"The greeter application appears to be crashing"

Abstract: Resolved "The greeter application appears to be crashing" problem after updating Linux system.

I don't know whether this could be called a solution... actually just a tip if you happened to come across a same problem. I updated my Arch Linux, and after reboot this dialog box jumps up and gdm could not start normally. I checked the system and found the problem was caused by a full filesystem. I cleaned up some old files and then everything goes right.

So I guess I should give more disk space to / in the future.