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- F. Stephan
- Computer ScienceParadoxes and Inconsistent Mathematics
- 21 October 2021
The study of set algebra provides a solid background to understanding of probability and statistics, which are important business decision-making tools.
Randomness, relativization and Turing degrees
It is shown that the notions of Martin-Löfrandomness, recursive randomness, and Schnorr randomness can be separated in every high degree while the same notions coincide in every non-high degree.
Using random sets as oracles
It is shown that the bases for 1‐randomness are exactly the K‐trivial sets, and several consequences of this result are discussed.
Automatic structures: richness and limitations
- B. Khoussainov, A. Nies, S. Rubin, F. Stephan
- MathematicsProceedings of the 19th Annual IEEE Symposium on…
- 13 July 2004
It is proven that the free Abelian group of infinite rank and many Fraisse limits do not have automatic presentations, and the complexity of the isomorphism problem for the class of all automatic structures is /spl Sigma//sub 1//sup 1/-complete.
Hausdorff dimension in exponential time
- K. Ambos-Spies, W. Merkle, Jan Reimann, F. Stephan
- Mathematics, Computer ScienceProceedings 16th Annual IEEE Conference on…
- 18 June 2001
By a new general invariance theorem for resource-bounded dimension, it is shown that the class of p-m-complete sets for E has dimension 1 in E and there are p- m-lower spans in E of dimension /spl Hscr/(/spl beta/) for any rational /spl beta/ between 0 and 1.
Deciding parity games in quasipolynomial time
- Cristian S. Calude, Sanjay Jain, B. Khoussainov, Wei Li, F. Stephan
- Computer Science, MathematicsSTOC
- 19 June 2017
It is shown that the parity game can be solved in quasipolynomial time and it is proven that coloured Muller games with n nodes and m colours can be decided in time O((mm · n)5); it is also shown that this bound cannot be improved to O((2m · n), for any c, unless FPT = W.
Kolmogorov Complexity and the Recursion Theorem
It is argued that for plain Kolmogorov complexity exactly the PA-complete sets compute incompressible words, while the class of sets that compute words of maximum complexity depends on the choice of the universal Turing machine, whereas for prefix-free Kolmogsorov simplicity exactly the complete sets allow to compute wordsof maximum complexity.
Lowness for the Class of Schnorr Random Reals
We answer a question of Ambos-Spies and Kucera in the affirmative. They asked whether, when a real is low for Schnorr randomness, it is already low for Schnorr tests.
Martin-Löf random and PA-complete sets
A set A is Martin-LL of random ii the class fAg does not have 0 1-measure 0. A set A is PA-complete if one can compute relative to A a consistent and complete extension of Peano Arithmetic. It is…