${\hat{I}}_2^P$ vs PSpace Dichotomy for the Quantified Constraint Satisfaction Problem

Abstract

The Quantified Constraint Satisfaction Problem is the problem of evaluating a sentence with both quantifiers, over relations from some constraint language, with conjunction as the only connective. We show that for any constraint language on a finite domain the Quantified Constraint Satisfaction Problem is either in ${\hat{I}}_2^P$, or PSpace-complete. Additionally, we build a constraint language on a 6-element domain such that the Quantified Constraint Satisfaction Problem over this language is ${\hat{I}}_2^P$-complete.

Some misprints were fixed, acknowledgements were added