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Download Knowledge Processing and Data Analysis: First International by Rudolf Wille (auth.), Karl Erich Wolff, Dmitry E. Palchunov, PDF

By Rudolf Wille (auth.), Karl Erich Wolff, Dmitry E. Palchunov, Nikolay G. Zagoruiko, Urs Andelfinger (eds.)

ISBN-10: 3642221394

ISBN-13: 9783642221392

This e-book constitutes the lawsuits of the 1st overseas convention on wisdom - Ontology - thought (KONT 2007) held in Novosibirsk, Russia, in September 2007 and the 1st foreign convention on wisdom Processing in perform (KPP 2007) held in Darmstadt, Germany, in September 2007. The 21 revised complete papers have been conscientiously reviewed and chosen from a number of submissions and canopy 4 major concentration components: functions of conceptual constructions; proposal established software program; ontologies as conceptual buildings; and information analysis.

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Extra resources for Knowledge Processing and Data Analysis: First International Conference, KONT 2007, Novosibirsk, Russia, September 14-16, 2007 and First International Conference, KPP 2007,Darmstadt, Germany, September 28-30, 2007. Revised Selected Papers

Sample text

Why should we consider pairs that do not actually approximate anything? It turns out that the properly generated elements are exceptional cases, caused by singleton equivalence classes. Simply doubling such points removes the problem, as will be shown below. Before going into more detail, we demonstrate the effect by continuing a small example from [2]. Figure 2 shows an ordered set, representing the indiscernibility (pre-)order of our example. Next to it there is the lattice of its order ideals.

For preorders we even have to distinguish two cases: Isolated points, that is, objects incomparable to all other objects. If u is isolated, then u ∈ R(A) ⇐⇒ u ∈ A ⇐⇒ u ∈ R(A). Moreover, (R(A ∪ {u}), R(A ∪ {u})) = (R(A) ∪ {u}, R(A) ∪ {u}). As a consequence, the lattice of rough set approximations splits into a direct product of the two-element lattice and the lattice of rough set approximations not containing u. , an element v = u such that u ≥ v 32 B. Ganter and v ≥ u. Let u be such an element, and assume that ↓ u = {u}.

Xk } and {Y1 , . . , Ym } is defined as {X1 , . . , Xk } {Y1 , . . t. ≤) elements of X. Here is an example of applying : ⎧ ⎪ CH3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ NH2 ⎪ ⎪ ⎩ ⎫ C C ⎪ OH ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎧ ⎪ CH3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ ⎪ OH ⎪ ⎪ ⎩ ⎫ C C Cl CH3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ C C Cl , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ = ⎪⎪⎪ OH ⎪ ⎪ ⎪ ⎩ C C C C , ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ CH3 ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ The similarity operation on graph sets is commutative: X Y = Y X and associative: (X Y ) Z = X (Y Z). , X X = X holds, is called a pattern of P .

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