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

TL;DR

It is shown that for any constraint language on a finite domain the Quantified Constraint Satisfaction Problem is either in $\Pi_{2}^{P}$, or PSpace-complete.

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 $Π_{2}^{P}$, 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 $Π_{2}^{P}$-complete.

Figures (5)

• Figure 1: Complexity classes expressible as $\mathop{\mathrm{QCSP}}\nolimits(\Gamma)$ for some $\Gamma$.
• Figure 2: A graph for $\exists u_1\exists u_2 \exists u_3 R_{1}(y_1,u_1,x_1)\wedge R_{0}(u_1,u_2,x_2)\wedge R_{1}(u_2,u_3,x_3)\wedge (u_3=+)$
• Figure 3: A graph expressing $\neg((x_1\vee \overline x_2\vee x_3)\wedge (\overline x_1\vee x_2\vee \overline x_3) \wedge (x_1\vee \overline x_2\vee \overline x_3)).$
• Figure 4: A graph expressing $\forall x_1\exists x_2\forall x_3\;\;\neg((x_1\vee \overline x_2\vee x_3)\wedge (\overline x_1\vee x_2\vee \overline x_3) \wedge (x_1\vee \overline x_2\vee \overline x_3))$
• Figure 5: Reduction from $\Pi_{2}$-$\mathop{\mathrm{QCSP}}\nolimits(\mathrm{1IN3})$ to $\mathop{\mathrm{QCSP}}\nolimits(\Gamma)$.

Theorems & Definitions (128)

• Theorem 1: Schaefer
• Theorem 2: QCSPMonstersSTOCQCSP_Monsters_JACM
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Lemma 8
• proof : Sketch of the proof:
• Lemma 9