Random numbers as probabilities of machine behavior

  title={Random numbers as probabilities of machine behavior},
  author={George Barmpalias and Douglas A. Cenzer and Christopher P. Porter},
The Probability of a Computable Output from a Random Oracle
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.
Randomness and uniform distribution modulo one
Aspects of Chaitin's Omega
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.
It is proved that the continuous function $X:2^\omega \to $ for all $X is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic.


Relativizing Chaitin's Halting Probability
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
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
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
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.
The Definition of Random Sequences
Minimum Message Length and Kolmogorov Complexity
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.
Universality probability of a prefix-free machine
  • George Barmpalias, D. Dowe
  • Computer Science
    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2012
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
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.