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A Geometry of ApproximationConcrete and Formal Information Constructions

A Geometry of Approximation: Concrete and Formal Information Constructions Chapter 2 Concrete and Formal Information Constructions 2.1 Concrete and Formal Observation Spaces Now let us come back to the observation systems. First of all, let us make an inventory of the basic actions one is allowed to do in a generic P-system G, M, : • Any point of G may be uniquely associated with a subset of M (collecting the observed properties of the given point). • Any property of M may be associated with a subset of G (col- lecting the points manifesting the given property). It seems to be a really poor stock of actions. Can we mine meaningful information by means of the mathematical tools so far introduced? Can one define a “conceptual geometry” on G and M using this machinery? It will be seen that observation systems together with the math- ematical machinery so far introduced are sufficient to define a rather complex and comprehensive theory. We shall start with an ‘observation function’ obs : G −→ ℘(M ), defined by setting: b ∈ obs(a) ⇔ a  b. (2.1.1) Technically, obs is what is called a constructor because it builds-up a set from a point. Obviously, for each point a, obs(a)is {b http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Geometry of ApproximationConcrete and Formal Information Constructions

Part of the Trends in Logic Book Series (volume 27)
Editors: Pagliani, Piero; Chakraborty, Mihir
Springer Journals — Jan 1, 2008
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Publisher
Springer Netherlands
Copyright
© Springer Netherlands 2008
ISBN
978-1-4020-8621-2
Pages
43 –71
DOI
10.1007/978-1-4020-8622-9_2
Publisher site
See Chapter on Publisher Site

Abstract

Chapter 2 Concrete and Formal Information Constructions 2.1 Concrete and Formal Observation Spaces Now let us come back to the observation systems. First of all, let us make an inventory of the basic actions one is allowed to do in a generic P-system G, M, : • Any point of G may be uniquely associated with a subset of M (collecting the observed properties of the given point). • Any property of M may be associated with a subset of G (col- lecting the points manifesting the given property). It seems to be a really poor stock of actions. Can we mine meaningful information by means of the mathematical tools so far introduced? Can one define a “conceptual geometry” on G and M using this machinery? It will be seen that observation systems together with the math- ematical machinery so far introduced are sufficient to define a rather complex and comprehensive theory. We shall start with an ‘observation function’ obs : G −→ ℘(M ), defined by setting: b ∈ obs(a) ⇔ a  b. (2.1.1) Technically, obs is what is called a constructor because it builds-up a set from a point. Obviously, for each point a, obs(a)is {b

Published: Jan 1, 2008

Keywords: Formal Operator; Functional Relation; Complete Lattice; Formal Concept Analysis; Nearness Relation

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