would be evaluated as:

$f($f($f($f($f(0, 1), 2), 3), 4), 5)
Examples
  • Product of a sequence of integers:
    fn:fold-left(1 to 5, 1,
      function($result, $curr) { $result * $curr }
    )
    

    Result: 120

  • Illustrating the evaluation order:
    fn:fold-left(1 to 5, '$seed',
      concat('$f(', ?, ', ', ?, ')')
    )
    

    Result: $f($f($f($f($f($seed, 1), 2), 3), 4), 5)

  • Building a decimal number from digits:
    let $from-digits := fold-left(?, 0,
      function($n, $d) { 10 * $n + $d }
    )
    return (
      $from-digits(1 to 5),
      $from-digits((4, 2))
    )
    

    Result: 12345 42

XQuery 1.0 As folds are more general than FLWOR expressions, the implementation isn't as concise as the former ones:
declare function local:fold-left(
  $seq as item()*,
  $seed as item()*,
  $function as function(item()*, item()) as item()*
) as item()* {
  if(empty($seq)) then $seed
  else local:fold-left(
    fn:tail($seq),
    $function($seed, fn:head($seq)),
    $function
  ) 
};

fn:fold-right[edit]

Signatures fn:fold-right($seq as item()*, $seed as item()*, $function as function(item(), item()*) as item()*) as item()*
Summary The right fold fn:fold-right($seq, $seed, $fun) traverses the sequence from the right.

The query fn:fold-right(1 to 5, 0, $f) for example would be evaluated as:

$f(1, $f(2, $f(3, $f(4, $f(5, 0)))))
Examples
  • Product of a sequence of integers:
    fn:fold-right(1 to 5, 1,
      function($curr, $result) { $result * $curr }
    )
    

    Result: 120

  • Illustrating the evaluation order:
    fn:fold-right(1 to 5, '$seed',
      concat('$f(', ?, ', ', ?, ')')
    )
    

    Result: $f(1, $f(2, $f(3, $f(4, $f(5, $seed)))))

  • Reversing a sequence of items:
    let $reverse := fn:fold-right(?, (),
      function($item, $rev) {
        $rev, $item
      }
    )
    return $reverse(1 to 10)
    

    Result: 10 9 8 7 6 5 4 3 2 1

XQuery 1.0
declare function local:fold-right(
  $seq as item()*,
  $seed as item()*,
  $function as function(item(), item()*) as item()*
) as item()* {
  if(empty($seq)) then $seed
  else $function(
    fn:head($seq),
    local:fold-right(tail($seq), $seed, $function)
  )
};

Note that the order of the arguments of $fun are inverted compared to that in fn:fold-left(...).