Abstract
This chapter introduces the notion of a cover or repertoire and its proper descriptions. Based on the new idea of relating covers and descriptions, some interesting properties of covers are defined.
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Notes
- 1.
In addition we usually may require that \(p(A)\neq 0\) for every \(A \in \alpha \).
- 2.
More exactly: for every \(B \in R(d)\) there is \(A \in \alpha \) with \(p(A\bigtriangleup B) = 0\).
- 3.
In line with our general strategy to disregard sets of probability 0, we can interpret \(A \subset d(x)\) as \(p(A \setminus d(x)) = 0\) and \(p(d(x) \setminus A) > 0\).
- 4.
In the definition of D(α), we understand a description d simply as a mapping \(d: \Omega \rightarrow \alpha \) with \(\omega \in d(\omega )\), i.e., without the additional requirement of Definition 2.3.
- 5.
If \(D(\alpha ) = \varnothing \) we obtain \({\alpha }_{c} = \varnothing \), so α c is not a cover. In this case, we add Ω to α c . To avoid this redefinition, one could define α c only for repertoires.
- 6.
- 7.
The flattening of an arbitrary cover may not exist, because \({\alpha }_{f}\) may not be a cover. An example for this is \(\alpha = R({X}^{\geq })\) for a random variable X with \(R(X) = \mathbb{R}\). In this case, α has no maximal elements. If the flattening exists, it is clearly flat. Usually we consider finite covers which guarantees that the flattening exists.
- 8.
Here we are using the countable version of Definition 2.4.
References
Chaitin, G. J. (1975). Randomness and mathematical proof. Scientific American, 232(5), 47–52.
Chaitin, G. J. (1977). Algorithmic information theory. IBM Journal of Research and Development, 21, 350–359.
Chaitin, G. J. (1987). Algorithmic information theory, volume I of Cambridge tracts in theoretical computer science. Cambridge: Cambridge University Press.
Kolmogorov, A. N. (1965). Three approaches to the quantitative definition of information. Problems in Information Transmission, 1, 4–7.
Kolmogorov, A. N. (1968). Logical basis for information theory and probability theory. IEEE Transactions on Information Theory, 14, 662–664.
Palm, G. (1981). Evidence, information and surprise. Biological Cybernetics, 42(1), 57–68.
Palm, G. (1985). Information und entropie. In H. Hesse (Ed.), Natur und Wissenschaft. Tubingen: Konkursbuch Tübingen.
Palm, G. (1996). Information and surprise in brain theory. In G. Rusch, S. J. Schmidt, & O. Breidbach (Eds.), Innere Repräsentationen—Neue Konzepte der Hirnforschung, DELFIN Jahrbuch (stw-reihe edition) (pp. 153–173). Frankfurt: Suhrkamp.
Palm, G. (2007). Information theory for the brain. In V. Braitenberg, & F. Radermacher (Eds.), Interdisciplinary approaches to a new understanding of cognition and consciousness: vol. 20 (pp. 215–244). Wissensverarbeitung und Gesellschaft: die Publikationsreihe des FAW/n Ulm, Ulm.
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Palm, G. (2012). Repertoires and Descriptions. In: Novelty, Information and Surprise. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29075-6_9
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DOI: https://doi.org/10.1007/978-3-642-29075-6_9
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