184 research outputs found
On implicational bases of closure system with unique critical sets
We show that every optimum basis of a nite closure system,
in D. Maier's sense, is also right-side optimum, which is a parameter of a
minimum CNF representation of a Horn Boolean function. New parameters
for the size of the binary part are also established. We introduce the K-basis
of a general closure system, which is a re nement of the canonical basis of
V. Duquenne and J.L. Guigues, and discuss a polynomial algorithm to obtain
it. We study closure systems with unique critical sets, and some subclasses
of these where the K-basis is unique. A further re nement in the form of the
E-basis is possible for closure systems without D-cycles. There is a polynomial
algorithm to recognize the D-relation from a K-basis. Thus, closure systems
without D-cycles can be e ectively recognized. While the E-basis achieves an
optimum in one of its parts, the optimization of the others is an NP-complete
proble
Discovery of the D-basis in binary tables based on hypergraph dualization
Discovery of (strong) association rules, or implications, is an important
task in data management, and it nds application in arti cial intelligence,
data mining and the semantic web. We introduce a novel approach
for the discovery of a speci c set of implications, called the D-basis, that provides
a representation for a reduced binary table, based on the structure of
its Galois lattice. At the core of the method are the D-relation de ned in
the lattice theory framework, and the hypergraph dualization algorithm that
allows us to e ectively produce the set of transversals for a given Sperner hypergraph.
The latter algorithm, rst developed by specialists from Rutgers
Center for Operations Research, has already found numerous applications in
solving optimization problems in data base theory, arti cial intelligence and
game theory. One application of the method is for analysis of gene expression
data related to a particular phenotypic variable, and some initial testing is
done for the data provided by the University of Hawaii Cancer Cente
Lattices of quasi-equational theories as congruence lattices of semilattices with operators, part II
Part I proved that for every quasivariety K of structures
(which may have both operations and relations) there is a semilattice
S with operators such that the lattice of quasi-equational theories of
K (the dual of the lattice of sub-quasivarieties of K) is isomorphic to
Con(S;+; 0; F). It is known that if S is a join semilattice with 0 (and no
operators), then there is a quasivariety Q such that the lattice of theories
of Q is isomorphic to Con(S;+; 0). We prove that if S is a semilattice
having both 0 and 1 with a group G of operators acting on S, and each
operator in G xes both 0 and 1, then there is a quasivariety W such
that the lattice of theories of W is isomorphic to Con(S;+; 0; G)
On implicational bases of closure system with unique critical sets
We show that every optimum basis of a nite closure system,
in D. Maier's sense, is also right-side optimum, which is a parameter of a
minimum CNF representation of a Horn Boolean function. New parameters
for the size of the binary part are also established. We introduce the K-basis
of a general closure system, which is a re nement of the canonical basis of
V. Duquenne and J.L. Guigues, and discuss a polynomial algorithm to obtain
it. We study closure systems with unique critical sets, and some subclasses
of these where the K-basis is unique. A further re nement in the form of the
E-basis is possible for closure systems without D-cycles. There is a polynomial
algorithm to recognize the D-relation from a K-basis. Thus, closure systems
without D-cycles can be e ectively recognized. While the E-basis achieves an
optimum in one of its parts, the optimization of the others is an NP-complete
proble
A class of infinite convex geometries
Various characterizations of finite convex geometries
are well known. This note provides similar characterizations for
possibly infinite convex geometries whose lattice of closed sets is
strongly coatomic and lower continuous. Some classes of examples
of such convex geometries are give
Discovery of the D-basis in binary tables based on hypergraph dualization
Discovery of (strong) association rules, or implications, is an important
task in data management, and it nds application in arti cial intelligence,
data mining and the semantic web. We introduce a novel approach
for the discovery of a speci c set of implications, called the D-basis, that provides
a representation for a reduced binary table, based on the structure of
its Galois lattice. At the core of the method are the D-relation de ned in
the lattice theory framework, and the hypergraph dualization algorithm that
allows us to e ectively produce the set of transversals for a given Sperner hypergraph.
The latter algorithm, rst developed by specialists from Rutgers
Center for Operations Research, has already found numerous applications in
solving optimization problems in data base theory, arti cial intelligence and
game theory. One application of the method is for analysis of gene expression
data related to a particular phenotypic variable, and some initial testing is
done for the data provided by the University of Hawaii Cancer Cente
Ordered direct implication basis of a finite closure system
Closure system on a nite set is a unifying concept in logic programming,
relational data bases and knowledge systems. It can also be presented
in the terms of nite lattices, and the tools of economic description of a
nite lattice have long existed in lattice theory. We present this approach by
describing the so-called D-basis and introducing the concept of ordered direct
basis of an implicational system. A direct basis of a closure operator, or an
implicational system, is a set of implications that allows one to compute the
closure of an arbitrary set by a single iteration. This property is preserved by
the D-basis at the cost of following a prescribed order in which implications
will be attended. In particular, using an ordered direct basis allows to optimize
the forward chaining procedure in logic programming that uses the Horn
fragment of propositional logic. One can extract the D-basis from any direct
unit basis in time polynomial in the size s( ), and it takes only linear time
of the cardinality of the D-basis to put it into a proper order. We produce
examples of closure systems on a 6-element set, for which the canonical basis
of Duquenne and Guigues is not ordered direc
On implicational bases of closure systems with unique critical sets
We show that every optimum basis of a finite closure system, in D.Maier's
sense, is also right-side optimum, which is a parameter of a minimum CNF
representation of a Horn Boolean function. New parameters for the size of the
binary part are also established. We introduce a K-basis of a general closure
system, which is a refinement of the canonical basis of Duquenne and Guigues,
and discuss a polynomial algorithm to obtain it. We study closure systems with
the unique criticals and some of its subclasses, where the K-basis is unique. A
further refinement in the form of the E-basis is possible for closure systems
without D-cycles. There is a polynomial algorithm to recognize the D-relation
from a K-basis. Thus, closure systems without D-cycles can be effectively
recognized. While E-basis achieves an optimum in one of its parts, the
optimization of the others is an NP-complete problem.Comment: Presented on International Symposium of Artificial Intelligence and
Mathematics (ISAIM-2012), Ft. Lauderdale, FL, USA Results are included into
plenary talk on conference Universal Algebra and Lattice Theory, June 2012,
Szeged, Hungary 29 pages and 2 figure
Ordered direct implicational basis of a finite closure system
Closure system on a finite set is a unifying concept in logic programming,
relational data bases and knowledge systems. It can also be presented in the
terms of finite lattices, and the tools of economic description of a finite
lattice have long existed in lattice theory. We present this approach by
describing the so-called D-basis and introducing the concept of ordered direct
basis of an implicational system. A direct basis of a closure operator, or an
implicational system, is a set of implications that allows one to compute the
closure of an arbitrary set by a single iteration. This property is preserved
by the D-basis at the cost of following a prescribed order in which
implications will be attended. In particular, using an ordered direct basis
allows to optimize the forward chaining procedure in logic programming that
uses the Horn fragment of propositional logic. One can extract the D-basis from
any direct unit basis S in time polynomial in the size of S, and it takes only
linear time of the cardinality of the D-basis to put it into a proper order. We
produce examples of closure systems on a 6-element set, for which the canonical
basis of Duquenne and Guigues is not ordered direct.Comment: 25 pages, 10 figures; presented at AMS conference,
TACL-2011,ISAIM-2012 and at RUTCOR semina
One-dimensional quantum channel and Hawking radiation of the Kerr and Kerr-Newman black holes
In this paper, we review the one-dimensional quantum channel and investigate
Hawking radiation of bosons and fermions in Kerr and Kerr-Newman black holes.
The result shows the Hawking radiation can be described by the quantum channel.
The thermal conductances are derived and related to the black holes'
temperatures.Comment: V2, 12 pages. Typo correcte
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