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.

翻译：序不变一阶逻辑是一阶逻辑（FO）的扩展，其中公式可以利用结构上的线性序，但需满足序不变性条件，即其真值对所有线性序均相同。我们延续Zeume与Harwath发起的关于序不变一阶逻辑双变量片段的研究，探讨其复杂性和表达能力。首先，我们证明判定给定双变量公式是否为序不变的问题属于coNExpTime完全问题，这一结果强化并显著简化了Zeume与Harwath的coN2ExpTime证明。其次，我们探讨序不变双变量逻辑中可表达的所有性质是否也能在不使用线性序的一阶逻辑中表达。尽管未能给出该问题的满意答案，我们推测答案是否定的。为佐证这一论断，我们构造了一类有限树状结构（无界度数），其中松弛变体的序不变双变量FO可表达普通FO无法定义的性质。另一方面，我们证明若将注意力限制在有界度数的结构类上，则序不变双变量FO的表达能力包含于FO之中。