We present first-order (FO) and monadic second-order (MSO) logics with predicates ‘between’ and ‘neighbour’ that characterise the class of regular languages that are closed under the reverse operation and its subclasses. The ternary between predicate bet(x, y, z) is true if the position y is strictly between the positions x and z. The binary neighbour predicate N(x, y) is true when the the positions x and y are adjacent. It is shown that the class of reversible regular languages is precisely the class definable in the logics MSO(bet) and MSO(N). Moreover the class is definable by their existential fragments EMSO(bet) and EMSO(N), yielding a normal form for MSO formulas. In the first-order case, the logic FO(bet) corresponds precisely to the class of reversible languages definable in FO(<). Every formula in FO(bet) is equivalent to one that uses at most 3 variables. However the logic FO(N) defines only a strict subset of reversible languages definable in FO(+1). A language-theoretic characterisation of the class of languages definable in FO(N), called locally-reversible threshold-testable (LRTT), is given. In the second part of the paper we show that the standard connections that exist between MSO and FO logics with order and successor predicates and varieties of finite semigroups extend to the new setting with the semigroups extended with an involution operation on its elements. The case is different for FO(N) where we show that one needs an additional equation that uses the involution operator to characterise the class. While the general problem of characterising FO(N) is open, an equational characterisation is shown for the case of neutral letter languages.

中文翻译：

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可逆正则语言：逻辑和代数特征

我们展示了带有谓词“介于”和“邻居”的一阶 (FO) 和一元二阶 (MSO) 逻辑，这些谓词表征了在反向操作及其子类下关闭的常规语言类。如果位置 y 严格位于位置 x 和 z 之间，则谓词 bet(x, y, z) 之间的三元组为真。当位置 x 和 y 相邻时，二元邻居谓词 N(x, y) 为真。结果表明，可逆正则语言的类正是在逻辑 MSO(bet) 和 MSO(N) 中可定义的类。此外，该类可以通过它们的存在片段 EMSO(bet) 和 EMSO(N) 来定义，从而产生 MSO 公式的范式。在一阶情况下，逻辑 FO(bet) 恰好对应于可在 FO(<) 中定义的可逆语言类。FO(bet) 中的每个公式都相当于一个最多使用 3 个变量的公式。然而，逻辑 FO(N) 仅定义了可在 FO(+1) 中定义的可逆语言的严格子集。给出了可在 FO(N) 中定义的语言类的语言理论特征，称为局部可逆阈值可测试 (LRTT)。在论文的第二部分，我们展示了 MSO 和 FO 逻辑之间存在的标准连接，带有顺序和后继谓词以及有限半群的变体，扩展到新设置，其中半群通过对其元素的对合运算扩展。FO(N) 的情况不同，我们表明需要一个额外的方程，使用对合算子来表征类。虽然表征 FO(N) 的一般问题是开放的，