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The complexity of quantified constraints using the algebraic formulation
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Existentially Restricted Quantified Constraint Satisfaction
The quantified constraint satisfaction problem (QCSP) is a powerful framework
for modelling computational problems. The general intractability of the QCSP
has motivated the pursuit of restricted cases that avoid its maximal
complexity. In this paper, we introduce and study a new model for investigating
QCSP complexity in which the types of constraints given by the existentially
quantified variables, is restricted. Our primary technical contribution is the
development and application of a general technology for proving positive
results on parameterizations of the model, of inclusion in the complexity class
coNP
Generalizing Consistency and other Constraint Properties to Quantified Constraints
Quantified constraints and Quantified Boolean Formulae are typically much
more difficult to reason with than classical constraints, because quantifier
alternation makes the usual notion of solution inappropriate. As a consequence,
basic properties of Constraint Satisfaction Problems (CSP), such as consistency
or substitutability, are not completely understood in the quantified case.
These properties are important because they are the basis of most of the
reasoning methods used to solve classical (existentially quantified)
constraints, and one would like to benefit from similar reasoning methods in
the resolution of quantified constraints. In this paper, we show that most of
the properties that are used by solvers for CSP can be generalized to
quantified CSP. This requires a re-thinking of a number of basic concepts; in
particular, we propose a notion of outcome that generalizes the classical
notion of solution and on which all definitions are based. We propose a
systematic study of the relations which hold between these properties, as well
as complexity results regarding the decision of these properties. Finally, and
since these problems are typically intractable, we generalize the approach used
in CSP and propose weaker, easier to check notions based on locality, which
allow to detect these properties incompletely but in polynomial time
The Complexity of Quantified Constraint Satisfaction: Collapsibility, Sink Algebras, and the Three-Element Case
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 quantification. The general intractability of these
problems has motivated research studying the complexity of these problems under
a restricted constraint language, which is a set of relations that can be used
to express constraints.
This paper introduces collapsibility, a technique for deriving positive
complexity results on the QCSP. In particular, this technique allows one to
show that, for a particular constraint language, the QCSP reduces to the CSP.
We show that collapsibility applies to three known tractable cases of the QCSP
that were originally studied using disparate proof techniques in different
decades: Quantified 2-SAT (Aspvall, Plass, and Tarjan 1979), Quantified
Horn-SAT (Karpinski, Kleine B\"{u}ning, and Schmitt 1987), and Quantified
Affine-SAT (Creignou, Khanna, and Sudan 2001). This reconciles and reveals
common structure among these cases, which are describable by constraint
languages over a two-element domain. In addition to unifying these known
tractable cases, we study constraint languages over domains of larger size
Existentially restricted quantified constraint satisfaction
AbstractThe quantified constraint satisfaction problem (QCSP) is a framework for modelling PSPACE computational problems. The general intractability of the QCSP has motivated the pursuit of restricted cases that avoid its maximal complexity. In this paper, we introduce and study a new model for investigating QCSP complexity in which the types of constraints given by the existentially quantified variables, is restricted. Our primary technical contribution is the development and application of a general technology for proving positive results on parameterizations of the model, of inclusion in the complexity class coNP
Generalized Satisfiability Problems via Operator Assignments
Schaefer introduced a framework for generalized satisfiability problems on
the Boolean domain and characterized the computational complexity of such
problems. We investigate an algebraization of Schaefer's framework in which the
Fourier transform is used to represent constraints by multilinear polynomials
in a unique way. The polynomial representation of constraints gives rise to a
relaxation of the notion of satisfiability in which the values to variables are
linear operators on some Hilbert space. For the case of constraints given by a
system of linear equations over the two-element field, this relaxation has
received considerable attention in the foundations of quantum mechanics, where
such constructions as the Mermin-Peres magic square show that there are systems
that have no solutions in the Boolean domain, but have solutions via operator
assignments on some finite-dimensional Hilbert space. We obtain a complete
characterization of the classes of Boolean relations for which there is a gap
between satisfiability in the Boolean domain and the relaxation of
satisfiability via operator assignments. To establish our main result, we adapt
the notion of primitive-positive definability (pp-definability) to our setting,
a notion that has been used extensively in the study of constraint satisfaction
problems. Here, we show that pp-definability gives rise to gadget reductions
that preserve satisfiability gaps. We also present several additional
applications of this method. In particular and perhaps surprisingly, we show
that the relaxed notion of pp-definability in which the quantified variables
are allowed to range over operator assignments gives no additional expressive
power in defining Boolean relations
State Merging with Quantifiers in Symbolic Execution
We address the problem of constraint encoding explosion which hinders the
applicability of state merging in symbolic execution. Specifically, our goal is
to reduce the number of disjunctions and if-then-else expressions introduced
during state merging. The main idea is to dynamically partition the symbolic
states into merging groups according to a similar uniform structure detected in
their path constraints, which allows to efficiently encode the merged path
constraint and memory using quantifiers. To address the added complexity of
solving quantified constraints, we propose a specialized solving procedure that
reduces the solving time in many cases. Our evaluation shows that our approach
can lead to significant performance gains
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