  Recently data trees and data words have received considerable amount of
attention in connection with XML reasoning and system verification. These are
trees or words that, in addition to labels from a finite alphabet, carry data
values from an infinite alphabet (data). In general it is rather hard to obtain
logics for data words and trees that are sufficiently expressive, but still
have reasonable complexity for the satisfiability problem. In this paper we
extend and study the notion of B\"uchi automata for omega-words with data. We
prove that the emptiness problem for such extension is decidable in elementary
complexity. We then apply our result to show the decidability of two kinds of
logics for omega-words with data: the two-variable fragment of first-order
logic and some extensions of classical linear temporal logic for omega-words
with data.
