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By Edsger W. Dijkstra, W. H. J. Feijen, Joke Sterringa

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A Method of Programming

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Using Euler's equation eix = cos x + i sin x. Such a description is not only compact, but it can be used to generate arbitrarily long trigonometric tables. The above method fails to be adequate for empirical data. For instance, consider the collection of gold medal winners in the Olympic Games since 1896 (see Rozenberg and Salomaa [199]). For such information the amount of compression is practically null, especially if attention is restricted to the least significant digits. Moreover, since the tendency is for (slow) improvement, the most significant digits have a kind of regularity which even makes predictions possible.

Accordingly, in view of the Invariance Theorem, for infinitely many i > 0, we have: o This yields a contradiction. 4 Quantitative Estimates In this section we derive some elementary estimations for (Chaitin) absolute complexities. Similar results can be obtained for the conditional complexities. Sharper estimations, deserving more involved proofs, will be presented later. 21. There exists a natural constant c > 0 such that for all x E A+, K(x) :::; Ixl + c, H(x) :::; Ixl + 2 log Ixl + C. 21) 34 3.

If x, y E A* are minimal free-strings and x Ixl ~ Iyl· « y, then Proof Assume, by absurdity, that Ixl < Iyl· Take x'

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