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，其公式可在结构上利用线性序，但需满足序不变条件，即其真值对所有线性序均相同。我们继续研究由Zeume与Harwath开创的序不变一阶逻辑的双变量片段，并探讨其复杂度与表达能力。首先，我们确定判定给定双变量公式是否具有序不变性的问题为coNExpTime完全问题，这强化并显著简化了Zeume与Harwath提出的coN2ExpTime证明。其次，我们探讨序不变双变量逻辑可表达的所有性质是否均能通过不依赖线性序的一阶逻辑表达。尽管未能给出满意解答，但我们推测答案为“否”。为佐证这一论断，我们构造了一类有限树状结构（无界度数），其中序不变双变量FO的松弛变体能表达普通FO不可定义的性质。另一方面，我们证明若将关注范围限于有界度数结构类，则序不变双变量FO的表达能力蕴含于FO之内。