Formal languages over infinite alphabets serve as abstractions of structures and processes carrying data. Automata models over infinite alphabets, such as classical register automata or, equivalently, nominal orbit-finite automata, tend to have computationally hard or even undecidable reasoning problems unless stringent restrictions are imposed on either the power of control or the number of registers. This has been shown to be ameliorated in automata models with name allocation such as regular nondeterministic nominal automata, which allow for deciding language inclusion in elementary complexity even with unboundedly many registers while retaining a reasonable level of expressiveness. In the present work, we demonstrate that elementary complexity survives under extending the power of control to alternation: We introduce regular alternating nominal automata (RANAs), and show that their non-emptiness and inclusion problems have elementary complexity even when the number of registers is unbounded. Moreover, we show that RANAs allow for nearly complete de-alternation, specifically de-alternation up to a single deadlocked universal state. As a corollary to our results, we improve the complexity of model checking for a flavour of Bar-µTL, a fixed-point logic with name allocation over finite data words, by one exponential level.

Positive data languages are languages over an infinite alphabet closed under possibly non-injective renamings of data values. Informally, they model properties of data words expressible by assertions about equality, but not inequality, of data values occurring in the word. We investigate the class of positive data languages recognizable by nondeterministic orbit-finite nominal automata, an abstract form of register automata introduced by Bojańczyk, Klin, and Lasota. As our main contribution we provide a number of equivalent characterizations of that class in terms of positive register automata, monadic second-order logic with positive equality tests, and finitely presentable nondeterministic automata in the categories of nominal renaming sets and of presheaves over finite sets.