With respect to the number of variables the border of decidability lies between 2 and 3: the two-variable fragment of first-order logic, FO2, has an exponential model property and hence NExpTime-complete satisfiability problem, while for the three-variable fragment, FO3, satisfiability is undecidable. In this paper we propose a rich subfragment of FO3, containing full FO2 (without equality), and show that it retains the finite model property and NExpTime complexity. Our fragment is obtained as an extension of the uniform one-dimensional variant of FO3.

Order-invariant first-order logic is an extension of first-order logic (FO) where formulae can make use of a linear order on the structures, under the proviso that they are order-invariant, i.e. that their truth value is the same for all linear orders. We continue the study of the two-variable fragment of order-invariant first-order logic initiated by Zeume and Harwath, and study its complexity and expressive power. We first establish coNExpTime-completeness for the problem of deciding if a given two-variable formula is order-invariant, which tightens and significantly simplifies the coN2ExpTime proof by Zeume and Harwath. Second, we address the question of whether every property expressible in order-invariant two-variable logic is also expressible in first-order logic without the use of a linear order. While we were not able to provide a satisfactory answer to the question, we suspect that the answer is "no". To justify our claim, we present a class of finite tree-like structures (of unbounded degree) in which a relaxed variant of order-invariant two-variable FO expresses properties that are not definable in plain FO. On the other hand, we show that if one restricts their attention to classes of structures of bounded degree, then the expressive power of order-invariant two-variable FO is contained within FO.