Normal and Martin-Löf Random Numbers
نویسندگان
چکیده
We survey the current known relations between four classes of real numbers: Liouville numbers, computable reals, Borel absolutely normal numbers and Martin-Löf random reals. The expansions of the reals will play an important role. The paper refers to the original material and does not repeat the proofs; a characterisation of Liouville numbers in terms of their expansions will be proved. Finally we mention two open problems: are there a) computable, absolutely Borel normal, non-Liouville numbers? and b) Martin-Löf random reals of arbitrary finite irrationality exponent?
منابع مشابه
Martin-löf Reducibility and Cost Functions
Martin-Löf (ML)-reducibility compares K-trivial sets by examining the Martin-Löf random sequences that compute them. We show that every K-trivial set is computable from a c.e. set of the same ML-degree. We investigate the interplay between ML-reducibility and cost functions, which are used to both measure the number of changes in a computable approximation, and the type of null sets used to cap...
متن کاملStrategies for interactive proof and program development in Martin-Löf Type Theory
We propose some strategies for automatic tools facilitating interactive proof and program development in the proof editor Alfa based on Martin-Löf Type Theory.
متن کاملTt-functionals and Martin-löf Randomness for Bernoulli Measures
For r ∈ [0, 1], the Bernoulli measure μr on the Cantor space {0, 1}N assigns measure r to the set of sequences with 1 at a fixed position. In Mauldin [5] it is shown that for r, s ∈ [0, 1], μs is continuously reducible to μr if any only if r and s satisfy certain purely number theoretic conditions (binomial reduciblility). We bring these results into the context of computability theory and Mart...
متن کاملA note on the Martin-Löf test for unidimensionality
One test which is often used to investigate fit of the Rasch model to a dataset, is the Martin-Löf test for unidimensionality. This paper investigates whether (and when) its asymptotic chi-square distribution can be assumed to be appropriate. We also study the power of this test.
متن کاملSome tools for computer-assisted theorem proving in Martin-Löf type theory
We propose some tools facilitating interactive proof and program development in the proof editor Alfa based on Martin-Löf Type Theory, in particular a tool for equality reasoning supported by tools for deriving equality (and proofs or its properties) for inductive datatypes as well as automated proof-search.
متن کامل