Decidability, complexity, and expressiveness of first-order logic over the subword ordering

We consider first-order logic over the subword ordering on finite words where each word is available as a constant. Our first result is that the Σ₁ theory is undecidable (already over two letters). We investigate the decidability border by considering fragments where all but a certain number of variables are alternation bounded, meaning that the variable must always be quantified over languages with a bounded number of letter alternations. We prove that when at most two variables are not alternation bounded, the Σ₁ fragment is decidable, and that it becomes undecidable when three variables are not alternation bounded. Regarding higher quantifier alternation depths, we prove that the Σ₂ fragment is undecidable already for one variable without alternation bound and that when all variables are alternation bounded, the entire first-order theory is decidable.