Aspects of Chaitin's Omega
- Computer ScienceAlgorithmic Randomness
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
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.
SHOWING 1-10 OF 42 REFERENCES
Compressibility of Infinite Binary Sequences
- Computer Science, Mathematicscomplexity, logic, and recursion theory
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
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
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.
Randomness and halting probabilities
- MathematicsJournal of Symbolic Logic
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.
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.
Anti-Complex Sets and Reducibilities with Tiny Use
- PhilosophyThe Journal of Symbolic Logic
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.