470,140 research outputs found
Counting Answers to Existential Positive Queries: A Complexity Classification
Existential positive formulas form a fragment of first-order logic that
includes and is semantically equivalent to unions of conjunctive queries, one
of the most important and well-studied classes of queries in database theory.
We consider the complexity of counting the number of answers to existential
positive formulas on finite structures and give a trichotomy theorem on query
classes, in the setting of bounded arity. This theorem generalizes and unifies
several known results on the complexity of conjunctive queries and unions of
conjunctive queries.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0719
FO-definable transformations of infinite strings
The theory of regular and aperiodic transformations of finite strings has
recently received a lot of interest. These classes can be equivalently defined
using logic (Monadic second-order logic and first-order logic), two-way
machines (regular two-way and aperiodic two-way transducers), and one-way
register machines (regular streaming string and aperiodic streaming string
transducers). These classes are known to be closed under operations such as
sequential composition and regular (star-free) choice; and problems such as
functional equivalence and type checking, are decidable for these classes. On
the other hand, for infinite strings these results are only known for
-regular transformations: Alur, Filiot, and Trivedi studied
transformations of infinite strings and introduced an extension of streaming
string transducers over -strings and showed that they capture monadic
second-order definable transformations for infinite strings. In this paper we
extend their work to recover connection for infinite strings among first-order
logic definable transformations, aperiodic two-way transducers, and aperiodic
streaming string transducers
Finite model theory for pseudovarieties and universal algebra: preservation, definability and complexity
We explore new interactions between finite model theory and a number of
classical streams of universal algebra and semigroup theory. A key result is an
example of a finite algebra whose variety is not finitely axiomatisable in
first order logic, but which has first order definable finite membership
problem. This algebra witnesses the simultaneous failure of the {\L}os-Tarski
Theorem, the SP-preservation theorem and Birkhoff's HSP-preservation theorem at
the finite level as well as providing a negative solution to a first order
formulation of the long-standing Eilenberg Sch\"utzenberger problem. The
example also shows that a pseudovariety without any finite pseudo-identity
basis may be finitely axiomatisable in first order logic. Other results include
the undecidability of deciding first order definability of the pseudovariety of
a finite algebra and a mapping from any fixed template constraint satisfaction
problem to a first order equivalent variety membership problem, thereby
providing examples of variety membership problems complete in each of the
classes , , ,
, and infinitely many others (depending on complexity-theoretic
assumptions)
Asymptotic Theories of Classes Defined by Forbidden Homomorphisms
We study the first-order almost-sure theories for classes of finite
structures that are specified by homomorphically forbidding a set
of finite structures. If consists of undirected graphs, a full
description of these theories can be derived from the
Kolaitis-Pr\"omel-Rothschild theorem, which treats the special case where
. The corresponding question for finite of
finite directed graphs is wide open. We present a full description of the
almost-sure theories of classes described by homomorphically forbidding finite
sets of oriented trees; all of them are -categorical. In
our proof, we establish a result of independent interest, namely that every
constraint satisfaction problem for a finite digraph has first-order
convergence, and that the corresponding asymptotic theory can be described as a
finite linear combination of -categorical theories
Inquisitive bisimulation
Inquisitive modal logic InqML is a generalisation of standard Kripke-style
modal logic. In its epistemic incarnation, it extends standard epistemic logic
to capture not just the information that agents have, but also the questions
that they are interested in. Technically, InqML fits within the family of
logics based on team semantics. From a model-theoretic perspective, it takes us
a step in the direction of monadic second-order logic, as inquisitive modal
operators involve quantification over sets of worlds. We introduce and
investigate the natural notion of bisimulation equivalence in the setting of
InqML. We compare the expressiveness of InqML and first-order logic in the
context of relational structures with two sorts, one for worlds and one for
information states. We characterise inquisitive modal logic, as well as its
multi-agent epistemic S5-like variant, as the bisimulation invariant fragment
of first-order logic over various natural classes of two-sorted structures.
These results crucially require non-classical methods in studying bisimulation
and first-order expressiveness over non-elementary classes of structures,
irrespective of whether we aim for characterisations in the sense of classical
or of finite model theory
Methods of class field theory to separate logics over finite residue classes and circuit complexity
This is a pre-copyedited, author-produced version of an article accepted for publication in Journal of logic and computation following peer review.Separations among the first-order logic Res(0,+,×) of finite residue classes, its extensions with generalized quantifiers, and in the presence of a built-in order are shown in this article, using algebraic methods from class field theory. These methods include classification of spectra of sentences over finite residue classes as systems of congruences, and the study of their h-densities over the set of all prime numbers, for various functions h on the natural numbers. Over ordered structures, the logic of finite residue classes and extensions are known to capture DLOGTIME-uniform circuit complexity classes ranging from AC to TC. Separating these circuit complexity classes is directly related to classifying the h-density of spectra of sentences in the corresponding logics of finite residue classes. General conditions are further shown in this work for a logic over the finite residue classes to have a sentence whose spectrum has no h-density. A corollary of this characterization of spectra of sentences is that in Res(0,+,×,<)+M, the logic of finite residue classes with built-in order and extended with the majority quantifier M, there are sentences whose spectrum have no exponential density.Peer ReviewedPostprint (author's final draft
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