555 research outputs found
Contamination-Free Measures and Algebraic Operations
An open concept of rough evolution and an axiomatic approach to granules was
also developed recently by the present author. Subsequently the concepts were
used in the formal framework of rough Y-systems (RYS) for developing on
granular correspondences by her. These have since been used for a new approach
towards comparison of rough algebraic semantics across different semantic
domains by way of correspondences that preserve rough evolution and try to
avoid contamination. In this research paper, new methods are proposed and a
semantics for handling possibly contaminated operations and structured bigness
is developed. These would also be of natural interest for relative consistency
of one collection of knowledge relative other.Comment: Preprint of FUZZIEEE'2013 Conference Pape
High Granular Operator Spaces, and Less-Contaminated General Rough Mereologies
Granular operator spaces and variants had been introduced and used in
theoretical investigations on the foundations of general rough sets by the
present author over the last few years. In this research, higher order versions
of these are presented uniformly as partial algebraic systems. They are also
adapted for practical applications when the data is representable by data
table-like structures according to a minimalist schema for avoiding
contamination. Issues relating to valuations used in information systems or
tables are also addressed. The concept of contamination introduced and studied
by the present author across a number of her papers, concerns mixing up of
information across semantic domains (or domains of discourse). Rough inclusion
functions (\textsf{RIF}s), variants, and numeric functions often have a direct
or indirect role in contaminating algorithms. Some solutions that seek to
replace or avoid them have been proposed and investigated by the present author
in some of her earlier papers. Because multiple kinds of solution are of
interest to the contamination problem, granular generalizations of RIFs are
proposed, and investigated. Interesting representation results are proved and a
core algebraic strategy for generalizing Skowron-Polkowski style of rough
mereology (though for a very different purpose) is formulated. A number of
examples have been added to illustrate key parts of the proposal in higher
order variants of granular operator spaces. Further algorithms grounded in
mereological nearness, suited for decision-making in human-machine interaction
contexts, are proposed by the present author. Applications of granular
\textsf{RIF}s to partial/soft solutions of the inverse problem are also
invented in this paper.Comment: Research paper: Preprint: Final versio
Dialectics of Counting and the Mathematics of Vagueness
New concepts of rough natural number systems are introduced in this research
paper from both formal and less formal perspectives. These are used to improve
most rough set-theoretical measures in general Rough Set theory (\textsf{RST})
and to represent rough semantics. The foundations of the theory also rely upon
the axiomatic approach to granularity for all types of general \textsf{RST}
recently developed by the present author. The latter theory is expanded upon in
this paper. It is also shown that algebraic semantics of classical \textsf{RST}
can be obtained from the developed dialectical counting procedures. Fuzzy set
theory is also shown to be representable in purely granule-theoretic terms in
the general perspective of solving the contamination problem that pervades this
research paper. All this constitutes a radically different approach to the
mathematics of vague phenomena and suggests new directions for a more realistic
extension of the foundations of mathematics of vagueness from both foundational
and application points of view. Algebras corresponding to a concept of
\emph{rough naturals} are also studied and variants are characterised in the
penultimate section.Comment: This paper includes my axiomatic approach to granules. arXiv admin
note: substantial text overlap with arXiv:1102.255
A Logic Approach to Granular computing
This article was originally published by the International Journal of Cognitive Informatics and Natural IntelligenceGranular computing is an emerging field of study that attempts to formalize and explore methods and
heuristics of human problem solving with multiple levels of granularity and abstraction. A fundamental
issue of granular computing is the representation and utilization of granular structures. The main objective
of this article is to examine a logic approach to address this issue. Following the classical interpretation
of a concept as a pair of intension and extension, we interpret a granule as a pair of a set of objects and a
logic formula describing the granule. The building blocks of granular structures are basic granules representing an elementary concept or a piece of knowledge. They are treated as atomic formulas of a logic
language. Different types of granular structures can be constructed by using logic connectives. Within
this logic framework, we show that rough set analysis (RSA) and formal concept analysis (FCA) can be
interpreted uniformly. The two theories use multilevel granular structures but differ in their choices of
definable granules and granular structures.NSERC Canada Discovery gran
Computing sets of graded attribute implications with witnessed non-redundancy
In this paper we extend our previous results on sets of graded attribute
implications with witnessed non-redundancy. We assume finite residuated
lattices as structures of truth degrees and use arbitrary idempotent
truth-stressing linguistic hedges as parameters which influence the semantics
of graded attribute implications. In this setting, we introduce algorithm which
transforms any set of graded attribute implications into an equivalent
non-redundant set of graded attribute implications with saturated consequents
whose non-redundancy is witnessed by antecedents of the formulas. As a
consequence, we solve the open problem regarding the existence of general
systems of pseudo-intents which appear in formal concept analysis of
object-attribute data with graded attributes and linguistic hedges.
Furthermore, we show a polynomial-time procedure for determining bases given by
general systems of pseudo-intents from sets of graded attribute implications
which are complete in data
Quantizing Euclidean motions via double-coset decomposition
Concepts from mathematical crystallography and group theory are used here to
quantize the group of rigid-body motions, resulting in a "motion alphabet" with
which to express robot motion primitives. From these primitives it is possible
to develop a dictionary of physical actions. Equipped with an alphabet of the
sort developed here, intelligent actions of robots in the world can be
approximated with finite sequences of characters, thereby forming the
foundation of a language in which to articulate robot motion. In particular, we
use the discrete handedness-preserving symmetries of macromolecular crystals
(known in mathematical crystallography as Sohncke space groups) to form a
coarse discretization of the space of rigid-body motions. This
discretization is made finer by subdividing using the concept of double-coset
decomposition. More specifically, a very efficient, equivolumetric quantization
of spatial motion can be defined using the group-theoretic concept of a
double-coset decomposition of the form ,
where is a Sohncke space group and is a finite group of
rotational symmetries such as those of the icosahedron. The resulting discrete
alphabet is based on a very uniform sampling of and is a tool for
describing the continuous trajectories of robots and humans. The general
"signals to symbols" problem in artificial intelligence is cast in this
framework for robots moving continuously in the world, and we present a
coarse-to-fine search scheme here to efficiently solve this decoding problem in
practice
Presheaves, Sheaves and their Topoi in Quantum Gravity and Quantum Logic
A brief synopsis of recent conceptions and results, the current status and
future outlook of our research program of applying sheaf and topos-theoretic
ideas to quantum gravity and quantum logic is presented.Comment: 12 pages; paper of a talk given at the 5th Biannual International
Quantum Structures Association Conference in Cesena, Italy (March-April 2001
Quantifying force networks in particulate systems
We present mathematical models based on persistent homology for analyzing
force distributions in particulate systems. We define three distinct chain
complexes: digital, position, and interaction, motivated by different
capabilities of collecting experimental or numerical data, e.g. digital images,
location of the particles, and normal forces between particles, respectively.
We describe how algebraic topology, in particular, homology allows one to
obtain algebraic representations of the geometry captured by these complexes.
To each complexes we define an associated force network from which persistent
homology is computed. Using numerical data obtained from molecular dynamics
simulations of a system of particles being slowly compressed we demonstrate how
persistent homology can be used to compare the geometries of the force
distributions in different granular systems. We also discuss the properties of
force networks as a function of the underlying complexes, and hence, as a
function of the type of experimental or numerical data provided
Network Analysis of Particles and Grains
The arrangements of particles and forces in granular materials have a complex
organization on multiple spatial scales that ranges from local structures to
mesoscale and system-wide ones. This multiscale organization can affect how a
material responds or reconfigures when exposed to external perturbations or
loading. The theoretical study of particle-level, force-chain, domain, and bulk
properties requires the development and application of appropriate physical,
mathematical, statistical, and computational frameworks. Traditionally,
granular materials have been investigated using particulate or continuum
models, each of which tends to be implicitly agnostic to multiscale
organization. Recently, tools from network science have emerged as powerful
approaches for probing and characterizing heterogeneous architectures across
different scales in complex systems, and a diverse set of methods have yielded
fascinating insights into granular materials. In this paper, we review work on
network-based approaches to studying granular matter and explore the potential
of such frameworks to provide a useful description of these systems and to
enhance understanding of their underlying physics. We also outline a few open
questions and highlight particularly promising future directions in the
analysis and design of granular matter and other kinds of material networks
Inference of hidden structures in complex physical systems by multi-scale clustering
We survey the application of a relatively new branch of statistical
physics--"community detection"-- to data mining. In particular, we focus on the
diagnosis of materials and automated image segmentation. Community detection
describes the quest of partitioning a complex system involving many elements
into optimally decoupled subsets or communities of such elements. We review a
multiresolution variant which is used to ascertain structures at different
spatial and temporal scales. Significant patterns are obtained by examining the
correlations between different independent solvers. Similar to other
combinatorial optimization problems in the NP complexity class, community
detection exhibits several phases. Typically, illuminating orders are revealed
by choosing parameters that lead to extremal information theory correlations.Comment: 25 pages, 16 Figures; a review of earlier work
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