3,428 research outputs found
Dual Logic Concepts based on Mathematical Morphology in Stratified Institutions: Applications to Spatial Reasoning
Several logical operators are defined as dual pairs, in different types of
logics. Such dual pairs of operators also occur in other algebraic theories,
such as mathematical morphology. Based on this observation, this paper proposes
to define, at the abstract level of institutions, a pair of abstract dual and
logical operators as morphological erosion and dilation. Standard quantifiers
and modalities are then derived from these two abstract logical operators.
These operators are studied both on sets of states and sets of models. To cope
with the lack of explicit set of states in institutions, the proposed abstract
logical dual operators are defined in an extension of institutions, the
stratified institutions, which take into account the notion of open sentences,
the satisfaction of which is parametrized by sets of states. A hint on the
potential interest of the proposed framework for spatial reasoning is also
provided.Comment: 36 page
Decoding by Embedding: Correct Decoding Radius and DMT Optimality
The closest vector problem (CVP) and shortest (nonzero) vector problem (SVP)
are the core algorithmic problems on Euclidean lattices. They are central to
the applications of lattices in many problems of communications and
cryptography. Kannan's \emph{embedding technique} is a powerful technique for
solving the approximate CVP, yet its remarkable practical performance is not
well understood. In this paper, the embedding technique is analyzed from a
\emph{bounded distance decoding} (BDD) viewpoint. We present two complementary
analyses of the embedding technique: We establish a reduction from BDD to
Hermite SVP (via unique SVP), which can be used along with any Hermite SVP
solver (including, among others, the Lenstra, Lenstra and Lov\'asz (LLL)
algorithm), and show that, in the special case of LLL, it performs at least as
well as Babai's nearest plane algorithm (LLL-aided SIC). The former analysis
helps to explain the folklore practical observation that unique SVP is easier
than standard approximate SVP. It is proven that when the LLL algorithm is
employed, the embedding technique can solve the CVP provided that the noise
norm is smaller than a decoding radius , where
is the minimum distance of the lattice, and . This
substantially improves the previously best known correct decoding bound . Focusing on the applications of BDD to decoding of
multiple-input multiple-output (MIMO) systems, we also prove that BDD of the
regularized lattice is optimal in terms of the diversity-multiplexing gain
tradeoff (DMT), and propose practical variants of embedding decoding which
require no knowledge of the minimum distance of the lattice and/or further
improve the error performance.Comment: To appear in IEEE Transactions on Information Theor
Emergence of Structures in Particle Systems: Mechanics, Analysis and Computation
The meeting focused on the last advances in particle systems. The talks covered a broad range of topics, ranging from questions in crystallization and atomistic systems to mesoscopic models of defects to machine learning approaches and computational aspects
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