230,301 research outputs found
Flow Logic
Flow networks have attracted a lot of research in computer science. Indeed,
many questions in numerous application areas can be reduced to questions about
flow networks. Many of these applications would benefit from a framework in
which one can formally reason about properties of flow networks that go beyond
their maximal flow. We introduce Flow Logics: modal logics that treat flow
functions as explicit first-order objects and enable the specification of rich
properties of flow networks. The syntax of our logic BFL* (Branching Flow
Logic) is similar to the syntax of the temporal logic CTL*, except that atomic
assertions may be flow propositions, like or , for
, which refer to the value of the flow in a vertex, and
that first-order quantification can be applied both to paths and to flow
functions. We present an exhaustive study of the theoretical and practical
aspects of BFL*, as well as extensions and fragments of it. Our extensions
include flow quantifications that range over non-integral flow functions or
over maximal flow functions, path quantification that ranges over paths along
which non-zero flow travels, past operators, and first-order quantification of
flow values. We focus on the model-checking problem and show that it is
PSPACE-complete, as it is for CTL*. Handling of flow quantifiers, however,
increases the complexity in terms of the network to , even
for the LFL and BFL fragments, which are the flow-counterparts of LTL and CTL.
We are still able to point to a useful fragment of BFL* for which the
model-checking problem can be solved in polynomial time. Finally, we introduce
and study the query-checking problem for BFL*, where under-specified BFL*
formulas are used for network exploration
Low dimensional manifolds for exact representation of open quantum systems
Weakly nonlinear degrees of freedom in dissipative quantum systems tend to
localize near manifolds of quasi-classical states. We present a family of
analytical and computational methods for deriving optimal unitary model
transformations based on representations of finite dimensional Lie groups. The
transformations are optimal in that they minimize the quantum relative entropy
distance between a given state and the quasi-classical manifold. This naturally
splits the description of quantum states into quasi-classical coordinates that
specify the nearest quasi-classical state and a transformed quantum state that
can be represented in fewer basis levels. We derive coupled equations of motion
for the coordinates and the transformed state and demonstrate how this can be
exploited for efficient numerical simulation. Our optimization objective
naturally quantifies the non-classicality of states occurring in some given
open system dynamics. This allows us to compare the intrinsic complexity of
different open quantum systems.Comment: Added section on semi-classical SR-latch, added summary of method,
revised structure of manuscrip
Tensor-Sparsity of Solutions to High-Dimensional Elliptic Partial Differential Equations
A recurring theme in attempts to break the curse of dimensionality in the
numerical approximations of solutions to high-dimensional partial differential
equations (PDEs) is to employ some form of sparse tensor approximation.
Unfortunately, there are only a few results that quantify the possible
advantages of such an approach. This paper introduces a class of
functions, which can be written as a sum of rank-one tensors using a total of
at most parameters and then uses this notion of sparsity to prove a
regularity theorem for certain high-dimensional elliptic PDEs. It is shown,
among other results, that whenever the right-hand side of the elliptic PDE
can be approximated with a certain rate in the norm of
by elements of , then the solution can be
approximated in from to accuracy
for any . Since these results require
knowledge of the eigenbasis of the elliptic operator considered, we propose a
second "basis-free" model of tensor sparsity and prove a regularity theorem for
this second sparsity model as well. We then proceed to address the important
question of the extent such regularity theorems translate into results on
computational complexity. It is shown how this second model can be used to
derive computational algorithms with performance that breaks the curse of
dimensionality on certain model high-dimensional elliptic PDEs with
tensor-sparse data.Comment: 41 pages, 1 figur
Structure and Complexity in Planning with Unary Operators
Unary operator domains -- i.e., domains in which operators have a single
effect -- arise naturally in many control problems. In its most general form,
the problem of STRIPS planning in unary operator domains is known to be as hard
as the general STRIPS planning problem -- both are PSPACE-complete. However,
unary operator domains induce a natural structure, called the domain's causal
graph. This graph relates between the preconditions and effect of each domain
operator. Causal graphs were exploited by Williams and Nayak in order to
analyze plan generation for one of the controllers in NASA's Deep-Space One
spacecraft. There, they utilized the fact that when this graph is acyclic, a
serialization ordering over any subgoal can be obtained quickly. In this paper
we conduct a comprehensive study of the relationship between the structure of a
domain's causal graph and the complexity of planning in this domain. On the
positive side, we show that a non-trivial polynomial time plan generation
algorithm exists for domains whose causal graph induces a polytree with a
constant bound on its node indegree. On the negative side, we show that even
plan existence is hard when the graph is a directed-path singly connected DAG.
More generally, we show that the number of paths in the causal graph is closely
related to the complexity of planning in the associated domain. Finally we
relate our results to the question of complexity of planning with serializable
subgoals
Large violation of Bell inequalities with low entanglement
In this paper we obtain violations of general bipartite Bell inequalities of
order with inputs, outputs and
-dimensional Hilbert spaces. Moreover, we construct explicitly, up to a
random choice of signs, all the elements involved in such violations: the
coefficients of the Bell inequalities, POVMs measurements and quantum states.
Analyzing this construction we find that, even though entanglement is necessary
to obtain violation of Bell inequalities, the Entropy of entanglement of the
underlying state is essentially irrelevant in obtaining large violation. We
also indicate why the maximally entangled state is a rather poor candidate in
producing large violations with arbitrary coefficients. However, we also show
that for Bell inequalities with positive coefficients (in particular, games)
the maximally entangled state achieves the largest violation up to a
logarithmic factor.Comment: Reference [16] added. Some typos correcte
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