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Knowledge Spaces

In: Learning Spaces

Author

Listed:
  • Jean-Claude Falmagne

    (University of California, Irvine, Department of Cognitive Sciences, Institute of Mathematical Behavioral Sciences)

  • Jean-Paul Doignon

    (Université Libre de Bruxelles, Département de Mathématique)

Abstract

We have learned from Theorem 2.2.4 that any learning space is a knowledge space, that is, a knowledge structure closed under union. The ∪-closure property is critical for the following reason. Certain knowledge spaces, and in particular the finite ones, can be faithfully summarized by a subfamily of their states. To wit, any state of the knowledge space can be generated by forming the union of some states in the subfamily. When such a subfamily exists and is minimal for inclusion, it is unique and is called the ‘base’ of the knowledge space. In some cases, the base can be considerably smaller than the knowledge space, which results in a substantial economy of storage in a computer memory. The extreme case is the power set of a set of n elements, where the 2n knowledge states can be subsumed by the family of the n singleton sets. This property inspires most of this chapter, beginning with the basic concepts of ‘base’ and ‘atoms’ in Sections 3.4 to 3.6. Other features of knowledge spaces are also important, however, and are dealt with in this chapter.

Suggested Citation

  • Jean-Claude Falmagne & Jean-Paul Doignon, 2011. "Knowledge Spaces," Springer Books, in: Learning Spaces, chapter 3, pages 43-60, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-01039-2_3
    DOI: 10.1007/978-3-642-01039-2_3
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