10 research outputs found
Optimum basis of finite convex geometry
Convex geometries form a subclass of closure systems with unique
criticals, or UC-systems. We show that the F-basis introduced in [6] for UC-
systems, becomes optimum in convex geometries, in two essential parts of the
basis: right sides (conclusions) of binary implications and left sides (premises)
of non-binary ones. The right sides of non-binary implications can also be
optimized, when the convex geometry either satis es the Carousel property,
or does not have D-cycles. The latter generalizes a result of P.L. Hammer
and A. Kogan for acyclic Horn Boolean functions. Convex geometries of order
convex subsets in a poset also have tractable optimum basis. The problem of
tractability of optimum basis in convex geometries in general remains to be
ope
Optimum basis of finite convex geometry
Convex geometries form a subclass of closure systems with unique
criticals, or UC-systems. We show that the F-basis introduced in [6] for UC-
systems, becomes optimum in convex geometries, in two essential parts of the
basis: right sides (conclusions) of binary implications and left sides (premises)
of non-binary ones. The right sides of non-binary implications can also be
optimized, when the convex geometry either satis es the Carousel property,
or does not have D-cycles. The latter generalizes a result of P.L. Hammer
and A. Kogan for acyclic Horn Boolean functions. Convex geometries of order
convex subsets in a poset also have tractable optimum basis. The problem of
tractability of optimum basis in convex geometries in general remains to be
ope
Optimization of convex geometries: component quadratic and general
In this Capstone Project, we worked with a class of closure systems called convex
geometries, which are closure systems with a closure operator that satisfies the
anti-exchange property. We first looked at the result of optimization algorithm of
component quadratic systems, which are discussed in [4], and reproved it for the
case of convex geometries. We then investigated the following question: if a convex
geometry is given by a set of implications, is it possible to find its optimum basis
in polynomial time when the convex geometry does not have particular properties
(for instance, not component quadratic)
Optimization of convex geometries: component quadratic and general
In this Capstone Project, we worked with a class of closure systems called convex
geometries, which are closure systems with a closure operator that satisfies the
anti-exchange property. We first looked at the result of optimization algorithm of
component quadratic systems, which are discussed in [4], and reproved it for the
case of convex geometries. We then investigated the following question: if a convex
geometry is given by a set of implications, is it possible to find its optimum basis
in polynomial time when the convex geometry does not have particular properties
(for instance, not component quadratic)
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
Quasi-closed elements in fuzzy posets
We generalize the notion of quasi-closed element to fuzzy posets in two stages: First, in the crisp style in which each element in a given universe either is quasi-closed or not. Second, in the graded style by defining degrees to which an element is quasi-closed. We discuss the different possible definitions and comparing them with each other. Finally, we show that the most general one has good properties to be used when we have a complete fuzzy lattice as a frame
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
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