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arXiv · 2408.08236

Derivatives on Graphs for the Positive Calculus of Relations with Transitive Closure

Yoshiki Nakamura
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Abstract

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.

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Yoshiki Nakamura. 2024-08-15. Derivatives on Graphs for the Positive Calculus of Relations with Transitive Closure. https://doi.org/10.46298/lmcs-21(4%3A27)2025

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