HigherOrder Functions
This page talks about higherorder functions introduced with XQuery 3.0. The BaseXspecific hof
module containing some more very usful functions can be found at HigherOrder Functions Module.
Contents
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 firstclass 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 $item instance of function(*) return fn:functionarity($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:charat( $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 placeholder 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:onsequences
, which takes a function defined on single items and makes it work on sequences as well:
declare function local:onsequences( $f as function(item()) as item()* ) as function(item()*) as item()* { fn:map($f, ?) };
We'll see later how fn:map(...)
works. The type of local:onsequences(...)
on the other hand is easily constructed, if a bit long:
function(function(item()) as item()*) as function(item()*) as item()*
.
HigherOrder Functions
A higherorder function is a function that takes other functions as arguments and/or returns them as results. fn:map
and local:onsequences
from the last chapter are nice examples.
With the help of higherorder functions, one can extract common patterns of behaviour and abstract them into a library function.
HigherOrder Functions on Sequences
Some usage patterns on sequences are so common that the higherorder 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 

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 

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:mappairs($f, $seq1, $seq2)
Signatures  fn:mappairs($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 

XQuery 1.0  declare function local:mappairs( $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 higherorder 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 userdefined 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:foldleft($f, $seed, $seq)
Signatures  fn:foldleft($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 $f($f($f($f($f(0, 1), 2), 3), 4), 5) 
Examples 

XQuery 1.0  As folds are more general than FLWOR expressions, the implementation isn't as concise as the former ones:
declare function local:foldleft( $f as function(item()*, item()) as item()*, $seed as item()*, $seq as item()* ) as item()* { if(empty($seq)) then $seed else local:foldleft( $f, $f($seed, fn:head($seq)), fn:tail($seq) ) }; 
fn:foldright($f, $seed, $seq)
Signatures  fn:foldright($f as function(item(), item()*) as item()*, $seed as item()*, $seq as item()*) as item()* 
Summary  The right fold fn:foldright($f, $seed, $seq) traverses the from the right.
The query $f(1, $f(2, $f(3, $f(4, $f(5, 0))))) 
Examples 

XQuery 1.0  declare function local:foldright( $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:foldright($f, $seed, tail($seq)) ) }; Note that the order of the arguments of 