Proper Mergings of Stars and Chains are Counted by Sums of Antidiagonals in Certain Convolution Arrays -- The Details

A proper merging of two disjoint quasi-ordered sets $P$ and $Q$ is a quasi-order on the union of $P$ and $Q$ such that the restriction to $P$ or $Q$ yields the original quasi-order again and such that no elements of $P$ and $Q$ are identified. In this article, we determine the number of proper mergings in the case where $P$ is a star (i.e. an antichain with a smallest element adjoined), and $Q$ is a chain. We show that the lattice of proper mergings of an $m$-antichain and an $n$-chain, previously investigated by the author, is a quotient lattice of the lattice of proper mergings of an $m$-star and an $n$-chain, and we determine the number of proper mergings of an $m$-star and an $n$-chain by counting the number of congruence classes and by determining their cardinalities. Additionally, we compute the number of Galois connections between certain modified Boolean lattices and chains.Comment: 27 pages, 7 figures, 1 table. Jonathan Farley has solved Problem 4.18; added Section 4.4 to describe his solutio

Towards Collaborative Conceptual Exploration

In domains with high knowledge distribution a natural objective is to create principle foundations for collaborative interactive learning environments. We present a first mathematical characterization of a collaborative learning group, a consortium, based on closure systems of attribute sets and the well-known attribute exploration algorithm from formal concept analysis. To this end, we introduce (weak) local experts for subdomains of a given knowledge domain. These entities are able to refute and potentially accept a given (implicational) query for some closure system that is a restriction of the whole domain. On this we build up a consortial expert and show first insights about the ability of such an expert to answer queries. Furthermore, we depict techniques on how to cope with falsely accepted implications and on combining counterexamples. Using notions from combinatorial design theory we further expand those insights as far as providing first results on the decidability problem if a given consortium is able to explore some target domain. Applications in conceptual knowledge acquisition as well as in collaborative interactive ontology learning are at hand.Comment: 15 pages, 2 figure

Formal Context Generation using Dirichlet Distributions

We suggest an improved way to randomly generate formal contexts based on Dirichlet distributions. For this purpose we investigate the predominant way to generate formal contexts, a coin-tossing model, recapitulate some of its shortcomings and examine its stochastic model. Building up on this we propose our Dirichlet model and develop an algorithm employing this idea. By comparing our generation model to a coin-tossing model we show that our approach is a significant improvement with respect to the variety of contexts generated. Finally, we outline a possible application in null model generation for formal contexts.Comment: 16 pages, 7 figure

Distributed Formal Concept Analysis Algorithms Based on an Iterative MapReduce Framework

While many existing formal concept analysis algorithms are efficient, they are typically unsuitable for distributed implementation. Taking the MapReduce (MR) framework as our inspiration we introduce a distributed approach for performing formal concept mining. Our method has its novelty in that we use a light-weight MapReduce runtime called Twister which is better suited to iterative algorithms than recent distributed approaches. First, we describe the theoretical foundations underpinning our distributed formal concept analysis approach. Second, we provide a representative exemplar of how a classic centralized algorithm can be implemented in a distributed fashion using our methodology: we modify Ganter's classic algorithm by introducing a family of MR* algorithms, namely MRGanter and MRGanter+ where the prefix denotes the algorithm's lineage. To evaluate the factors that impact distributed algorithm performance, we compare our MR* algorithms with the state-of-the-art. Experiments conducted on real datasets demonstrate that MRGanter+ is efficient, scalable and an appealing algorithm for distributed problems.Comment: 17 pages, ICFCA 201, Formal Concept Analysis 201

On the Usability of Probably Approximately Correct Implication Bases

We revisit the notion of probably approximately correct implication bases from the literature and present a first formulation in the language of formal concept analysis, with the goal to investigate whether such bases represent a suitable substitute for exact implication bases in practical use-cases. To this end, we quantitatively examine the behavior of probably approximately correct implication bases on artificial and real-world data sets and compare their precision and recall with respect to their corresponding exact implication bases. Using a small example, we also provide qualitative insight that implications from probably approximately correct bases can still represent meaningful knowledge from a given data set.Comment: 17 pages, 8 figures; typos added, corrected x-label on graph

Revisiting Pattern Structure Projections

International audienceFormal concept analysis (FCA) is a well-founded method for data analysis and has many applications in data mining. Pattern structures is an extension of FCA for dealing with complex data such as sequences or graphs. However the computational complexity of computing with pattern structures is high and projections of pattern structures were introduced for simplifying computation. In this paper we introduce o-projections of pattern structures, a generalization of projections which defines a wider class of projections preserving the properties of the original approach. Moreover, we show that o-projections form a semilattice and we discuss the correspondence between o-projections and the representation contexts of o-projected pattern structures

Towards a generalisation of formal concept analysis for data mining purposes

In this paper we justify the need for a generalisation of Formal Concept Analysis for the purpose of data mining and begin the synthesis of such theory. For that purpose, we first review semirings and semimodules over semirings as the appropriate objects to use in abstracting the Boolean algebra and the notion of extents and intents, respectively. We later bring to bear powerful theorems developed in the field of linear algebra over idempotent semimodules to try to build a Fundamental Theorem for K-Formal Concept Analysis, where K is a type of idempotent semiring. Finally, we try to put Formal Concept Analysis in new perspective by considering it as a concrete instance of the theory developed

Особенности инженерных изысканий на нефтезагрязненных территориях на примере г. Томска

В диссертации рассмотрены особенности проведения инженерных изысканий на нефтезагрязненных территориях на примере г. Томска. Целью работы является рассмотрение особенностей инженерных изысканий для разработки проекта строительства АЗС в проектируемом районе «Новый город» на территории бывшего промышленного объекта, где содержание нефтепродуктов в грунтах и подземных водах превышает фоновые значения. В работе проведен анализ большого числа литературных источников, нормативных документов по теме исследований, обобщены данные о содержании нефтепродуктов в грунтах и подземных водах в пределах АЗК г. Томска. В результате анализа предложен новый подход к выбору площадки АЗС с учетом оценки степени защищенности подземных вод от воздействия нефтепродуктов, составлен проект инженерно-геологических изысканий для проектирования АЗС микрорайоне «Новый город». Тщательно проработан вопрос лабораторного определения нефтепродуктов в грунтах и подземных водах и предложен наиболее оптимальный вариант. Полученные результаты и выводы рекомендуется использовать в практике инженерных изысканий.The thesis describes the features of engineering surveys on contaminated territories on the example of Tomsk. The aim of this work is the consideration of the characteristics of engineering survey for development of gas station construction project in the planned area of "New town" in the territory of the former industrial site, where the content of oil products in soils and groundwater exceed background values. In the work the analysis of a large number of literary sources and normative documents on the topic of the research, summarizes data on the content of oil products in soils and groundwater within the filling station in Tomsk. The analysis suggests a new approach to the siting of filling stations based on an assessment of the degree of protection of groundwater from exposure to petroleum products, drafted engineering-geological surveys for design of the filling station neighborhood "New city". Carefully crafted question for laboratory determination of petroleum products in soils and groundwater, and suggested the best option. The obtained results and conclusions it is recommended to use in the practice of engineering surveys

Practical Use of Formal Concept Analysis in Service-Oriented Computing

International audiencePervasive applications are encountered in a number of settings, including smart houses, intelligent buildings or connected plants. Service-Oriented Computing is today the technology of choice for implementing and exposing resources in such environments. The selection of appropriate services at the right moment in order to compose meaningful applications is however a real issue. In this paper, we propose a FCA-based solution to this problem.We have integrated FCA algorithms in our pervasive gateways and adapted them in order to allow efficient runtime selection of heterogeneous and dynamic services. This work has been applied to realistic use cases in the scope of a European project