This might be a way to go to start going about it:

Conway's Game of Life is [Turing complete](http://en.wikipedia.org/wiki/Turing_complete): it is possible to simulate a universal turing machine within the Game of Life.

  Deciding whether a Turing machine will halt or continue infinitely for an input is the ["Halting Problem"](http://en.wikipedia.org/wiki/Halting_problem).  It is not possible to have a general algorithm that decides the Halting Problem for all possible inputs to a Turing machine simulated on the Game of Life.  

Thus the Halting Problem is also undecidable for arbitrary inputs on particular subsets of initial patterns on the Game of Life: specifically those which implement a Turing machine simulation.