A generalization of Chaitin's halting probability $\Omega$ and halting self-similar sets
نویسندگان
چکیده
منابع مشابه
A Generalization of Chaitin's Halting Probability \Omega and Halting Self-Similar Sets
We generalize the concept of randomness in an infinite binary sequence in order to characterize the degree of randomness by a real number D > 0. Chaitin’s halting probability Ω is generalized to ΩD whose degree of randomness is precisely D. On the basis of this generalization, we consider the degree of randomness of each point in Euclidean space through its base-two expansion. It is then shown ...
متن کاملThe Halting Probability Omega: Irreducible Complexity in Pure Mathematics∗
Some Gödel centenary reflections on whether incompleteness is really serious, and whether mathematics should be done somewhat differently, based on using algorithmic complexity measured in bits of information. Introduction: What is mathematics? It is a pleasure for me to be here today giving this talk in a lecture series in honor of Frederigo Enriques. Enriques was a great believer in mathemati...
متن کاملRelativizing Chaitin’s Halting Probability
As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U . Let ΩU be the halting probability of U A; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω . It is this operator which is our primary object of study. We can...
متن کاملChaitin Omega Numbers and Halting Problems
Chaitin [G. 1975] introduced Ω number as a concrete example of random real. The real Ω is defined as the probability that an optimal computer halts, where the optimal computer is a universal decoding algorithm used to define the notion of program-size complexity. Chaitin showed Ω to be random by discovering the property that the first n bits of the base-two expansion of Ω solve the halting prob...
متن کاملHalting probability amplitude of quantum computers
The classical halting probability introduced by Chaitin is generalized to quantum computations. Chaitin's [1, 2, 3] is a magic number. It is a measure for arbitrary programs to take a nite number of execution steps and then halt. It contains the solution for all halting problems, and hence to questions codable into halting problems, such as Fermat's theorem. It contains the solution for the que...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Hokkaido Mathematical Journal
سال: 2002
ISSN: 0385-4035
DOI: 10.14492/hokmj/1350911778