@Article{	  dk11tocs,
  author	= "Diekert, Volker and Kufleitner, Manfred",
  affiliation	= "FMI, Universit{\"a}t Stuttgart, Universit{\"a}tsstra{\ss}e
		  38, 70569 Stuttgart, Germany",
  title		= "Fragments of First-Order Logic over Infinite Words",
  journal	= "Theory of Computing Systems",
  publisher	= "Springer New York",
  issn		= "1432-4350",
  keywords	= "Computer Science",
  pages		= "486--516",
  volume	= "48",
  number	= "3",
  year		= "2011",
  url		= "http://dx.doi.org/10.1007/s00224-010-9266-7",
  abstract	= "We give topological and algebraic characterizations as
		  well as language theoretic descriptions of the following
		  subclasses of first-order logic $\mathrm{FO}[<]$ for
		  $\omega$-languages: $\Sigma_{2}$, $\mathrm{FO}_{2}$,
		  $\mathrm{FO}_{2} \cap \Sigma_{2}$, and $\Delta_{2}$ (and by
		  duality $\Pi_{2}$ and $\mathrm{FO}_{2} \cap \Pi_{2}$).
		  These descriptions extend the respective results for finite
		  words. In particular, we relate the above fragments to
		  language classes of certain (unambiguous) polynomials. An
		  immediate consequence is the decidability of the membership
		  problem of these classes, but this was shown before by
		  Wilke (Classifying Discrete Temporal Properties.
		  Habilitationsschrift, Universit{\"a}t Kiel, April 1998) and
		  Boja{\'n}czyk (Lecture Notes in Computer Science, vol.
		  4962, pp. 172 -- 185, 2008) and is therefore not our main
		  focus. The paper is about the interplay of algebraic,
		  topological, and language theoretic properties.",
  doi		= "10.1007/s00224-010-9266-7"
}