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I was wondering whether something like elimination of second-order quantifiers exist, and indeed it seems it does. I've found there's a workshop on this topic, and the webpage describes exactly what I need:

Second-order quantifier elimination (SOQE) is the problem of equivalently reducing a formula with quantifiers upon second-order objects such as predicates to a formula in which these quantified second-order objects no longer occur.

Unexpectedly (at least for me) the topic seems to be related to a lot of other things that I need for what I'm doing: Craig interpolants, uniform interpolants, Beth definability.

So is there a reference (maybe a survey?) about second-order quantifier elimination and its connections with the topics above?

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Since there are no answer I think it may be useful to post what I've found:

Dov M. Gabbay, Renate A. Schmidt, Andrzej Szalas
Second-Order Quantifier Elimination - Foundations, Computational Aspects and Applications.
Studies in logic : Mathematical logic and foundations 12, College Publications 2008,
ISBN 978-1-904987-56-7, pp. I-VIII, 1-308

It seems quite complete and well-written. That's what I was looking for.

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