Preliminaries (original) (raw)

Abstract

Suppose we want to describe a given object by a finite binary string. We do not care whether the object has many descriptions; however, each description should describe but one object. From among all descriptions of an object we can take the length of the shortest description as a measure of the object's complexity. It is natural to call an object ‘simple’ if it has at least one short description, and to call it ‘complex’ if all of its descriptions are long.

Notes

    1. If we regard a symbol as literally printed on a square, we may suppose that the square is 0 < x < 1, 0 < y < 1. The symbol is defined as the set of points in this square, viz., the set occupied by printer's ink. If these sets are restricted to be measurable, we can define the ‘distance’ between two symbols as the cost of transforming one symbol into the other if the cost of moving a unit area of printer's ink unit distance is unity, and there is an infinite supply of ink at x = 2, y = 0. With this topology the symbols form a conditionally compact space [Turing's note].

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Authors and Affiliations

  1. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada
    Ming Li
  2. Centrum voor Wiskunde en Informatica, Kruislaan 413, 1098 SJ, Amsterdam, The Netherlands
    Paul Vitányi

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  1. Ming Li
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  2. Paul Vitányi
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Correspondence toMing Li .

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Li, M., Vitányi, P. (2008). Preliminaries. In: An Introduction to Kolmogorov Complexity and Its Applications. Texts in Computer Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-49820-1\_1

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