In this paper, we define k-counting automata as recognizers for
ω-languages, i.e. languages of infinite words. We
prove that the class of ω-languages they recognize is a proper extension
of the ω-regular languages. In addition we prove that languages
recognized by k-counting automata are closed under Boolean operations. It
remains an open problem whether or not emptiness is decidable for
k-counting automata. However, we conjecture strongly that it is decidable
and give formal reasons why we believe so.