Higher-Order Functions

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

Contents

[edit] 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. 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.

[edit] Function Types

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()*.

[edit] Higher-Order Functions

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.

[edit] Sequences

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.

[edit] fn:for-each

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)
};

[edit] fn:filter

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(.)]
};

[edit] fn:for-each-pair

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(?, ': ', ?)
        ),
        '
    '
      )
    }
    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])
};

[edit] Folds

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:

[edit] fn:fold-left

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
  ) 
};

[edit] fn:fold-right

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(...).

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