<p>If you say something like “It is their dog”, their is correct.
If you use it as a pronoun, as in “It is theirs”, then theirs is the proper form.</p>
<p>So “their” is an adjective and “theirs” is a pronoun.</p>
<p>“their” as being the same pronoun category as “my”
ie. “It is my dog” and “It is their dog” </p>
<p>“theirs” as being the same pronoun category as “mine”
ie. “This dog is mine” and “This dog is theirs”</p>
<p>For the sentence you provided, you can invert the second clause to read:
“The oldest known writing system is (their)” which is obviously incorrect. It should read “The oldest known writing system is (theirs)”</p>