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We consider fragments of first-order logic over finite words. In particular, we deal with first-order logic with a restricted number of variables and with the lower levels of the alternation hierarchy. We use the algebraic approach to show decidability of expressibility within these fragments. As a byproduct, we survey several characterizations of the respective fragments. We give complete proofs for all characterizations and we provide all necessary background. Some of the proofs seem to be new and simpler than those which can be found elsewhere. We also give a proof of Simon's theorem on factorization forests restricted to aperiodic monoids because this is simpler and sufficient for our purpose. PreambleThere are many brilliant surveys on formal language theory [36,41,48,85,86]. Quite many surveys cover first-order and monadic second-order definability. But there are also nuggets below. There are deep theorems on proper fragments of firstorder definability. The most prominent fragment is FO 2 ; it is the class of languages which are defined by first-order sentences which do not use more than two names for variables. Although various characterizations are known for this class, there seems to be little knowledge in a broad community. A reason for this is that the proofs are spread over the literature and even in the survey [76] many proofs are referred to the original literature which in turn is sometimes quite difficult to read. This is our starting point. We restrict our attention to fragments strictly below first-order definability. We concentrate on algebraic and formal language theoretic characterizations for those fragments where decidability results are known, although we do not discuss complexity issues here. We give a clear preference to full proofs rather than to state all results. In our proofs we tried to be minimalistic. All technical concepts which are introduced are also used in the proofs for the main results.

We give topological and algebraic characterizations as well as language theoretic descriptions of the following subclasses of first-order logic FO [<] for ω-languages: Σ2, FO 2 , FO 2 ∩ Σ2, and ∆2 (and by duality Π2 and FO 2 ∩ Π2). 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 [31] and Bojańczyk [2] and is therefore not our main focus. The paper is about the interplay of algebraic, topological, and language theoretic properties.

We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Jeż, and a new way to integrate solutions of linear Diophantine equations into the process.As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to NSPACE(n log n) here.

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