2 Citations
Aspects of Chaitin's Omega
- Computer ScienceAlgorithmic Randomness
- 2020
The purpose of this survey is to expose developments and tell a story about Omega, which outlines its multifaceted mathematical properties and roles in algorithmic randomness.
Compression of Data Streams Down to Their Information Content
- Computer ScienceIEEE Transactions on Information Theory
- 2019
A new coding method is devised that uniformly codes every stream into an algorithmically random stream, providing a strong analogue of Shannon’s source coding theorem for the algorithmic information theory.
References
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Compressibility of Infinite Binary Sequences
- Computer Science, Mathematicscomplexity, logic, and recursion theory
- 2019
This work proposes some definitions, based on Kobayashi's notion of compressibility, of the polynomial-time computable sequences, and compares them to both the standard resource-bounded Kolmogorov complexity of infinite strings, and the uniform complexity.
On Innnite Sequences (almost) as Easy As
- Computer Science, Mathematics
- 1994
This work proposes some deeni-tions, based on Kobayashi's notion of compressibility, and compares them to the standard resource-bounded Kolmogorov complexity of innnite strings, and some non-trivial coincidences and disagreements are proved.
Chaitin Ω numbers and halting problems
- Computer Science
- 2009
The relative computational power between the base-two expansion of Ω and the halting problem by imposing the restriction to finite size on both the problems is considered.
Optimal asymptotic bounds on the oracle use in computations from Chaitin's Omega
- Mathematics, Computer ScienceJ. Comput. Syst. Sci.
- 2016
Randomness and halting probabilities
- MathematicsJournal of Symbolic Logic
- 2006
It follows that for any optimal machine U and any sufficiently small real r, there is a set X ⊆ 2≤ω recursive in ∅′ ⊕ r, such that ΩU[X] = r.
Kolmogorov Complexity and Instance Complexity of Recursively Enumerable Sets
- Computer Science, MathematicsSIAM J. Comput.
- 1996
The main part of the paper is concerned with instance complexity, introduced by Ko, Orponen, Schoning, and Watanabe in 1986, as a measure of the complexity of individual instances of a decision problem, and it is shown that for every r.
Computing halting probabilities from other halting probabilities
- Computer Science, MathematicsTheor. Comput. Sci.
- 2017
Anti-Complex Sets and Reducibilities with Tiny Use
- PhilosophyThe Journal of Symbolic Logic
- 2013
This work shows the equivalence of anti-complexity and other lowness notions such as r.e. traceability or being weak truth-table reducible to a Schnorr trivial set, and investigates its range and the range of its uniform counterpart.
Algorithmic Randomness and Complexity
- Computer Science, MathematicsTheory and Applications of Computability
- 2010
This chapter discusses Randomness-Theoretic Weakness, Omega as an Operator, Complexity of C.E. Sets, and other Notions of Effective Randomness.