Optimal Representation in Average Using Kolmogorov Complexity

Abstract One knows from the Algorithmic Complexity Theory 1 [2–5, 8, 14] that a word is incompressible on average. For words of pattern xm, it is natural to believe that providing x and m is an optimal average representation. On the contrary, for words like x ⊕ y (i.e., the bit to bit x or between x and y), providing x and y is not an optimal description on average. In this work, we sketch a theory of average optimal representation that formalizes natural ideas and operates where intuition does not suffice. First, we formulate a definition of K-optimality on average for a pattern, then demonstrate results that corroborate intuitive ideas, and give worthy insights into the best compression in more complex cases.

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