2,707 research outputs found
Inductive Logic Programming in Databases: from Datalog to DL+log
In this paper we address an issue that has been brought to the attention of
the database community with the advent of the Semantic Web, i.e. the issue of
how ontologies (and semantics conveyed by them) can help solving typical
database problems, through a better understanding of KR aspects related to
databases. In particular, we investigate this issue from the ILP perspective by
considering two database problems, (i) the definition of views and (ii) the
definition of constraints, for a database whose schema is represented also by
means of an ontology. Both can be reformulated as ILP problems and can benefit
from the expressive and deductive power of the KR framework DL+log. We
illustrate the application scenarios by means of examples. Keywords: Inductive
Logic Programming, Relational Databases, Ontologies, Description Logics, Hybrid
Knowledge Representation and Reasoning Systems. Note: To appear in Theory and
Practice of Logic Programming (TPLP).Comment: 30 pages, 3 figures, 2 tables
Inseparable local uniformization
It is known since the works of Zariski in early 40ies that desingularization
of varieties along valuations (called local uniformization of valuations) can
be considered as the local part of the desingularization problem. It is still
an open problem if local uniformization exists in positive characteristic and
dimension larger than three. In this paper, we prove that Zariski local
uniformization of algebraic varieties is always possible after a purely
inseparable extension of the field of rational functions, i.e. any valuation
can be uniformized by a purely inseparable alteration.Comment: 66 pages, final version, the paper was seriously revise
Building Rules on Top of Ontologies for the Semantic Web with Inductive Logic Programming
Building rules on top of ontologies is the ultimate goal of the logical layer
of the Semantic Web. To this aim an ad-hoc mark-up language for this layer is
currently under discussion. It is intended to follow the tradition of hybrid
knowledge representation and reasoning systems such as -log that
integrates the description logic and the function-free Horn
clausal language \textsc{Datalog}. In this paper we consider the problem of
automating the acquisition of these rules for the Semantic Web. We propose a
general framework for rule induction that adopts the methodological apparatus
of Inductive Logic Programming and relies on the expressive and deductive power
of -log. The framework is valid whatever the scope of induction
(description vs. prediction) is. Yet, for illustrative purposes, we also
discuss an instantiation of the framework which aims at description and turns
out to be useful in Ontology Refinement.
Keywords: Inductive Logic Programming, Hybrid Knowledge Representation and
Reasoning Systems, Ontologies, Semantic Web.
Note: To appear in Theory and Practice of Logic Programming (TPLP)Comment: 30 pages, 6 figure
3-manifold groups are virtually residually p
Given a prime , a group is called residually if the intersection of
its -power index normal subgroups is trivial. A group is called virtually
residually if it has a finite index subgroup which is residually . It is
well-known that finitely generated linear groups over fields of characteristic
zero are virtually residually for all but finitely many . In particular,
fundamental groups of hyperbolic 3-manifolds are virtually residually . It
is also well-known that fundamental groups of 3-manifolds are residually
finite. In this paper we prove a common generalization of these results: every
3-manifold group is virtually residually for all but finitely many .
This gives evidence for the conjecture (Thurston) that fundamental groups of
3-manifolds are linear groups
A magnetic model with a possible Chern-Simons phase
An elementary family of local Hamiltonians , is described for a dimensional quantum mechanical system of spin
particles. On the torus, the ground state space is
extensively degenerate but should collapse under \lperturbation" to
an anyonic system with a complete mathematical description: the quantum double
of the Chern-Simons modular functor at which
we call . The Hamiltonian defines a
\underline{quantum} \underline{loop}\underline{gas}. We argue that for and 2, is unstable and the collapse to can occur truly by perturbation. For ,
is stable and in this case finding must require either , help from finite
system size, surface roughening (see section 3), or some other trick, hence the
initial use of quotes {\l}\quad". A hypothetical phase diagram is included in
the introduction.Comment: Appendix by F. Goodman and H. Wenz
Recursive Solution of Initial Value Problems with Temporal Discretization
We construct a continuous domain for temporal discretization of differential
equations. By using this domain, and the domain of Lipschitz maps, we formulate
a generalization of the Euler operator, which exhibits second-order
convergence. We prove computability of the operator within the framework of
effectively given domains. The operator only requires the vector field of the
differential equation to be Lipschitz continuous, in contrast to the related
operators in the literature which require the vector field to be at least
continuously differentiable. Within the same framework, we also analyze
temporal discretization and computability of another variant of the Euler
operator formulated according to Runge-Kutta theory. We prove that, compared
with this variant, the second-order operator that we formulate directly, not
only imposes weaker assumptions on the vector field, but also exhibits superior
convergence rate. We implement the first-order, second-order, and Runge-Kutta
Euler operators using arbitrary-precision interval arithmetic, and report on
some experiments. The experiments confirm our theoretical results. In
particular, we observe the superior convergence rate of our second-order
operator compared with the Runge-Kutta Euler and the common (first-order) Euler
operators.Comment: 50 pages, 6 figure
Estimation under group actions: recovering orbits from invariants
Motivated by geometric problems in signal processing, computer vision, and
structural biology, we study a class of orbit recovery problems where we
observe very noisy copies of an unknown signal, each acted upon by a random
element of some group (such as Z/p or SO(3)). The goal is to recover the orbit
of the signal under the group action in the high-noise regime. This generalizes
problems of interest such as multi-reference alignment (MRA) and the
reconstruction problem in cryo-electron microscopy (cryo-EM). We obtain
matching lower and upper bounds on the sample complexity of these problems in
high generality, showing that the statistical difficulty is intricately
determined by the invariant theory of the underlying symmetry group.
In particular, we determine that for cryo-EM with noise variance
and uniform viewing directions, the number of samples required scales as
. We match this bound with a novel algorithm for ab initio
reconstruction in cryo-EM, based on invariant features of degree at most 3. We
further discuss how to recover multiple molecular structures from heterogeneous
cryo-EM samples.Comment: 54 pages. This version contains a number of new result
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