We study first-order logic (FO) over the structure consisting of finite words over some alphabet $A$, together with the (non-contiguous) subword ordering. In terms of decidability of quantifier alternation fragments, this logic is well-understood: If every word is available as a constant, then even the $\Sigma_1$ (i.e., existential) fragment is undecidable, already for binary alphabets $A$. However, up to now, little is known about the expressiveness of the quantifier alternation fragments: For example, the undecidability proof for the existential fragment relies on Diophantine equations and only shows that recursively enumerable languages over a singleton alphabet (and some auxiliary predicates) are definable. We show that if $|A|\ge 3$, then a relation is definable in the existential fragment over $A$ with constants if and only if it is recursively enumerable. This implies characterizations for all fragments $\Sigma_i$: If $|A|\ge 3$, then a relation is definable in $\Sigma_i$ if and only if it belongs to the $i$-th level of the arithmetical hierarchy. In addition, our result yields an analogous complete description of the $\Sigma_i$-fragments for $i\ge 2$ of the pure logic, where the words of $A^*$ are not available as constants.

翻译：我们研究了第一阶逻辑（FO）在某一字母表 $A$ 上的有限词结构（包括非连续的子词序）中的性质。在量词交替片段的可判定性方面，这一逻辑已被充分理解：如果每个词都可以作为常数使用，那么即使 $\Sigma_1$（即存在性）片段对于二元字母表 $A$ 也是不可判定的。然而，迄今为止，关于量词交替片段的表达力所知甚少：例如，存在性片段的不可判定性证明依赖于丢番图方程，仅表明单字母字母表（及某些辅助谓词）上的递归可枚举语言是可定义的。我们证明，如果 $|A|\ge 3$，则一个关系在包含常数的字母表 $A$ 上的存在性片段中可定义当且仅当它是递归可枚举的。这推导出对所有片段 $\Sigma_i$ 的描述：如果 $|A|\ge 3$，则一个关系在 $\Sigma_i$ 中可定义当且仅当它属于算术层次的第 $i$ 级。此外，我们的结果对纯逻辑（其中 $A^*$ 的词不作为常数可用）中 $i\ge 2$ 的 $\Sigma_i$ 片段给出了完整的类比描述。