Related papers: Russell's typicality as another randomness notion
We study randomness beyond $\Pi^1_1$-randomness and its Martin-L\"of type variant, introduced in \cite{MR2340241} and further studied in \cite{Continuous-higher-randomness}. The class given by the infinite time Turing machines (\ITTM s),…
This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for…
We adjust the notion of typicality originated with Russell, which was introduced and studied in a previous paper for general first-order structures, to make it expressible in the language of set theory. The adopted definition of the class…
We investigate the role of continuous reductions and continuous relativisation in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with…
This expository paper advocates an approach to physics in which ``typicality" is identified with a suitable form of algorithmic randomness. To this end various theorems from mathematics and physics are reviewed. Their original versions…
In this paper we examine the natural interpretation of a ramified type hierarchy into Martin-L\"of type theory with an infinite sequence of universes. It is shown that under this predicative interpretation some useful special cases of…
A real number \alpha is called recursively enumerable if there exists a computable, increasing sequence of rational numbers which converges to \alpha. The randomness of a recursively enumerable real \alpha can be characterized in various…
A concept of randomness for infinite time register machines (ITRMs), resembling Martin-L\"of-randomness, is defined and studied. In particular, we show that for this notion of randomness, computability from mutually random reals implies…
We investigate the strength of a randomness notion $\mathcal R$ as a set-existence principle in second-order arithmetic: for each $Z$ there is an $X$ that is $\mathcal R$-random relative to $Z$. We show that the equivalence between…
The Turing degree of a real measures the computational difficulty of producing its binary expansion. Since Turing degrees are tailsets, it follows from Kolmogorov's 0-1 law that for any property which may or may not be satisfied by any…
We generalise the randomness test definitions in the literature for both the Martin-L\"of and Schnorr randomness of a series of binary outcomes, in order to allow for interval-valued rather than merely precise forecasts for these outcomes,…
A real \alpha is called recursively enumerable ("r.e." for short) if there exists a computable, increasing sequence of rationals which converges to \alpha. It is known that the randomness of an r.e. real \alpha can be characterized in…
We study the statistical properties of random numbers under the Martin-L\"of definition of randomness, proving that random numbers obey analogues of Strong Law of Large Numbers, the Law of the Iterated Logarithm, and that they are normal.…
We study generalizations of Demuth's Theorem, which states that the image of a Martin-L\"of random real under a tt-reduction is either computable or Turing equivalent to a Martin-L\"of random real. We show that Demuth's Theorem holds for…
We correct Miyabe's proof of van Lambalgen's Theorem for truth-table Schnorr randomness (which we will call uniformly relative Schnorr randomness). An immediate corollary is one direction of van Lambalgen's theorem for Schnorr randomness.…
In previous work we defined and studied a notion of typicality, originated with B. Russell, for properties and objects in the context of general infinite first-order structures. In this paper we consider this notion in the context of finite…
Martin-Lof's definition of random sequences of cbits as those not belonging to any set of constructive zero Lebesgue measure is reformulated in the language of Algebraic Probability Theory. The adoption of the Pour-El Richards theory of…
The van Lambalgen theorem is a surprising result in algorithmic information theory concerning the symmetry of relative randomness. It establishes that for any pair of infinite sequences $A$ and $B$, $B$ is Martin-L\"of random and $A$ is…
In this paper, we will generalize the definition of partially random or complex reals, and then show the duality of random and complex, i.e., a generalized version of Levin-Schnorr's theorem. We also study randomness from the view point of…
In characteristic zero, we construct relative principalization of ideals for logarithmically regular morphisms of logarithmic schemes, and use it to construct logarithmically regular desingularization of morphisms. These constructions are…