Immunity and Hyperimmunity for Generalized Random Strings
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2007-05-17T09:35:39Z
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Abstract
We introduce generalizations of the Kolmogorov random strings, called minimal index sets. The set of Kolmogorov random strings are known to be immune but not hyperimmune. In contrast, minimal index sets are not only immune but also immune against high levels of the arithmetic hierarchy. We give optimal immunity results with respect to the arithmetic hierarchy and we consider the set MIN* as an intuitive counter-example. Of particular note here are the fact that there are three minimal index sets located in Pi-3 but not in Sigma-3 with distinct immunities and that certain immunity properties depend on the choice of Goedel numbering. We show that minimal index sets are not hyperimmune. As a consequence of this fact, we are able to construct a set which neither contains nor is disjoint from any arithmetic set, yet it is majorized by a recursive function and contains a minimal index set. Lastly, we present generalizations of the Kolmogorov random strings which, unlike the more familiar random strings, need not be Turing complete.