Enumeration Complexity of Logical Query Problems with Second-order Variables

We consider query problems defined by first order formulas of the form F(x,T) with free first order and second order variables and study the data complexity of enumerating results of such queries. By considering the number of alternations in the quantifier prefixes of formulas, we show that such query problems either admit a constant delay or a polynomial delay enumeration algorithm or are hard to enumerate. We also exhibit syntactically defined fragments inside the hard cases that still admit good enumeration algorithms and discuss the case of some restricted classes of database structures as inputs.

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