Definition of Rabin acceptance condition for omega automatons I've been trying hard to understand something. According to wikipedia and this paper, the definition of the Rabin acceptance condition involves a set of pairs of states. I've been told that the left set in a pair must not be empty, but I cannot figure out why. Since the condition requires that at least one of the pairs will have none of the states on the left being visited infinitely often, and at least one of the state on the right be visited infinitely often, an empty left set seems to mean just that for every pair in the condition it is true that none of it's states is being visited infinitely often.
So my question is: why can't the left side of the pair be empty?
As Emil explained in a comment:
The sides are interchangable when defining the Rabin condition - my professor defined the left side as the one that is required to be visited infinitely often while Wikipedia and the paper I've mentioned define the right side as the one that is required to be visited infinitely often.
Whichever the side that has to be visited infinitely often is, it can be required to be non-empty, since if it is empty, the pair is meaningless - it cannot be visited infinitely often.
It is okay not to require it to be non-empty, because if it is empty, the pair simply accepts the empty language without causing any contradiction or changing any of the desired results of the definition.