Decidability conditions for problems about automata reading infinite words

We consider two algorithmic problems related to the automata on infinite words: does an automaton reach an accepting state while reading an infinite word and does this event occurs infinitely often. The former problem is related to generalized nondeterminism models and the latter, to decidability of monadic second order theories. The main contribution is a new decidability condition for these problems. We also show that decidability of a regular realizability problem (satisfiability of a regular property on a set of words) is equivalent to decidability of the first problem. Bibliogr. 11.