To make it clearer about what the two parameters of the callback are for, and what "reduce to a single value" actually means (using associative and commutative operators as examples may obscure this).

The first parameter to the callback is an accumulator where the result-in-progress is effectively assembled. If you supply an $initial value the accumulator starts out with that value, otherwise it starts out null.
The second parameter is where each value of the array is passed during each step of the reduction.
The return value of the callback becomes the new value of the accumulator. When the array is exhausted, array_reduce() returns accumulated value.

If you carried out the reduction by hand, you'd get something like the following lines, every one of which therefore producing the same result:
<?php
array_reduce(array(1,2,3,4), 'f',         99             );
array_reduce(array(2,3,4),   'f',       f(99,1)          );
array_reduce(array(3,4),     'f',     f(f(99,1),2)       );
array_reduce(array(4),       'f',   f(f(f(99,1),2),3)    );
array_reduce(array(),        'f', f(f(f(f(99,1),2),3),4) );
f(f(f(f(99,1),2),3),4)
?>

If you made function f($v,$w){return "f($v,$w)";} the last line would be the literal result.

A PHP implementation might therefore look something like this (less details like error checking and so on):
<?php
function array_reduce($array, $callback, $initial=null)
{
    $acc = $initial;
    foreach($array as $a)
        $acc = $callback($acc, $a);
    return $acc;
}
?>

So, if you were wondering how to use this where key and value are passed in to the function. I've had success with the following (this example generates formatted html attributes from an associative array of attribute => value pairs):

<?php

    // Attribute List
    $attribs = [
        'name' => 'first_name',
        'value' => 'Edward'
    ];

    // Attribute string formatted for use inside HTML element
    $formatted_attribs = array_reduce(
        array_keys($attribs),                       // We pass in the array_keys instead of the array here
        function ($carry, $key) use ($attribs) {    // ... then we 'use' the actual array here
            return $carry . ' ' . $key . '="' . htmlspecialchars( $attribs[$key] ) . '"';
        },
        ''
    );

echo $formatted_attribs;

?>

This will output:
 name="first_name" value="Edward"

Sometimes we need to go through an array and group the indexes so that it is easier and easier to extract them in the iteration.

<?php

$people = [
    ['id' => 1, 'name' => 'Hayley'],
    ['id' => 2, 'name' => 'Jack', 'dad' => 1],
    ['id' => 3, 'name' => 'Linus', 'dad'=> 4],
    ['id' => 4, 'name' => 'Peter' ],
    ['id' => 5, 'name' => 'Tom', 'dad' => 4],
];

$family = array_reduce($people, function($accumulator, $item) {
    // if you don't have a dad you are probably a dad
    if (!isset($item['dad'])) {
        $id = $item['id'];
        $name = $item['name'];
        // take the children if you already have 
        $children = $accumulator[$id]['children'] ?? [];
        // add dad
        $accumulator[$id] = ['id' => $id, 'name' => $name,'children' => $children];
        return $accumulator;
    }

    // add a new dad if you haven't already 
    $dad = $item['dad'];
    if (!isset($accumulator[$dad])) {
        // how did you find the dad will first add only with children 
        $accumulator[$dad] = ['children' => [$item]];
        return $accumulator;
    }

    //  add a son to his dad who has already been added
    //  by the first or second conditional "if"
    
    $accumulator[$dad]['children'][] = $item;
    return $accumulator;
}, []);

var_export(array_values($family));

?>

OUTPUT

array (
  0 =>
  array (
    'id' => 1,
    'name' => 'Hayley',
    'children' =>
    array (
      0 =>
      array (
        'id' => 2,
        'name' => 'Jack',
        'dad' => 1,
      ),
    ),
  ),
  1 =>
  array (
    'id' => 4,
    'name' => 'Peter',
    'children' =>
    array (
      0 =>
      array (
        'id' => 3,
        'name' => 'Linus',
        'dad' => 4,
      ),
      1 =>
      array (
        'id' => 5,
        'name' => 'Tom',
        'dad' => 4,
      ),
    ),
  ),
)

<?php
$array = [
  [
    "menu_id" => "1",
    "menu_name" => "Clients",
    "submenu_name" => "Add",
    "submenu_link" => "clients/add"
  ],
  [
    "menu_id" => "1",
    "menu_name" => "Clients",
    "submenu_name" => "List",
    "submenu_link" => "clients"
  ],
  [
    "menu_id" => "2",
    "menu_name" => "Products",
    "submenu_name" => "List",
    "submenu_link" => "products"
  ],
];

//Grouping submenus to their menus

$menu =  array_reduce($array, function($accumulator, $item){
  $index = $item['menu_name'];

  if (!isset($accumulator[$index])) {
    $accumulator[$index] = [
      'menu_id' => $item['menu_id'],
      'menu_name' => $item['menu_name'],
      'submenu' => []    
    ];
  }

  $accumulator[$index]['submenu'][] = [
    'submenu_name' => $item['submenu_name'],
    'submenu_link' => $item['submenu_link']
  ];

  return $accumulator;
}, []);

var_export(array_values($menu));

?>

OUTPUT

array (
  0 =>
  array (
    'menu_id' => '1',
    'menu_name' => 'Clients',
    'submenu' =>
    array (
      0 =>
      array (
        'submenu_name' => 'Add',
        'submenu_link' => 'clients/add',
      ),
      1 =>
      array (
        'submenu_name' => 'List',
        'submenu_link' => 'clients',
      ),
    ),
  ),
  1 =>
  array (
    'menu_id' => '2',
    'menu_name' => 'Products',
    'submenu' =>
    array (
      0 =>
      array (
        'submenu_name' => 'List',
        'submenu_link' => 'products',
      ),
    ),
  ),
)

If you want to reindex an array, say to add an ID to each item as a key, you can use the following:

$array = [['id' => 3, 'name' => 'bob'], ['id' => 4, 'name' => 'alice']];
array_reduce($array, fn($new_array, $item) => $new_array + [$item['id'] => $item], []);

Result:

[ 3 => ["id" => 3, "name" => "bob"], 4 => ['id' => 4, 'name' => 'alice'] ]

Let's see an example of array_reduce() to get the frequency of letters.

<?php

$items = "Hello";

$frequencies = array_reduce(str_split($items), 
    function($result, $item) {
        if (isset($result[$item])) {
            $result[$item] += 1;
        } else { 
            $result[$item] = 1;
        }
        return $result;
    },
    [] // note the initial is an array
);

print_r($frequencies);

?>

and output should be like:

Array
(
    [H] => 1
    [e] => 1
    [l] => 2
    [o] => 1
)

Array reduce offers a way to transform data. 
Please look at the array below. The array has 4 nested array's.
The nested array's have the same keys. Only the value is different. 

This code transforms the whole array. See below.  

$array = array(  
    0 => array('id' => '100', 'name' => 'Henk', 'age' => '30'),   
    1 => array('id' => '101', 'name' => 'Piet', 'age' => '33'),   
    2 => array('id' => '102', 'name' => 'Wim', 'age' => '43'),   
    3 => array('id' => '103', 'name' => 'Jaap', 'age' => '53'),  
);  
 
$arr = array_reduce($array, function($carry, $item){ 
 
    $arr = array(  
        'id' => $item['id'],  
        'value' => $item['name'],  
    );  
      
    $id = $item['id'];  
    $carry[$id] = $arr;  
      
    return $carry;  
}, array());  
 
var_dump($arr);
 
 
// OUTPUT

array (size=4)  
100 => array (size=2)         
   'id' => string '100' (length=3)         
   'value' => string 'Henk' (length=4)  
101 => array (size=2)          
   'id' => string '101' (length=3)          
   'value' => string 'Piet' (length=4)  
102 => array (size=2)          
   'id' => string '102' (length=3)          
   'value' => string 'Wim' (length=3)  
103 => array (size=2)          
   'id' => string '103' (length=3)          
   'value' => string 'Jaap' (length=4)

The first parameter $array can be also be functions, which produces very interesting and powerful result, which can be used to make an union of middlewares.

<?php

$f1 = function($x, $f){
    echo 'middleware 1 begin.'.PHP_EOL;
    $x += 1;
    $x = $f($x);
    echo 'middleware 1 end.'.PHP_EOL;
    return $x;    
};
    

$f2 = function($x, $f){
    echo 'middleware 2 begin: '.PHP_EOL;
    $x += 2;
    $x = $f($x);
    echo 'middleware 2 end.'.PHP_EOL;
    return $x;
};

$respond = function($x){
    echo 'Generate some response.'.PHP_EOL;
    return $x;
};

$middlewares = [$f1, $f2];
$initial = $respond;
$foo = array_reduce($middlewares, function($stack, $item){
    return function($request) use ($stack, $item){
        return $item($request, $stack);
    };
}, $initial);

$x = 1;
echo $foo($x);

?>

//output:
middleware 2 begin: 
middleware 1 begin.
Generate some response.
middleware 1 end.
middleware 2 end.
4

You can reduce a two-dimensional array into one-dimensional using array_reduce and array_merge. (PHP>=5.3.0)

<?php

$two_dimensional = array();
$two_dimensional['foo'] = array(1, 2, 3);
$two_dimensional['bar'] = array(4, 5, 6);

$one_dimensional = array_reduce($two_dimensional, 'array_merge', array());
# becomes array(1, 2, 3, 4, 5, 6)

You can effectively ignore the fact $result is passed into the callback by reference. Only the return value of the callback is accounted for.

<?php

$arr = [1,2,3,4];

var_dump(array_reduce(
    $arr,
    function(&$res, $a) { $res += $a; }, 
    0
));

# NULL

?>

<?php

$arr = [1,2,3,4];

var_dump(array_reduce(
    $arr, 
    function($res, $a) { return $res + $a;  }, 
    0
));

# int(10)
?>

Be warned, though, that you *can* accidentally change $res if it's not a simple scalar value, so despite the examples I'd recommend not writing to it at all.

If you need access to the keys of the array in your callback, this alternative uses the same signatures as array_reduce() and its callback, with the addition of a $key argument:
<?php
function array_reduce_assoc(array $array, callable $callback, mixed $initial = null): mixed
{
    foreach ($array as $key => $item)
    {
        $initial = call_user_func($callback, $initial, $item, $key);
    }
    return $initial;
}

If you want something elegant in your code, when dealing with reducing array, just unshift first element, and use it as initial, because if you do not do so, you will + first element with first element:

<?php
$arr = array(
    array('min' => 1.5456, 'max' => 2.28548, 'volume' => 23.152),
    array('min' => 1.5457, 'max' => 2.28549, 'volume' => 23.152),
    array('min' => 1.5458, 'max' => 2.28550, 'volume' => 23.152),
    array('min' => 1.5459, 'max' => 2.28551, 'volume' => 23.152),
    array('min' => 1.5460, 'max' => 2.28552, 'volume' => 23.152),
);

$initial = array_shift($arr);

$t = array_reduce($arr, function($result, $item) {
    $result['min'] = min($result['min'], $item['min']);
    $result['max'] = max($result['max'], $item['max']);
    $result['volume'] += $item['volume'];

    return $result;
}, $initial);
?>

If you do not provide $initial, the first value used in the iteration is NULL. This is not a problem for callback functions that treat NULL as an identity (e.g. addition), but is a problem for cases when NULL is not identity (such as boolean context).

Compare:

<?php
function andFunc($a, $b) {
  return $a && $b;
}
$foo = array(true, true, true);
var_dump(array_reduce($foo, "andFunc"));
?>

returns false! One would expect that it would return true because `true && true && true == true`!

Adding diagnostic output to andFunc() shows that the first call to andFunc is with the arguments (NULL, true). This resolves to false (as `(bool) null == false`) and thereby corrupts the whole reduction.

So in this case I have to set `$initial = true` so that the first call to andFunc() will be (true, true). Now, if I were doing, say, orFunc(), I would have to set `$initial = false`. Beware.

Note that the "rmul" case in the example sneakily hides this defect! They use an $initial of 10 to get `10*1*2*3*4*5 = 12000`. So you would assume that without an initial, you would get `1200/10 = 120 = 1*2*3*4*5`. Nope! You get big fat zero, because `int(null)==0`, and `0*1*2*3*4*5 = 0`!

I don't honestly see why array_reduce starts with a null argument. The first call to the callback should be with arguments ($initial[0],$initial[1]) [or whatever the first two array entries are], not (null,$initial[0]). That's what one would expect from the description.

Incidentally this also means that under the current implementation you will incur `count($input)` number of calls to the callback, not `count($input) - 1` as you might expect.

The single value returned by array_reduce() can be an array -- as illustrated in the following example:
<?php
# calculate the average of an array
function calculate_sum_and_count($sum_and_count, $item)
{
  list($sum, $count) = $sum_and_count;
  $sum += $item;
  $count += 1;
  return [$sum, $count];
}

$a = array(1, 2, 3, 4, 5, 6, 7, 8, 9, 10);
$initial_sum_and_count = [0, 0];
list($sum, $count) = array_reduce($a, "calculate_sum_and_count", $initial_sum_and_count);
echo $sum / $count;
?>

The above code works better this way.

<?php
function reduceToTable($html, $p) {
   $html .= "<TR><TD><a href=\"$p.html\">$p</a></td></tr>\n";
   return $html;
}

$list = Array("page1", "page2", "page3");

$tab = array_reduce($list, "reduceToTable");
echo "<table>".$tab . "</table>\n";
?>

This code will reduce array deeply.

<?php
function print_s($s) {
    return is_null($s) ? "NULL" : (is_array($s) ? "Array" : ($s ? "TRUE" : "FALSE"));
}
function r_and_dp($a, $b) {
    echo "phase1:" . print_s($a) . "," . print_s($b) . "<br>\n";
    if(is_array($a)) {
        $a = array_reduce($a, "r_and_dp");
    }
    if(is_array($b)) {
        $b = array_reduce($b, "r_and_dp");
    }
    echo "phase2:" . print_s($a) . "," . print_s($b) . "<br>\n";
    $a = is_null($a) ? TRUE : $a;
    $b = is_null($b) ? TRUE : $b;
    echo "phase3:" . print_s($a) . "," . print_s($b) . "<br>\n";
    return $a && $b;
}
$bools = array(TRUE, array(FALSE, TRUE), TRUE);
echo print_s(array_reduce($bools, "r_and_dp")) . "<br>\n";

// result: FALSE
?>

When using boolean, you have to carefully set an "initial" argument.

<?php
function r_or_dp($a, $b) {
    if(is_array($a)) {
        $a = array_reduce($a, "r_or_dp");
    }
    if(is_array($b)) {
        $b = array_reduce($b, "r_or_dp");
    }
    return (is_null($a) ? FALSE : $a) || (is_null($b) ? FALSE : $b);
}
?>

The code posted below by bishop to count the characters of an array is simply... erm... well useless to me...

$array=Array("abc","de","f");
strlen(implode("",$array)); //6

works; and is much smaller. Probably much faster too.

Walking down related object's properties using array_reduce:

<?php
  $a=new stdClass;
  $a->b=new stdClass;
  $a->b->c="Hello World!\n";

  $reductionPath=array("b","c");

  print_r(
    array_reduce(
      $reductionPath,
      function($result, $item){
        return $result->$item;
      },
      $a
    )
  );
?>