Graded Monads in the Semantics of Nominal Automata

Nominal automata models serve as a formalism for data languages, and in fact often relate closely to classical register models. The paradigm of name allocation in nominal automata helps alleviate the pervasive computational hardness of register models in a tradeoff between expressiveness and computational tractability. For instance, regular nondeterministic nominal automata (RNNAs) correspond, under their local freshness semantics, to a form of lossy register automata, and unlike the full register automaton model allow for inclusion checking in elementary complexity. The semantic framework of graded monads provides a unified algebraic treatment of spectra of behavioural equivalences in the setting of universal coalgebra. In the present work, we extend the associated notion of graded algebraic theory to the nominal setting. In the arising framework of graded nominal algebra, we give an algebraic theory capturing the local freshness semantics of RNNAs.