Difference between revisions of "Higher-Order Functions"

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| valign='top' | '''XQuery 1.0'''
 
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<pre class="brush:xquery">
 
declare function local:fold-left(
 
declare function local:fold-left(

Revision as of 16:51, 15 March 2012

This page talks about higher-order functions introduced with XQuery 3.0. The BaseX-specific hof module containing some more very usful functions can be found at Higher-Order Functions Module.

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.

The XQuery 3.0 page goes into details on how function items can be obtained.

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(
  $f as function(item()) as item()*
) as function(item()*) as item()* {
  fn:map($f, ?)
};

We'll see later how fn:map(...) 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

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

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

Higher-Order Functions on 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.

fn:map($f, $seq)

Signatures fn:map($f as function(item()) as item()*, $seq as item()*) as item()*
Summary Applies the function item $f to every element of the sequence $seq and returns all of the results as a sequence.
Examples
  • Squaring all numbers from 1 to 10:
    fn:map(math:pow(?, 2), 1 to 10)
    

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

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

    Result: FOOBAR bar 6

XQuery 1.0
declare function local:map(
  $f as function(item()) as item()*,
  $seq as item()* 
) as item()* {
  for $x in $seq
  return $f($seq)
};

fn:filter($pred, $seq)

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

    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($first-upper, ('FooBar', 'foo', 'BAR'))
    

    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 $y ne 0)
    }
    return filter($is-prime, 1 to 20)
    

    Result: 2 3 5 7 11 13 17 19

Note fn:filter can be easily implemented with fn:map:
declare function local:filter($pred, $seq) {
  map(
    function($x) {
      if($pred($x)) then $x else ()
    },
    $seq
  )
};
XQuery 1.0
declare function local:filter(
  $pred as function(item()) as xs:boolean,
  $seq as item()* 
) as item()* {
  $seq[$pred(.)]
};

fn:map-pairs($f, $seq1, $seq2)

Signatures fn:map-pairs($f as function(item(), item()) as item()*, $seq1 as item()*, $seq2 as item()*) as item()*
Summary zips the elements from the two sequences $seq1 and $seq2 together with the function $f. It stops after the shorter sequence ends.
Examples
  • Adding one to the numbers at odd positions:
    fn:map-pairs(
      function($a, $b) { $a + $b },
      fn:map(function($x) { $x mod 2 }, 1 to 10),
      (1, 1, 1, 1, 1)
    )
    

    Result: 2 1 2 1 2

  • Line numbering:
    let $number-lines := function($str) {
      fn:string-join(
        fn:map-pairs(
          concat(?, ': ', ?),
          1 to 1000,
          tokenize($str, '\r?\n|\r')
        ),
        '&#xa;'
      )
    }
    return $number-lines(
      'hello world,
      how are you?'
    )
    

    Result:

    1: hello world,
    2: how are you?
    
  • Checking if a sequence is sorted:
    let $is-sorted := function($seq) {
      every $b in
        fn:map-pairs(
          function($a, $b) { $a le $b },
          $seq,
          fn:tail($seq)      
        )
      satisfies $b
    }
    return (
      $is-sorted(1 to 10),
      $is-sorted((1, 2, 42, 4, 5))
    )
    

    Result: true false

XQuery 1.0
declare function local:map-pairs(
  $f as function(item(), item()) as item()*,
  $seq1 as item()*,
  $seq2 as item()*
) as item()* {
  for $pos in 1 to min(length($seq1), length($seq2))
  return $f($seq1[$pos], $seq2[$pos])
};

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 with 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 flavours. They differ in the direction in which they traverse the sequence:

fn:fold-left($f, $seed, $seq)

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

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

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

    Result: 120

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

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

  • Building a decimal number from digits:
    let $from-digits := fold-left(
      function($n, $d) { 10 * $n + $d },
      0,
      ?
    )
    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(
  $f as function(item()*, item()) as item()*,
  $seed as item()*,
  $seq as item()*
) as item()* {
  if(empty($seq)) then $seed
  else local:fold-left(
    $f,
    $f($seed, fn:head($seq)),
    fn:tail($seq)
  ) 
};

fn:fold-right($f, $seed, $seq)

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

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

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

    Result: 120

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

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

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