65 research outputs found
On the number of simple arrangements of five double pseudolines
We describe an incremental algorithm to enumerate the isomorphism classes of
double pseudoline arrangements. The correction of our algorithm is based on the
connectedness under mutations of the spaces of one-extensions of double
pseudoline arrangements, proved in this paper. Counting results derived from an
implementation of our algorithm are also reported.Comment: 24 pages, 16 figures, 6 table
COMs: Complexes of Oriented Matroids
In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured
them as asymmetric counterparts of oriented matroids, both sharing the key
property of strong elimination. Moreover, symmetry of faces holds in both
structures as well as in the so-called affine oriented matroids. These two
fundamental properties (formulated for covectors) together lead to the natural
notion of "conditional oriented matroid" (abbreviated COM). These novel
structures can be characterized in terms of three cocircuits axioms,
generalizing the familiar characterization for oriented matroids. We describe a
binary composition scheme by which every COM can successively be erected as a
certain complex of oriented matroids, in essentially the same way as a lopsided
set can be glued together from its maximal hypercube faces. A realizable COM is
represented by a hyperplane arrangement restricted to an open convex set. Among
these are the examples formed by linear extensions of ordered sets,
generalizing the oriented matroids corresponding to the permutohedra. Relaxing
realizability to local realizability, we capture a wider class of combinatorial
objects: we show that non-positively curved Coxeter zonotopal complexes give
rise to locally realizable COMs.Comment: 40 pages, 6 figures, (improved exposition
Rough matroids based on coverings
The introduction of covering-based rough sets has made a substantial
contribution to the classical rough sets. However, many vital problems in rough
sets, including attribution reduction, are NP-hard and therefore the algorithms
for solving them are usually greedy. Matroid, as a generalization of linear
independence in vector spaces, it has a variety of applications in many fields
such as algorithm design and combinatorial optimization. An excellent
introduction to the topic of rough matroids is due to Zhu and Wang. On the
basis of their work, we study the rough matroids based on coverings in this
paper. First, we investigate some properties of the definable sets with respect
to a covering. Specifically, it is interesting that the set of all definable
sets with respect to a covering, equipped with the binary relation of inclusion
, constructs a lattice. Second, we propose the rough matroids based
on coverings, which are a generalization of the rough matroids based on
relations. Finally, some properties of rough matroids based on coverings are
explored. Moreover, an equivalent formulation of rough matroids based on
coverings is presented. These interesting and important results exhibit many
potential connections between rough sets and matroids.Comment: 15page
How to say greedy in fork algebras
Because of their expressive power, binary relations are widely used in program specification and development within formal calculi. The existence of a finite equational axiomatization for algebras of binary relations with a fork operation guarantees that the heuristic power coming from binary relations is captured inside an abstract equational calculus.
In this paper we show how to express the greedy program design strategy into the first order theory of fork algebras.Eje: TeoríaRed de Universidades con Carreras en Informática (RedUNCI
How to say greedy in fork algebras
Because of their expressive power, binary relations are widely used in program specification and development within formal calculi. The existence of a finite equational axiomatization for algebras of binary relations with a fork operation guarantees that the heuristic power coming from binary relations is captured inside an abstract equational calculus.
In this paper we show how to express the greedy program design strategy into the first order theory of fork algebras.Eje: TeoríaRed de Universidades con Carreras en Informática (RedUNCI
The complexity of reasoning with relative directions
© 2014 The Authors and IOS Press. Whether reasoning with relative directions can be performed in NP has been an open problem in qualitative spatial reasoning. Efficient reasoning with relative directions is essential, for example, in rule-compliant agent navigation. In this paper, we prove that reasoning with relative directions is ∃ℝ-complete. As a consequence, reasoning with relative directions is not in NP, unless NP=∃ℝ
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