Word equations are equations \(\alpha \doteq \beta \) where \(\alpha \) and \(\beta \) are words consisting of letters from some alphabet \(\Sigma \) and variables from a set X. Recently, there has been substantial interest in the context of string solving in logics combining word equations with other kinds of constraints on words such as (regular) language membership (regular constraints) and arithmetic over string lengths (length constraints). We consider the expressive power of such logics by looking at the set of all values a single variable might take as part of a satisfying assignment for a given formula. Hence, each formula-variable pair defines a formal language, and each logic defines a class of formal languages. We consider logics arising from combining word equations with either length constraints, regular constraints, or both. We also consider word equations with visibly pushdown language membership constraints as a generalisation of the combination of regular and length constraints. We show that word equations with visibly pushdown membership constraints are sufficient to express all recursively enumerable languages and hence satisfiability is undecidable in this case. We then establish a strict hierarchy involving the other combinations. We also provide a complete characterisation of when a thin regular language is expressible by word equations (alone) and some further partial results for regular languages in the general case.

字方程是方程\(\alpha \doteq \beta \)，其中\(\alpha \)和\(\beta \)是由某个字母表\(\Sigma \)中的字母和集合X中的变量组成的单词。最近，人们对将词方程与其他类型的词约束相结合的逻辑中的字符串求解背景产生了浓厚的兴趣，例如（常规）语言成员资格（常规约束）和字符串长度算术（长度约束）。我们通过查看单个变量可能采用的所有值的集合来考虑这种逻辑的表达能力，作为给定公式的令人满意的赋值的一部分。因此，每个公式-变量对定义一种形式语言，每个逻辑定义一类形式语言。我们考虑将词方程与长度约束、常规约束或两者结合而产生的逻辑。我们还将具有明显下推语言成员资格约束的词方程视为常规约束和长度约束组合的概括。我们证明，具有明显下推隶属度约束的词方程足以表达所有递归可枚举语言，因此在这种情况下可满足性是不可判定的。然后我们建立一个涉及其他组合的严格层次结构。我们还提供了何时可以通过单词方程（单独）表达薄正则语言的完整特征，以及一般情况下正则语言的一些进一步的部分结果。