How can I filter an array of strings to remove elements that are prefixes of other elements?

$old_array = array('123','234','1234','2345','123456','345','1','12');

1, 12, 123, 1234, 123456
234, 2345
345

foreach ($old_array as $old) {

    // set a flag to indicate whether the value has already been added
    $matched = false;

    // loop over the new values by reference
    foreach ($new_array as &$new) {

        if ($new == $old) {  // this should prevent duplicate values from being added
            $matched = true; 

        } elseif ((strlen($old) > strlen($new)) && strpos($old, $new) === 0) {
            // if the old string starts with the new string, replace the value
            $new = $old;
            $matched = true;

        } elseif ((strlen($new) > strlen($old)) && strpos($new, $old) === 0) {
            // if the new string starts with the old string don't replace it and don't add it
            $matched = true;
        }
    }
    unset($new); // unset the reference

    if (!$matched) {
        // if no matching strings were found, add the string
        $new_array[] = $old;
    }
}

sort($old_array, SORT_STRING);
$old_array[] = null;  // append a null to force comparison of the last item
foreach ($old_array as $value) {
    if (isset($current) && !(strpos($value, $current) === 0)) {
        $new_array[] = $current;
    }
    $current = $value;
}