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Using Closed Sets to Model Cognitive Behavior

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Artificial Life and Computational Intelligence (ACALCI 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9592))

Abstract

We introduce closed sets, which we will call knowledge units, to represent tight collections of experience, facts, or skills, etc. Associated with each knowledge unit is the notion of its generators consisting of those attributes which characterize it.

Using these closure concepts, we then provide a rigorous mathematical model of learning in terms of continuous transformations. We illustrate the behavior of transformations by means of closure lattices, and provide necessary and sufficient criteria for simple transformations to be continuous. By using a rigorous definition, one can derive necessary alternative properties of cognition which may be more easily observed in experimental situations.

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Notes

  1. 1.

    This finiteness constraint can be relaxed somewhat, but there is relatively little yield for the resulting complexity.

  2. 2.

    If for all closed sets K, there is a unique minimal generating set, the closure operator is said to be antimatroid. While antimatroid knowledge systems, such as [4, 5], are mathematically most interesting, they seem, in practice, to be most rare.

  3. 3.

    Over 400 references can be found at the web site \(<\)cord.hockemeyer@uni-graz.at\(>\).

  4. 4.

    Because U is discrete, there always is a “next” set above \(K_i\) in \(\mathcal{L}\), unless \(K_i = U\), the maximal element.

  5. 5.

    In artificial intelligence (A.I.), learning is said to be “monotonic” if no new piece of information can invalidate any existing “knowledge” as represented by a set of rules. That concept of knowledge involves a notion of logical contradiction, not just the simple inclusion or deletion of experiential input. There is an abundance of literature about A.I. architectures which support both monotonic and non-monotonic reasoning [13, 17]. Our use of the term is rather different.

  6. 6.

    A real function \(y = f(x)\) is said to be continuous if for any open set \(O_y\) containing y, there exists an open set \(O_x\) containing x such that \(f(O_x) \subseteq O_y = O_{f(x)}\), or using suffix notation \(x.O.f \subseteq y.f.O'\).

  7. 7.

    Note that the \(\eta \) operator is normally neither expansive nor monotone.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Virginia, Charlottesville, USA

    John L. Pfaltz

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  1. John L. Pfaltz
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Correspondence to John L. Pfaltz .

Editor information

Editors and Affiliations

  1. Canberra, Aust Capital Terr, Australia

    Tapabrata Ray

  2. University of New South Wales, Canberra, Aust Capital Terr, Australia

    Ruhul Sarker

  3. RMIT University, Melbourne, Victoria, Australia

    Xiaodong Li

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Pfaltz, J.L. (2016). Using Closed Sets to Model Cognitive Behavior. In: Ray, T., Sarker, R., Li, X. (eds) Artificial Life and Computational Intelligence. ACALCI 2016. Lecture Notes in Computer Science(), vol 9592. Springer, Cham. https://doi.org/10.1007/978-3-319-28270-1_2

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  • DOI: https://doi.org/10.1007/978-3-319-28270-1_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-28269-5

  • Online ISBN: 978-3-319-28270-1

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