The Probability of a Computable Output from a Random Oracle
- Computer Science, MathematicsACM Trans. Comput. Log.
Surprisingly, it is found that these probabilities are the entire class of real numbers in (0,1) that can be written as the difference of two halting probabilities relative to the halting problem.
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.
SHOWING 1-10 OF 54 REFERENCES
Relativizing Chaitin's Halting Probability
- MathematicsJ. Math. Log.
A comparison is drawn between the jump operator from computability theory and this Omega operator, which is a natural uniform way of producing an A-random real for every A ∈ 2ω, and many other interesting properties of Omega operators.
A Highly Random Number
- Computer Science, MathematicsDMTCS
It is proved that α is a random number that goes beyond Ω, the probability that a universal self delimiting machine halts, and similar to the algorithmic complexity of Ω', the halting probability of an oracle machine.
Series Recursively Enumerable Reals and Chaitin Numbers
A real is called recursively enumerable if it can be approximated by an increasing, recursive sequence of rationals. The halting probability of a universal selfdelimiting Turing machine (Chaitin's…
From index sets to randomness in ∅n: random reals and possibly infinite computations part II
- Mathematics, Computer ScienceThe Journal of Symbolic Logic
A large class of significant examples of n-random reals (i.e., Martin-Löf random in oracle ∅(n−1)) à la Chaitin are obtained and methods to transfer many-one completeness results of index sets to n- randomness of associated probabilities are developed.
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.
Minimum Message Length and Kolmogorov Complexity
- Computer ScienceComput. J.
This work attempts to establish a parallel between a restricted (two-part) version of the Kolmogorov model and the minimum message length approach to statistical inference and machine learning of Wallace and Boulton (1968), in which an ‘explanation’ of a data string is modelled as a two-part message.
Information and Randomness: An Algorithmic Perspective
- Computer Science
Universality probability of a prefix-free machine
- Computer SciencePhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
The notion of universality probability of a universal prefix-free machine is random relative to the third iterate of the halting problem and its Turing degree and its place in the arithmetical hierarchy of complexity is determined.
The typical Turing degree
- Computer Science
A large number of results are described and proved in a new programme of research which aims to establish the (order theoretically) definable properties of the typical Turing degree, and the level of randomness required in order to guarantee typicality.