"ed" problem

1. The message posted to Hacker News used a search syntax <--this can be an independent sentence.....
first, this sentence can be simplified to
The message used a search syntax .
for easier understanding.
it is clear that you wont use "using", because
-the message itself cannot use sth , its not a person

2. the complete sentence should be
THe message posted to Hacker News using a search syntax expose a system used by Facebook that lets users quickly log back in to their account.

or the original sentence you post.

The message posted to Hacker News" which" is used a search syntax is wrong.
it should be
The message posted to Hacker News which used a search syntax exposed......
coz used is already a verb in past tense, an extra "is" (verb-to-be) is not needed