339 research outputs found
Tractability of Theory Patching
In this paper we consider the problem of `theory patching', in which we are
given a domain theory, some of whose components are indicated to be possibly
flawed, and a set of labeled training examples for the domain concept. The
theory patching problem is to revise only the indicated components of the
theory, such that the resulting theory correctly classifies all the training
examples. Theory patching is thus a type of theory revision in which revisions
are made to individual components of the theory. Our concern in this paper is
to determine for which classes of logical domain theories the theory patching
problem is tractable. We consider both propositional and first-order domain
theories, and show that the theory patching problem is equivalent to that of
determining what information contained in a theory is `stable' regardless of
what revisions might be performed to the theory. We show that determining
stability is tractable if the input theory satisfies two conditions: that
revisions to each theory component have monotonic effects on the classification
of examples, and that theory components act independently in the classification
of examples in the theory. We also show how the concepts introduced can be used
to determine the soundness and completeness of particular theory patching
algorithms.Comment: See http://www.jair.org/ for any accompanying file
The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme
The Traveling Salesman Problem (TSP) is among the most famous NP-hard
optimization problems. We design for this problem a randomized polynomial-time
algorithm that computes a (1+eps)-approximation to the optimal tour, for any
fixed eps>0, in TSP instances that form an arbitrary metric space with bounded
intrinsic dimension.
The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the
above result holds in the special case of TSP in a fixed-dimensional Euclidean
space. Thus, our algorithm demonstrates that the algorithmic tractability of
metric TSP depends on the dimensionality of the space and not on its specific
geometry. This result resolves a problem that has been open since the
quasi-polynomial time algorithm of Talwar (T-04)
A Goodwillie-type Theorem for Milnor K-Theory
Goodwillie's rational isomorphism between relative algebraic K-theory and
relative cyclic homology, together with the lambda decomposition of cyclic
homology, illustrates the close relationships among algebraic K-theory, cyclic
homology, and differential forms. In this paper, I prove a Goodwillie-type
theorem for relative Milnor -theory, working over a very general class of
commutative rings, defined via the stability criterion of Van der Kallen. Early
results of Van der Kallen and Bloch are special cases. The result likely
generalizes in terms of de Rahm-Witt complexes, by weakening some invertibility
assumptions, but the class of rings considered is already more than
sufficiently general for the intended applications. The main motivation for
this paper arises from applications to the infinitesimal theory of Chow groups,
first pointed out by Bloch in the 1970's, and prominent in recent work of Green
and Griffiths. Related results and geometric applications are discussed in the
final section.Comment: 34 page
Practical learning method for multi-scale entangled states
We describe a method for reconstructing multi-scale entangled states from a
small number of efficiently-implementable measurements and fast
post-processing. The method only requires single particle measurements and the
total number of measurements is polynomial in the number of particles. Data
post-processing for state reconstruction uses standard tools, namely matrix
diagonalisation and conjugate gradient method, and scales polynomially with the
number of particles. Our method prevents the build-up of errors from both
numerical and experimental imperfections
Easy plane baby skyrmions
The baby Skyrme model is studied with a novel choice of potential, . This "easy plane" potential vanishes at the equator of the target
two-sphere. Hence, in contrast to previously studied cases, the boundary value
of the field breaks the residual SO(2) internal symmetry of the model.
Consequently, even the unit charge skyrmion has only discrete symmetry and
consists of a bound state of two half lumps. A model of long-range
inter-skyrmion forces is developed wherein a unit skyrmion is pictured as a
single scalar dipole inducing a massless scalar field tangential to the vacuum
manifold. This model has the interesting feature that the two-skyrmion
interaction energy depends only on the average orientation of the dipoles
relative to the line joining them. Its qualitative predictions are confirmed by
numerical simulations. Global energy minimizers of charges B=1,...,14,18,32 are
found numerically. Up to charge B=6, the minimizers have 2B half lumps
positioned at the vertices of a regular 2B-gon. For charges B >= 7, rectangular
or distorted rectangular arrays of 2B half lumps are preferred, as close to
square as possible.Comment: v3: replaced with journal version, one new reference, one deleted
reference; 8 pages, 5 figures v2: fixed some typos and clarified the
relationship with condensed matter systems 8 pages, 5 figure
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