# 19.8 Logic and automata for multiply nested-words

Nested-word automata (NWA) are essentially visibly pushdown automata with multiple stacks. They accept languages of (multiply) nested-words, which are words equipped with a fixed number of binary nesting relations, connecting matching push and pop events. In the case of a single stack (that is, a single nesting relation), NWA are equivalent to monadic second-order logic (MSO). The binary predicates used in the logic are the direct successor relation, and the nesting relations. However, in the general case, and even with only two stacks, NWA are not closed under complementation, and strictly less expressive than MSO. In fact, for two stacks, NWA are equivalent to the existential fragment of MSO (EMSO). The equivalence with MSO can be recovered when restricting the set of possible behaviors, one classical example being phase-bounded nested words.

In the general case (more than two stack, arbitrary nested words), the exact expressive power of NWA is not known. In particular, the following question is open: can every first-order formula be translated into an equivalent NWA ?