# Quantified Constraint Satisfaction and the Polynomially Generated Powers Property

```@inproceedings{Chen2008QuantifiedCS,
title={Quantified Constraint Satisfaction and the Polynomially Generated Powers Property},
author={Hubie Chen},
booktitle={ICALP},
year={2008}
}```
• Hubie Chen
• Published in ICALP 7 July 2008
• Mathematics, Computer Science
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 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…
The complexity of quantified constraints: collapsibility, switchability and the algebraic formulation
• Mathematics
ArXiv
• 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.
The Complexity of Quantified Constraints Using the Algebraic Formulation
• Mathematics
MFCS
• 2017
A full dichotomy for the QCSP is derived, justifying the moral correctness of the Chen Conjecture, and it is proved that for every finite subset Delta of Inv(A), Pol(Delta) is Collapsible.
Meditations on Quantified Constraint Satisfaction
• Hubie Chen
Logic and Program Semantics
• 2012
A viewpoint on the research program of understanding the complexity of the problems QCSP( B ) on finite structures is offered and a group of conjectures are proposed and discussed.
Asking the Metaquestions in Constraint Tractability
• Computer Science
TOCT
• 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).
Decomposing Quantified Conjunctive (or Disjunctive) Formulas
• Computer Science
2012 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.
The complexity of the Quantified CSP having the polynomially generated powers property
This work drastically simplified the reduction of the Quantified CSP to the CSP and generalized it for constraint languages without constants, completely classified the complexity of the QCSP for constraint language having the PGP property.
From Complexity to Algebra and Back: Digraph Classes, Collapsibility, and the PGP
• Mathematics
2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science
• 2015
It is proved that partially reflexive paths bequeath a set of idem potent polymorphisms whose associated clone algebra has: either the polynomially generated powers property (PGP), or the exponentially generated power property (EGP).
The Constraint Satisfaction Problem: Complexity and Approximability
• Computer Science
The 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
• Mathematics
SIAM J. Discret. Math.
• 2015
It is observed that a single counting quantifier strictly between \$\exists\$ and \$\forall\$ already affords the maximal possible complexity of QCSPs, namely, being Pspace-complete for a suitably chosen template.
An Algebraic Preservation Theorem for Aleph-Zero Categorical Quantified Constraint Satisfaction
• Mathematics
2012 27th Annual IEEE Symposium on Logic in Computer Science
• 2012
The preservation theorem states that, over an aleph-zero categorical structure, a relation is positive Horn definable if and only if it is preserved by all periomorphisms of the structure.

## References

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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 Computational Complexity of Quantified Constraint Satisfaction
This dissertation investigates the computational complexity of cases of the QCSP where the types of constraints that may appear are restricted and introduces a new concept for proving QCSP tractability results called collapsibility.
Quantified Constraints: Algorithms and Complexity
• Mathematics
CSL
• 2003
This paper considers a more general framework for constraint satisfaction problems which allows arbitrary quantifiers over constrained variables, rather than just existential quantifiers, and shows that the complexity of such extended problems is determined by the surjective polymorphisms of the constraint predicates.
Classifying the Complexity of Constraints Using Finite Algebras
• Mathematics
SIAM J. Comput.
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It is shown that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra is explored.
The Complexity of Quantified Constraint Satisfaction Problems under Structural Restrictions
• Mathematics
IJCAI
• 2005
We give a clear picture of the tractability/intractability frontier for quantified constraint satisfaction problems (QCSPs) under structural restrictions. On the negative side, we prove that checking
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An infinite class of satisfiability problems is considered which contains these two particular problems as special cases, and it is shown that every member of this class is either polynomial-time decidable or NP-complete.
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• Mathematics
21st Annual IEEE Symposium on Logic in Computer Science (LICS'06)
• 2006
It is proved that constraint languages consisting of relations that are invariant under a short sequence of Jonsson terms are tractable by showing that such languages have bounded width.
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Every subproblem of the CSP is either tractable or NP-complete, and the criterion separating them is that conjectured in Bulatov et al.
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• A. Bulatov
• Computer Science
18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings.
• 2003
This work completely characterize conservative constraint languages that give rise to CSP classes solvable in polynomial time, and obtains a complete description of those (directed) graphs H for which the List H-Coloring problem is poynomial time solvable.