We show that the existence of a first-order formula separating two monadic second order formulas over countable ordinal words is decidable. This extends the work of Henckell and Almeida on finite words, and of Place and Zeitoun on $\omega$-words. For this, we develop the algebraic concept of monoid (resp. $\omega$-semigroup, resp. ordinal monoid) with aperiodic merge, an extension of monoids (resp. $\omega$-semigroup, resp. ordinal monoid) that explicitly includes a new operation capturing the loss of precision induced by first-order indistinguishability. We also show the computability of FO-pointlike sets, and the decidability of the covering problem for first-order logic on countable ordinal words.
翻译:我们证明存在将两个monadi 二级公式与可计数的单词区分开来的第一阶公式的存在是可以分辨的。 这延伸了Henckell和Almeida关于限定词的工作,以及Place和Zeitoun关于$\omega$-字词的工作。 为此,我们开发了单体( resp. $\omega$- semigroup, resp. ordinal oneid) 的代数概念, 其周期性合并, 单体( resp. $\omega$- semicroup, resp. ordinal monid) 的扩展, 明确包括一项新操作, 捕捉到因一阶不可调和性导致的精确性损失。 我们还展示了FO类集的可折合性, 以及覆盖问题在可计数或非词上的第一阶逻辑的衰减性。