Nominal automata models and transition systems serve as formalisms for languages and processes carrying data, and as such relate closely to classical register-based models. The paradigm of name allocation in nominal systems helps alleviate the pervasive computational hardness of register-based 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. Unlike the full register automaton model, RNNAs allow for inclusion checking in elementary complexity (parametrized PSPACE); similarly, trace inclusion in the underlying nominal transition systems is in parametrized PSPACE. In the present work, we develop a unified algebraic treatment of spectra of behavioural equivalences on nominal systems in the framework of graded monads, working in the setting of universal coalgebra. In particular, we extend the associated notion of graded algebraic theory to the nominal setting, and use this to give an algebraic axiomatization of the (linear-time) global and local freshness semantics of nominal systems with name allocation. As an illustration of the benefits of graded monads, we develop a nominal version of the generic game characterization of graded semantics, which we instantiate to obtain trace equivalence games for nominal transition systems under global and local freshness semantics.

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