Quantified constraint satisfaction and the polynomially generated powers property

The quantified constraint satisfaction probem (QCSP) is the problem of deciding, given a relational structure and a sentence consisting of a quantifier prefix followed by a conjunction of atomic formulas, whether or not the sentence is true in the structure. The general computational intractability of the QCSP has led to the study of restricted versions of this problem. In this article, we study restricted versions of the QCSP that arise from prespecifying the relations that may occur via a set of relations called a constraint language. A basic tool used is a correspondence that associates an algebra to each constraint language; this algebra can be used to derive information on the behavior of the constraint language. We identify a new combinatorial property on algebras, the polynomially generated powers (PGP) property, which we show is tightly connected to QCSP complexity. We also introduce another new property on algebras, switchability, which both implies the PGP property and implies positive complexity results on the QCSP. Our main result is a classification theorem on a class of three-element algebras: each algebra is either switchable and hence has the PGP, or provably lacks the PGP. The description of non-PGP algebras is remarkably simple and robust.