scholarly journals Convex Geometries are Extremal for the Generalized Sauer-Shelah Bound

10.37236/7217 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Bogdan Chornomaz
Keyword(s):  
Upper Bound ◽  
Closure Systems ◽  
Np Complete ◽  
Convex Geometries

The Sauer-Shelah lemma provides an exact upper bound on the size of set families with bounded Vapnik-Chervonekis dimension. When applied to lattices represented as closure systems, this lemma outlines a class of extremal lattices obtaining this bound. Here we show that the Sauer-Shelah bound can be easily generalized to arbitrary antichains, and extremal objects for this generalized bound are exactly convex geometries. We also show that the problem of classification of antichains admitting such extremal objects is NP-complete. 

scholarly journals On the Classification of NP Complete Problems and Their Duality Feature

2018 ◽  
Vol 10 (1) ◽  
pp. 67-78 ◽  
Author(s):  
Wenhong Tian ◽  
Wenxia Guo ◽  
Majun He
Keyword(s):  
Complete Problems ◽  
Np Complete

scholarly journals A Complete Classification of Tractability in RCC-5

1997 ◽  
Vol 6 ◽  
pp. 211-221 ◽  
Author(s):  
T. Drakengren ◽  
P. Jonsson
Keyword(s):  
Spatial Reasoning ◽  
Complete Classification ◽  
Satisfiability Problem ◽  
Restricted Version ◽  
Np Complete ◽  
Computational Properties

We investigate the computational properties of the spatial algebra RCC-5 which is a restricted version of the RCC framework for spatial reasoning. The satisfiability problem for RCC-5 is known to be NP-complete but not much is known about its approximately four billion subclasses. We provide a complete classification of satisfiability for all these subclasses into polynomial and NP-complete respectively. In the process, we identify all maximal tractable subalgebras which are four in total.

Supervised Adaptive Hamming Net for Classification of Multiple-Valued Patterns

1997 ◽  
Vol 08 (02) ◽  
pp. 181-200
Author(s):  
Cheng-An Hung ◽  
Sheng-Fuu Lin
Keyword(s):  
Upper Bound ◽  
Incremental Learning ◽  
Arbitrary Sequence ◽  
Input Output ◽  
Multiple Valued Logic

A Supervised Adaptive Hamming Net (SAHN) is introduced for incremental learning of recognition categories in response to arbitrary sequence of multiple-valued or binary-valued input patterns. The binary-valued SAHN derived from the Adaptive Hamming Net (AHN) is functionally equivalent to a simplified ARTMAP, which is specifically designed to establish many-to-one mappings. The generalization to learning multiple-valued input patterns is achieved by incorporating multiple-valued logic into the AHN. In this paper, we examine some useful properties of learning in a P-valued SAHN. In particular, an upper bound is derived on the number of epochs required by the P-valued SAHN to learn a list of input-output pairs that is repeatedly presented to the architecture. Furthermore, we connect the P-valued SAHN with the binary-valued SAHN via the thermometer code.

scholarly journals DENSE CODING WITH MULTIPARTITE QUANTUM STATES

2006 ◽  
Vol 04 (03) ◽  
pp. 415-428 ◽  
Author(s):  
DAGMAR BRUß ◽  
MACIEJ LEWENSTEIN ◽  
ADITI SEN(DE) ◽  
UJJWAL SEN ◽  
GIACOMO MAURO D'ARIANO ◽  
...  
Keyword(s):  
Quantum State ◽  
Upper Bound ◽  
Arbitrary Number ◽  
Information Transfer ◽  
Quantum States ◽  
Dense Coding ◽  
Joint Operations ◽  
Single Receiver

We consider generalizations of the dense coding protocol with an arbitrary number of senders and either one or two receivers, sharing a multiparty quantum state, and using a noiseless channel. For the case of a single receiver, the capacity of such information transfer is found exactly. It is shown that the capacity is not enhanced by allowing the senders to perform joint operations. We provide a nontrivial upper bound on the capacity in the case of two receivers. We also give a classification of the set of all multiparty states in terms of their usefulness for dense coding. We provide examples for each of these classes, and discuss some of their properties.

scholarly journals The Prime Stems of Rooted Circuits of Closure Spaces and Minimum Implicational Bases

10.37236/3068 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Masataka Nakamura ◽  
Kenji Kashiwabara
Keyword(s):  
Convex Geometry ◽  
Closure Operators ◽  
Optimal Basis ◽  
Closed Set ◽  
Closed Sets ◽  
Closure Systems ◽  
Closure Spaces ◽  
Convex Geometries ◽  
Necessary And Sufficient ◽  
Special Case

A rooted circuit is firstly introduced for convex geometries (antimatroids). We generalize it for closure systems or equivalently for closure operators. A rooted circuit is a specific type of a pair $(X,e)$ of a subset $X$, called a stem, and an element $e\not\in X$, called a root. We introduce a notion called a 'prime stem', which plays the key role in this article. Every prime stem is shown to be a pseudo-closed set of an implicational system. If the sizes of stems are all the same, the stems are all pseudo-closed sets, and they give rise to a canonical minimum implicational basis. For an affine convex geometry, the prime stems determine a canonical minimum basis, and furthermore  gives rise to an optimal basis. A 'critical rooted circuit' is a special case of a rooted circuit defined for an antimatroid. As a precedence structure, 'critical rooted circuits' are necessary and sufficient to fix an antimatroid whereas critical rooted circuits are not necessarily sufficient to restore the original antimatroid as an implicational system. It is shown through an example.

scholarly journals On the Number of Similar Instances of a Pattern in a Finite Set

10.37236/4972 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Bernardo M. Ábrego ◽  
Silvia Fernández-Merchant ◽  
Daniel J. Katz ◽  
Levon Kolesnikov
Keyword(s):  
Arithmetic Progression ◽  
Finite Subset ◽  
Upper Bound ◽  
Point Subset ◽  
Finite Set ◽  
Equilateral Triangles ◽  
Optimal Configurations ◽  
Full Classification ◽  
General Method

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is shown to be no more than $\lfloor{(4 n-1)(n-1)/18}\rfloor$. The number of $k$-term arithmetic progressions that lie within an $n$-point subset of the line is shown to be at most $(n-r)(n+r-k+1)/(2 k-2)$, where $r$ is the remainder when $n$ is divided by $k-1$. This upper bound is achieved when the $n$ points themselves form an arithmetic progression, but for some values of $k$ and $n$, it can also be achieved for other configurations of the $n$ points, and a full classification of such optimal configurations is given. These results are achieved using a new general method based on ordering relations.

ON AN INVERSE PROBLEM TO FROBENIUS' THEOREM II

2012 ◽  
Vol 11 (05) ◽  
pp. 1250092 ◽  
Author(s):  
WEI MENG ◽  
JIANGTAO SHI ◽  
KELIN CHEN
Keyword(s):  
Inverse Problem ◽  
Finite Group ◽  
Positive Integer ◽  
Upper Bound ◽  
Complete Classification ◽  
Frobenius Theorem ◽  
A Finite Group

Let G be a finite group and e a positive integer dividing |G|, the order of G. Denoting Le(G) = {x ∈ G|xe = 1}. Frobenius proved that |Le(G)| = ke for some positive integer k ≥ 1. Let k(G) be the upper bound of the set {k||Le(G)| = ke, ∀ e ||G|}. In this paper, a complete classification of the finite group G with k(G) = 3 is obtained.

scholarly journals An improvement of a transcendence measure of Galochkin and Mahler's S-numbers

1992 ◽  
Vol 52 (1) ◽  
pp. 130-140 ◽  
Author(s):  
Masaaki Amou
Keyword(s):  
Upper Bound ◽  
Functional Equations ◽  
Special Values ◽  
Transcendental Numbers ◽  
Transcendence Measure

AbstractWe give a transcendence measure of special values of functions satisfying certain functional equations. This improves an earlier result of Galochkin, and gives a better upper bound of the type for such a number as an S-number in the classification of transcendental numbers by Mahler.

scholarly journals Wilf-Classification of Mesh Patterns of Short Length

10.37236/4678 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Ísak Hilmarsson ◽  
Ingibjörg Jónsdóttir ◽  
Steinunn Sigurðardóttir ◽  
Lína Viðarsdóttir ◽  
Henning Ulfarsson
Keyword(s):  
Upper Bound ◽  
Short Length ◽  
Actual Number ◽  
Automatic Methods

This paper starts the Wilf-classification of mesh patterns of length 2. Although there are initially 1024 patterns to consider we introduce automatic methods to reduce the number of potentially different Wilf-classes to at most 65. By enumerating some of the remaining classes we bring that upper-bound further down to 56. Finally, we conjecture that the actual number of Wilf-classes of mesh patterns of length 2 is 46.

scholarly journals The Complexity of Reasoning about Spatial Congruence

1999 ◽  
Vol 11 ◽  
pp. 361-390 ◽  
Author(s):  
M. Cristani
Keyword(s):  
Artificial Intelligence ◽  
Recent Literature ◽  
Research Effort ◽  
Complete Classification ◽  
Spatial Knowledge ◽  
Np Complete ◽  
Temporal And Spatial ◽  
Spatial Congruence ◽  
Full Algebra

In the recent literature of Artificial Intelligence, an intensive research effort has been spent, for various algebras of qualitative relations used in the representation of temporal and spatial knowledge, on the problem of classifying the computational complexity of reasoning problems for subsets of algebras. The main purpose of these researches is to describe a restricted set of maximal tractable subalgebras, ideally in an exhaustive fashion with respect to the hosting algebras. In this paper we introduce a novel algebra for reasoning about Spatial Congruence, show that the satisfiability problem in the spatial algebra MC-4 is NP-complete, and present a complete classification of tractability in the algebra, based on the individuation of three maximal tractable subclasses, one containing the basic relations. The three algebras are formed by 14, 10 and 9 relations out of 16 which form the full algebra.

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