# The Complexity of Quantified Constraint Satisfaction: Collapsibility, Sink Algebras, and the Three-Element Case

@article{Chen2008TheCO, title={The Complexity of Quantified Constraint Satisfaction: Collapsibility, Sink Algebras, and the Three-Element Case}, author={Hubie Chen}, journal={SIAM J. Comput.}, year={2008}, volume={37}, pages={1674-1701} }

The constraint satisfaction probem (CSP) is a well-acknowledged framework in which many combinatorial search problems can be naturally formulated. The CSP may be viewed as the problem of deciding the truth of a logical sentence consisting of a conjunction of constraints, in front of which all variables are existentially quantified. The quantified constraint satisfaction problem (QCSP) is the generalization of the CSP where universal quantification is permitted in addition to existential…

## 56 Citations

Quantified Constraint Satisfaction and the Polynomially Generated Powers Property

- Mathematics, Computer ScienceICALP
- 2008

This article studies restricted versions of the quantified constraint satisfaction probem that arise from prespecifying the relations that may occur via a set of relations called a constraint language, and identifies a new combinatorial property on algebras, the polynomially generated powers(PGP) property, which it is shown is tightly connected to QCSP complexity.

Quantified constraint satisfaction and the polynomially generated powers property

- Mathematics, Computer Science
- 2011

This article identifies a new combinatorial property on algebras, the polynomially generated powers (PGP) property, which it is shown is tightly connected to QCSP complexity, and introduces another new property, switchability, which both implies the PGP property and implies positive complexity results on the QCSP.

Quantified Constraints and Containment Problems

- Mathematics, Philosophy2008 23rd Annual IEEE Symposium on Logic in Computer Science
- 2008

This work gives an algorithm that determines whether "phi implies psi" is true in all structures (models) and shows that the entailment problem for positive first-order logic is undecidable.

Quantified Constraints and Containment Problems

- Mathematics, PhilosophyLICS
- 2008

This work gives an algorithm that determines whether "phi implies psi" is true in all structures (models) and shows that the entailment problem for positive first-order logic is undecidable.

The complexity of quantified constraints: collapsibility, switchability and the algebraic formulation

- MathematicsArXiv
- 2021

It is proved a complexity-theoretic converse in the case of infinite constraint languages encoded in propositional logic, that if Inv(A) satisfies the exponentially generated powers property (EGP), then QCSP(Inv(A)) is co-NP-hard.

On the complexity of the model checking problem

- MathematicsSIAM J. Comput.
- 2018

The model checking problem for various fragments of first-order logic has attracted much attention over the last two decades: in particular, for the primitive positive and the positive Horn…

The Constraint Satisfaction Problem: Complexity and Approximability

- Computer ScienceThe Constraint Satisfaction Problem
- 2017

This report documents the material presented during the course of the Dagstuhl Seminar 18231 “The Constraint Satisfaction Problem: Complexity and Approximability”, aimed at bringing together researchers using all the different techniques in the study of the CSP to share their insights obtained.

Constraint Satisfaction with Counting Quantifiers

- MathematicsCSR
- 2012

This work completes the classification of clique templates by proving a full trichotomy for all possible combinations of counting quantifiers and clique sizes, and obtains a full dichotomy for ∃ and ∃ ≥ 2 quantifiers on finite graphs.

Relatively quantified constraint satisfaction

- BusinessConstraints
- 2008

This paper gives a complete complexity classification of the cases of the RQCSP where the types of constraints that may appear are specified by a constraint language.

A Proof of the CSP Dichotomy Conjecture

- Mathematics, Computer ScienceJ. ACM
- 2020

This article presents an algorithm that solves Constraint Satisfaction Problem in polynomial time for constraint languages having a weak near unanimity polymorphism, which proves the remaining part of the conjecture.

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