By Bernhard Ganter, Sergei Obiedkov

This is the 1st textbook on characteristic exploration, its concept, its algorithms forapplications, and a few of its many attainable generalizations. characteristic explorationis important for buying established wisdom via an interactive strategy, byasking queries to knowledgeable. Generalizations that deal with incomplete, defective, orimprecise info are mentioned, however the concentration lies on wisdom extraction from areliable info source.The process relies on Formal proposal research, a mathematical conception ofconcepts and proposal hierarchies, and makes use of its expressive diagrams. The presentationis self-contained. It presents an creation to Formal proposal Analysiswith emphasis on its skill to derive algebraic constructions from qualitative data,which will be represented in significant and certain graphics.

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4 The closure system of closure systems A closure system on M is a set of subsets of M which is closed under arbitrary intersections. There are many closure systems, even when the base set M is of modest size. Their numbers are known up to |M | = 7 [121, 64]: Size of the base set M Number of closure systems |M | # closure syst. 0 1 6 75 973 751 474 1 2 2 7 3 61 4 2 480 5 1 385 552 7 14 087 648 235 707 352 472 The intersection of arbitrary many closure systems (on the same base set) is again a closure system.

11 and follow the descending paths from the corresponding nodes in the diagram. There is always a unique highest point where these paths meet, that is, the highest concept that is below both, namely, the infimum. Any other concept below both can be reached from the highest one on a descending path. Similarly, for any two formal concepts there is always the lowest node (the supremum of the two) that can be reached from both concepts via ascending paths. And any common superconcept of the two is on an ascending path from their supremum.

Chapter 2 An algorithm for closure systems The knowledge acquisition algorithm that is the central theme of our book can most naturally be phrased in the language of Formal Concept Analysis. It essentially relies on a single property of concept lattices, namely, that the set of concept intents is closed under intersection. The technique works for an arbitrary intersection-closed family of sets, that is, for a closure system. Readers who are familiar with closure systems but not with Formal Concept Analysis may prefer this approach.