Genericity and measure for exponential time

AbstractRecently, Lutz [14, 15] introduced a polynomial time bounded version of Lebesgue measure. He and others (see e.g. [11, 13–18, 20]) used this concept to investigate the quantitative structure of Exponential Time (E = DTIME(2lin)). Previously, Ambos-Spies et al. [2, 3] introduced polynomial time bounded genericity concepts and used them for the investigation of structural properties of NP (under appropriate assumptions) and E. Here we relate these concepts to each other. We show that, for any c ⩾ 1, the class of nc-generic sets has p-measure 1. This allows us to simplify and extend certain p-measure 1-results. To illustrate the power of generic sets we take the Small Span Theorem of Juedes and Lutz [11] as an example and prove a generalization for bounded query reductions

Relatively computably enumerable reals

A real X is defined to be relatively c.e. if there is a real Y such that X is c.e.(Y) and Y does not compute X. A real X is relatively simple and above if there is a real Y <_T X such that X is c.e.(Y) and there is no infinite subset Z of the complement of X such that Z is c.e.(Y). We prove that every nonempty Pi^0_1 class contains a member which is not relatively c.e. and that every 1-generic real is relatively simple and above.Comment: 5 pages. Significant changes from earlier versio

Computable randomness is about more than probabilities

We introduce a notion of computable randomness for infinite sequences that generalises the classical version in two important ways. First, our definition of computable randomness is associated with imprecise probability models, in the sense that we consider lower expectations (or sets of probabilities) instead of classical 'precise' probabilities. Secondly, instead of binary sequences, we consider sequences whose elements take values in some finite sample space. Interestingly, we find that every sequence is computably random with respect to at least one lower expectation, and that lower expectations that are more informative have fewer computably random sequences. This leads to the intriguing question whether every sequence is computably random with respect to a unique most informative lower expectation. We study this question in some detail and provide a partial answer

A note on the differences of computably enumerable reals

We show that given any non-computable left-c.e. real α there exists a left-c.e. real β such that α≠β+γ for all left-c.e. reals and all right-c.e. reals γ. The proof is non-uniform, the dichotomy being whether the given real α is Martin-Loef random or not. It follows that given any universal machine U, there is another universal machine V such that the halting probability of U is not a translation of the halting probability of V by a left-c.e. real. We do not know if there is a uniform proof of this fact

Computable randomness is about more than probabilities

We introduce a notion of computable randomness for infinite sequences that generalises the classical version in two important ways. First, our definition of computable randomness is associated with imprecise probability models, in the sense that we consider lower expectations (or sets of probabilities) instead of classical 'precise' probabilities. Secondly, instead of binary sequences, we consider sequences whose elements take values in some finite sample space. Interestingly, we find that every sequence is computably random with respect to at least one lower expectation, and that lower expectations that are more informative have fewer computably random sequences. This leads to the intriguing question whether every sequence is computably random with respect to a unique most informative lower expectation. We study this question in some detail and provide a partial answer

On the structure of the polynomial time degrees of recursive sets

SIGLETIB: RN 4237 (206) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman