Visibly pushdown automata are input-driven pushdown automata that recognize some non-regular context-free languages while preserving the nice closure and decidability properties of finite automata.Visibly pushdown automata with multiple stacks have been considered recently by La Torre, 3-Wheel Rollators Madhusudan, and Parlato, who exploit the concept of visibility further to obtain a rich automata class that can even express properties beyond the class of context-free languages.At the same time, their automata are closed under boolean operations, have a decidable emptiness and inclusion problem, and enjoy a logical characterization in terms of a monadic second-order logic over words with an additional nesting structure.These results require a restricted version of visibly pushdown automata with multiple stacks whose behavior can be split up into a fixed number of phases.
In this paper, we consider 2-stack visibly pushdown automata (i.e., visibly pushdown automata with two stacks) in their unrestricted form.We show that they are expressively equivalent to the existential fragment of monadic second-order logic.
Furthermore, it turns out that monadic second-order quantifier alternation forms an infinite hierarchy wrt words with multiple nestings.Combining these results, we conclude that 2-stack visibly pushdown automata are not closed under complementation.Finally, we discuss the expressive power of B"{u}chi 2-stack visibly pushdown automata AEG BP831660KM Electric Single Multifunction Oven running on infinite (nested) words.Extending the logic by an infinity quantifier, we can likewise establish equivalence to existential monadic second-order logic.