Regular Monoidal Languages
Matthew Earnshaw, Pawe{\l} Soboci\'nski
TL;DR
This paper extends the concept of regular languages to morphisms in free monoidal categories, introducing new grammars and automata that generalize classical word and tree languages to string diagrams.
Contribution
It develops a framework for regular monoidal languages using monoidal and cartesian restriction categories, and establishes conditions for their recognizability by deterministic monoidal automata.
Findings
Regular monoidal languages generalize classical regular languages.
Sufficient conditions for recognizability by deterministic monoidal automata are provided.
The framework unifies languages of words, trees, and string diagrams.
Abstract
We introduce regular languages of morphisms in free monoidal categories, with their associated grammars and automata. These subsume the classical theory of regular languages of words and trees, but also open up a much wider class of languages over string diagrams. We use the algebra of monoidal and cartesian restriction categories to investigate the properties of regular monoidal languages, and provide sufficient conditions for their recognizability by deterministic monoidal automata.
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