$\Pi_{2}^{P}$ vs PSpace Dichotomy for the Quantified Constraint Satisfaction Problem

TL;DR

This paper establishes a clear dichotomy for the complexity of the Quantified Constraint Satisfaction Problem (QCSP) over finite domains, showing it is either in ^P or PSpace-complete, and constructs a specific language where QCSP is ^P-complete.

Contribution

The paper proves a dichotomy theorem for QCSP complexity over finite domains and provides an explicit example of a language with ^P-complete QCSP.

Findings

QCSP over finite domains is either in ^P or PSpace-complete

Constructed a 6-element domain language with ^P-complete QCSP

Established a complexity classification for QCSP based on the constraint language

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 $\Pi_{2}^{P}$, 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 $\Pi_{2}^{P}$-complete.