# Meditations on Quantified Constraint Satisfaction

@inproceedings{Chen2012MeditationsOQ, title={Meditations on Quantified Constraint Satisfaction}, author={Hubie Chen}, booktitle={Logic and Program Semantics}, year={2012} }

The quantified constraint satisfaction problem (QCSP) is the problem of deciding, given a structure and a first-order prenex sentence whose quantifier-free part is the conjunction of atoms, whether or not the sentence holds on the structure. One obtains a family of problems by defining, for each structure B , the problem QCSP( B ) to be the QCSP where the structure is fixed to be B . In this article, we offer a viewpoint on the research program of understanding the complexity of the problems…

## 39 Citations

Asking the Metaquestions in Constraint Tractability

- Computer ScienceTOCT
- 2017

This article systematically studies—for various classes of polymorphisms—the computational complexity of deciding whether or not a given structure ℍ admits a polymorphism from the class, and proves the NP-completeness of deciding a condition conjectured to characterize the tractable problems CSP(ℍ).

Containment, Equivalence and Coreness from CSP to QCSP and beyond

- Computer ScienceCP
- 2012

Previous results on containment, equivalence and "coreness" for QCSP are recalled and extended before initiating a preliminary study of cores for QC SP which turns out to be more elusive.

Tractability Frontier for Dually-Closed Temporal Quantified Constraint Satisfaction Problems

- Computer ScienceArXiv
- 2021

It is proved that QCSP for such a language is either solvable in polynomial time or it is hard for NP or coNP, a dichotomy for the restricted class of duallyclosed temporal languages.

Decomposing Quantified Conjunctive (or Disjunctive) Formulas

- Computer Science2012 27th Annual IEEE Symposium on Logic in Computer Science
- 2012

A comprehensive classification of the sets of prefixed graphs on which model checking is tractable is given, based on a novel generalization of treewidth, that generalizes and places into a unified framework a number of existing results.

Fixed-Template Promise Model Checking Problems

- Mathematics, Computer ScienceArXiv
- 2022

A class of problems that generalizes the CSP simultaneously in two directions is studied: a set L of quantiﬁers and Boolean connectives, and two versions of each constraint are specified, one strong and one weak.

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.

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.

A Proof of CSP Dichotomy Conjecture

- Mathematics, Computer Science2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
- 2017

An algorithm is presented 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.

QCSP monsters and the demise of the chen conjecture

- Mathematics, Computer ScienceSTOC
- 2020

A surprising classification for the computational complexity of the Quantified Constraint Satisfaction Problem over a constraint language Γ, QCSP(Γ), where Γ is a finite language over 3 elements which contains all constants and the Chen Conjecture is refuted.

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.

## References

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- Mathematics, Computer Science
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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.

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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.

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A complete classification of 2-semilattice polymorphisms is proved, demonstrating that each gives rise to a case of the QCSP that is either tractable in polynomial time, or coNP-hard.

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The quantified constraint satisfaction problem (QCSP), a more general framework in which variables can be quantified both universally and existentially, is concerned with and a complete complexity classification of maximal constraint languages is given.

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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…

The complexity of temporal constraint satisfaction problems

- Computer ScienceJACM
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This work presents a complete complexity classification of the constraint satisfaction problem (CSP) for temporal constraint languages: if the constraint language is contained in one out of nine temporal constraint language, then the CSP can be solved in polynomial time; otherwise, the C SP is NP-complete.

Universal Algebra and Hardness Results for Constraint Satisfaction Problems

- Mathematics, Computer ScienceICALP
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Algebraic conditions on constraint languages Γ are presented that ensure the hardness of the constraint satisfaction problem CSP(Γ) for complexity classes L, NL, P, NP and ModpL and it is shown that if C SP( Γ) is not first-order definable then it is L-hard.

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