121 research outputs found
On the Complexity of Existential Positive Queries
We systematically investigate the complexity of model checking the
existential positive fragment of first-order logic. In particular, for a set of
existential positive sentences, we consider model checking where the sentence
is restricted to fall into the set; a natural question is then to classify
which sentence sets are tractable and which are intractable. With respect to
fixed-parameter tractability, we give a general theorem that reduces this
classification question to the corresponding question for primitive positive
logic, for a variety of representations of structures. This general theorem
allows us to deduce that an existential positive sentence set having bounded
arity is fixed-parameter tractable if and only if each sentence is equivalent
to one in bounded-variable logic. We then use the lens of classical complexity
to study these fixed-parameter tractable sentence sets. We show that such a set
can be NP-complete, and consider the length needed by a translation from
sentences in such a set to bounded-variable logic; we prove superpolynomial
lower bounds on this length using the theory of compilability, obtaining an
interesting type of formula size lower bound. Overall, the tools, concepts, and
results of this article set the stage for the future consideration of the
complexity of model checking on more expressive logics
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
Parameter Compilation
In resolving instances of a computational problem, if multiple instances of
interest share a feature in common, it may be fruitful to compile this feature
into a format that allows for more efficient resolution, even if the
compilation is relatively expensive. In this article, we introduce a formal
framework for classifying problems according to their compilability. The basic
object in our framework is that of a parameterized problem, which here is a
language along with a parameterization---a map which provides, for each
instance, a so-called parameter on which compilation may be performed. Our
framework is positioned within the paradigm of parameterized complexity, and
our notions are relatable to established concepts in the theory of
parameterized complexity. Indeed, we view our framework as playing a unifying
role, integrating together parameterized complexity and compilability theory
Beyond Q-Resolution and Prenex Form: A Proof System for Quantified Constraint Satisfaction
We consider the quantified constraint satisfaction problem (QCSP) which is to
decide, given a structure and a first-order sentence (not assumed here to be in
prenex form) built from conjunction and quantification, whether or not the
sentence is true on the structure. We present a proof system for certifying the
falsity of QCSP instances and develop its basic theory; for instance, we
provide an algorithmic interpretation of its behavior. Our proof system places
the established Q-resolution proof system in a broader context, and also allows
us to derive QCSP tractability results
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
Beyond Hypertree Width: Decomposition Methods Without Decompositions
The general intractability of the constraint satisfaction problem has
motivated the study of restrictions on this problem that permit polynomial-time
solvability. One major line of work has focused on structural restrictions,
which arise from restricting the interaction among constraint scopes. In this
paper, we engage in a mathematical investigation of generalized hypertree
width, a structural measure that has up to recently eluded study. We obtain a
number of computational results, including a simple proof of the tractability
of CSP instances having bounded generalized hypertree width
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