There are a few open source chess engines out there, like Gnu Chess. If you can figure out how to read the board position in physical space, and transfer that to text format, then you can feed that text into GNU Chess as moves. (like A2-B2 or whatever). It will respond with text representing the computer move (C5-D4), which can be translated into the robot arm motion. However this would require an attached "real computer" as stated above, GNU Chess is probably at least a megabyte of RAM.
However there is a chess engine called 'Nano Chess', which takes a very small amount of memory in C (there is even a wiki page). You might be able to fit this onto an arduino with a lot of work, or by using multiple arduinos (one for chess, one for motion control, one for reading position)
