I am looking for a good introductory text to the theory of quantifier complexity. It can be a chapter in a larger text on logic. I have an understanding of logic already.
Everything I could find online was either advanced, or only a few slides in a powerpoint presentation.
This question is motivated by this answer to an earlier question. I am interested in simplifying formulae, specifically in getting rid of quantifiers over functions $A\to B$, but also more generally in understanding when formulae can and cannot be reduced to simpler ones (e.g. see this question). e.g. I am interested in the arithmetic hierarchy (though I'm not sure how well it captures the specific case where there are quantifiers like $f:A\to A$, which I think would require a multi-sorted theory?)