We prove that the equational theory of the positive calculus of relations with transitive closure (PCoR*) is EXPSPACE-complete. Here, PCoR* terms consist of the following standard operators on binary relations: identity, empty, universality, union, intersection, composition, converse, and reflexive transitive closure (so, PCoR* terms subsume Kleene algebra and allegory terms as fragments). Additionally, we show that the equational theory of PCoR* extended with tests and nominals (in hybrid logic) is still EXPSPACE-complete; moreover, it is PSPACE-complete for its intersection-free fragment. To this end, we design derivatives on graphs by extending derivatives on words for regular expressions. The derivatives give a finite automata construction on path decompositions, like those on words. Because the equational theory has a linearly bounded pathwidth model property, we can decide the equational theory of PCoR* using these automata.

@article{paperbot3343,
  title = {Derivatives on Graphs for the Positive Calculus of Relations with Transitive Closure},
  author = {Yoshiki Nakamura},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 21, Issue 4},
  year = {2025},
  doi = {10.46298/lmcs-21(4:27)2025}
}