153,866 research outputs found
An Algebraic Preservation Theorem for Aleph-Zero Categorical Quantified Constraint Satisfaction
We prove an algebraic preservation theorem for positive Horn definability in
aleph-zero categorical structures. In particular, we define and study a
construction which we call the periodic power of a structure, and define a
periomorphism of a structure to be a homomorphism from the periodic power of
the structure to the structure itself. Our 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. We give
applications of this theorem, including a new proof of the known complexity
classification of quantified constraint satisfaction on equality templates
A Fine-Grained Hierarchy of Hard Problems in the Separated Fragment
Recently, the separated fragment (SF) has been introduced and proved to be
decidable. Its defining principle is that universally and existentially
quantified variables may not occur together in atoms. The known upper bound on
the time required to decide SF's satisfiability problem is formulated in terms
of quantifier alternations: Given an SF sentence
in which is quantifier free, satisfiability can be decided in
nondeterministic -fold exponential time. In the present paper, we conduct a
more fine-grained analysis of the complexity of SF-satisfiability. We derive an
upper and a lower bound in terms of the degree of interaction of existential
variables (short: degree)}---a novel measure of how many separate existential
quantifier blocks in a sentence are connected via joint occurrences of
variables in atoms. Our main result is the -NEXPTIME-completeness of the
satisfiability problem for the set of all SF sentences that have
degree or smaller. Consequently, we show that SF-satisfiability is
non-elementary in general, since SF is defined without restrictions on the
degree. Beyond trivial lower bounds, nothing has been known about the hardness
of SF-satisfiability so far.Comment: Full version of the LICS 2017 extended abstract having the same
title, 38 page
Computabilities of Validity and Satisfiability in Probability Logics over Finite and Countable Models
The -logic (which is called E-logic in this paper) of
Kuyper and Terwijn is a variant of first order logic with the same syntax, in
which the models are equipped with probability measures and in which the
quantifier is interpreted as "there exists a set of measure
such that for each , ...." Previously, Kuyper and
Terwijn proved that the general satisfiability and validity problems for this
logic are, i) for rational , respectively
-complete and -hard, and ii) for ,
respectively decidable and -complete. The adjective "general" here
means "uniformly over all languages."
We extend these results in the scenario of finite models. In particular, we
show that the problems of satisfiability by and validity over finite models in
E-logic are, i) for rational , respectively
- and -complete, and ii) for , respectively
decidable and -complete. Although partial results toward the countable
case are also achieved, the computability of E-logic over countable
models still remains largely unsolved. In addition, most of the results, of
this paper and of Kuyper and Terwijn, do not apply to individual languages with
a finite number of unary predicates. Reducing this requirement continues to be
a major point of research.
On the positive side, we derive the decidability of the corresponding
problems for monadic relational languages --- equality- and function-free
languages with finitely many unary and zero other predicates. This result holds
for all three of the unrestricted, the countable, and the finite model cases.
Applications in computational learning theory, weighted graphs, and neural
networks are discussed in the context of these decidability and undecidability
results.Comment: 47 pages, 4 tables. Comments welcome. Fixed errors found by Rutger
Kuype
On the uniform one-dimensional fragment
The uniform one-dimensional fragment of first-order logic, U1, is a recently
introduced formalism that extends two-variable logic in a natural way to
contexts with relations of all arities. We survey properties of U1 and
investigate its relationship to description logics designed to accommodate
higher arity relations, with particular attention given to DLR_reg. We also
define a description logic version of a variant of U1 and prove a range of new
results concerning the expressivity of U1 and related logics
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
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