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：我们研究有限字母表 $A$ 上的有限词结构（包含（非连续）子词序）的一阶逻辑。在量词交替片段可判定性方面，该逻辑已得到充分理解：若每个词均可作为常数使用，则即使 $\Sigma_1$（即存在性）片段也是不可判定的，且对二元字母表 $A$ 亦然。然而，迄今为止，关于量词交替片段的表达能力知之甚少：例如，存在性片段的不可判定性证明依赖于丢番图方程，仅表明单元素字母表上的递归可枚举语言（及某些辅助谓词）是可定义的。我们证明：若 $|A|\ge 3$，则一个关系在带常数的 $A$ 上存在性片段中可定义当且仅当该关系是递归可枚举的。这蕴含了所有 $\Sigma_i$ 片段的特征刻画：若 $|A|\ge 3$，则一个关系在 $\Sigma_i$ 中可定义当且仅当该关系属于算术层级中的第 $i$ 层。此外，我们的结果还给出了纯逻辑（即 $A^*$ 中的词不作为常数可用）中 $i\ge 2$ 的 $\Sigma_i$ 片段的完整类似描述。