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Border algorithms for computing Hasse diagrams of arbitrary lattices

機(jī)譯：用于計(jì)算任意格的Hasse圖的邊界算法

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摘要

Lattices are mathematical structures with many applications in computer science; among these, we are interested in fields like data mining, machine learning, or knowledge discovery in databases. One well-established use of lattice theory is in formal concept analysis (FCA), where the concept lattice with its diagram graph allows the visualization and summarization of data in a more concise representation. In the Data Mining community, the same mathematical notions (often under additional “frequency” constraints that bound from below the size of the support set) are studied under the banner of Closed-Set Mining. In these applications, each dataset consists of transactions, also called objects, each of which, besides having received a unique identifier, consists of a set of items or attributes taken from a previously agreed finite set. A concept is a pair formed by a set of transactions —the extent set or support set of the concept— and a set of attributes —the intent set of the concept— defined as the set of all those attributes that are shared by all the transactions present in the extent. Some data analysis processes are based on the family of all intents (the “closures” stemming from the dataset); but others require to determine also their order relation, which is a finite lattice, in the form of a line graph (the Hasse diagram).
機(jī)譯：格是數(shù)學(xué)結(jié)構(gòu)，在計(jì)算機(jī)科學(xué)中有許多應(yīng)用。其中，我們對(duì)數(shù)據(jù)挖掘，機(jī)器學(xué)習(xí)或數(shù)據(jù)庫(kù)中的知識(shí)發(fā)現(xiàn)等領(lǐng)域感興趣。格網(wǎng)理論的一種公認(rèn)的用法是形式概念分析（FCA），其中概念格及其圖表使您能夠以更簡(jiǎn)潔的表示形式對(duì)數(shù)據(jù)進(jìn)行可視化和匯總。在數(shù)據(jù)挖掘社區(qū)中，在封閉集挖掘的旗幟下研究了相同的數(shù)學(xué)概念（通常在支持集大小以下的附加“頻率”約束下）。在這些應(yīng)用程序中，每個(gè)數(shù)據(jù)集均由事務(wù)（也稱為對(duì)象）組成，每個(gè)事務(wù)除已接收到唯一標(biāo)識(shí)符外，還包括從先前約定的有限集中獲取的一組項(xiàng)或?qū)傩?。概念是由一組事務(wù)（概念的范圍集或支持集）和一組屬性（概念的意圖集）形成的一對(duì)，定義為所有事務(wù)共享的所有那些屬性的集合存在的程度。一些數(shù)據(jù)分析過(guò)程基于所有意圖的族（源自數(shù)據(jù)集的“封閉”）；但是其他人也需要以線圖（Hasse圖）的形式確定其順序關(guān)系，即有限晶格。

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