Graph logics with rational relations: the role of word combinatorics

Graph databases make use of logics that combine traditional first-order features with navigation on paths, in the same way logics for model checking do. However, modern applications of graph databases impose a new requirement on the expressiveness of the logics: they need comparing labels of paths based on word relations (such as prefix, subword, or subsequence). This has led to the study of logics that extend basic graph languages with features for comparing labels of paths based on regular relations, or the strictly more powerful rational relations. The evaluation problem for the former logic is decidable (and even tractable in data complexity), but already extending this logic with such a common rational relation as subword or suffix turns evaluation undecidable. In practice, however, it is rare to have the need for such powerful logics. Therefore, it is more realistic to study the complexity of less expressive logics that still allow comparing paths based on practically motivated rational relations. Here we concentrate on the most basic such languages, which extend graph pattern logics with path comparisons based only on suffix, subword or subsequence. We pinpoint the complexity of evaluation for each one of these logics, which shows that all of them are decidable in elementary time (PSpace or NExpTime). Furthermore, the extension with suffix is even tractable in data complexity (but the other two are not). In order to obtain our results we establish a link between the evaluation problem for graph logics and two important problems in word combinatorics: word equations with regular constraints and square shuffling.