It seems that your form submits a json object to the server so use json_decode with second parameter set to true to convert it to an array.

$php_array=  json_decode($_POST["questionandanswers"],true)

Is this what you are trying to achieve?

$incoming is the jsonString coming from the javascript code

$incoming = '{
    "name":"A Name",
    "_typography": {
      "color": {
        "hex":"#2196f3"
      },
      "text-transform":"uppercase",
      "font-weight":"600"
      }
    }';

$a = json_decode($incoming, true);
echo var_export($a);

RESULT

array (
  'name' => 'A Name',
  '_typography' => 
  array (
    'color' => 
    array (
      'hex' => '#2196f3',
    ),
    'text-transform' => 'uppercase',
    'font-weight' => '600',
  ),
)

I had to reformat your JSON which was not valid (checked on https://jsonlint.com/).

I voluntarily used multiple str_replace() so you better understand the process, however you can optimize the code below by making multiple replacements at the same time within the same str_replace().

This works:

<?php

$to_decode = "var BASE_DATA = {
userInfo: {
  id: 0,
  userName: 'no-needed',
  avatarUrl: 'no-needed',
  isPgc: false,
  isOwner: false
},
headerInfo: {
  id: 0,
  isPgc: false,
  userName: 'no-needed',
  avatarUrl: 'no-needed',
  isHomePage: false,
  crumbTag: 'no-needed',
  hasBar: true
},
articleInfo: 
{
  title: 'needed',
  content: 'needed',
  groupId: 'needed',
  itemId: 'needed',
  type: 1,
  subInfo: {
    isOriginal: false,
    source: 'needed',
    time: 'needed'
  },
  tagInfo: {
    tags: [{\"name\":\"no-needed 1\"},{\"name\":\"no-needed 2\"},{\"name\":\"no-needed 3\"}],
    groupId: 'no-needed',
    itemId: 'no-needed',
    repin: 0,
  },
  has_extern_link: 0,
  coverImg: 'no-needed'
},
commentInfo:
{
  groupId: 'no-needed',
  itemId: 'no-needed',
  comments_count: 151,
  ban_comment: 0
},};";

/* Clean JSON and encapsulate in brackets */
$to_decode = str_replace('var BASE_DATA = {', '', $to_decode);
$to_decode = '{'.substr($to_decode, 0, -3).'}';

/* Remove spaces, tabs, new lines, etc. */
$to_decode = str_replace(' ', '', $to_decode);
$to_decode = str_replace("\n", '', $to_decode);
$to_decode = str_replace("\t", '', $to_decode);
$to_decode = str_replace("\r", '', $to_decode);

/* Encapsulate keys with quotes */
$to_decode = preg_replace('/([a-z_]+)\:/ui', '"{$1}":', $to_decode);
$to_decode = str_replace('"{', '"', $to_decode);
$to_decode = str_replace('}"', '"', $to_decode);
$to_decode = str_replace('\'', '"', $to_decode);

/* Remove unecessary trailing commas */
$to_decode = str_replace(',}', '}', $to_decode);

echo '<pre>';
var_dump(json_decode($to_decode));

Result using print_r :

(I added true/false for clarity, these will only show using var_dump() otherwise)

stdClass Object
(
    [userInfo] => stdClass Object
        (
            [id] => 0
            [userName] => no-needed
            [avatarUrl] => no-needed
            [isPgc] => false
            [isOwner] => false
        )

    [headerInfo] => stdClass Object
        (
            [id] => 0
            [isPgc] => false
            [userName] => no-needed
            [avatarUrl] => no-needed
            [isHomePage] => false
            [crumbTag] => no-needed
            [hasBar] => true
        )

    [articleInfo] => stdClass Object
        (
            [title] => needed
            [content] => needed
            [groupId] => needed
            [itemId] => needed
            [type] => 1
            [subInfo] => stdClass Object
                (
                    [isOriginal] => false
                    [source] => needed
                    [time] => needed
                )

            [tagInfo] => stdClass Object
                (
                    [tags] => Array
                        (
                            [0] => stdClass Object
                                (
                                    [name] => no-needed1
                                )

                            [1] => stdClass Object
                                (
                                    [name] => no-needed2
                                )

                            [2] => stdClass Object
                                (
                                    [name] => no-needed3
                                )

                        )

                    [groupId] => no-needed
                    [itemId] => no-needed
                    [repin] => 0
                )

            [has_extern_link] => 0
            [coverImg] => no-needed
        )

    [commentInfo] => stdClass Object
        (
            [groupId] => no-needed
            [itemId] => no-needed
            [comments_count] => 151
            [ban_comment] => 0
        )

)

Post your javascript array as a JSON string via ajax and process it server-side.

Javascript

var str_array = ["aa", "bb"];
var request = $.ajax({
  url: "test.php",
  method: "POST",
  data: { myData : JSON.stringify(str_array) },
  dataType: "html"
});
request.done(function( msg ) {
  // ajax response
});
request.fail(function( jqXHR, textStatus ) {
  alert( "Request failed: " + textStatus );
});

PHP (test.php)

$json = $_POST['myData'];
$myDataArray = json_decode($json,true);
print_r($myDataArray);

I'm going to assume that the two arrays you've given for PHP and JS are not related, and they're just examples of how arrays look in the two languages. Clearly you're not going to be able to convert those sequences of letters and numbers into those city names.

PHP provides a function to convert PHP arrays into Javascript code: json_encode(). (technically, it's JSON format; JSON stands for JavaScript Object Notation)

Use it like this:

<script type='text/javascript'>
<?php
$php_array = array('abc','def','ghi');
$js_array = json_encode($php_array);
echo "var javascript_array = ". $js_array . ";\n";
?>
</script>

See also the manual page I linked above for more information.

Note that json_encode() is only available in PHP 5.2 and up, so if you're using an older version, you'll need to use an existing one -- the PHP manual page also includes comments with functions written by people who needed it. (but that said, if you're using anything older than PHP 5.2 you should upgrade ASAP)