In this survey we discuss work of Levin and V’yugin on collections of sequences that are non-negligible in the sense that they can be computed by a probabilistic algorithm with positive probability. More precisely, Levin and V’yugin introduced an ordering on collections of sequences that are closed under Turing equivalence. Roughly speaking, given two such collections $\mathcal {A}$ and $\mathcal {B}$, $\mathcal {A}$ is below $\mathcal {B}$ in this ordering if $\mathcal {A}\setminus \mathcal {B}$ is negligible. The degree structure associated (...) with this ordering, the Levin–V’yugin degrees, can be shown to be a Boolean algebra, and in fact a measure algebra. We demonstrate the interactions of this work with recent results in computability theory and algorithmic randomness: First, we recall the definition of the Levin–V’yugin algebra and identify connections between its properties and classical properties from computability theory. In particular, we apply results on the interactions between notions of randomness and Turing reducibility to establish new facts about specific LV-degrees, such as the LV-degree of the collection of 1-generic sequences, that of the collection of sequences of hyperimmune degree, and those collections corresponding to various notions of effective randomness. Next, we provide a detailed explanation of a complex technique developed by V’yugin that allows the construction of semi-measures into which computability-theoretic properties can be encoded. We provide two examples of the use of this technique by explicating a result of V’yugin’s about the LV-degree of the collection of Martin-Löf random sequences and extending the result to the LV-degree of the collection of sequences of DNC degree. (shrink)

The strong measure zero sets of reals have been widely studied in the context of set theory of the real line. The notion of strong measure zero is straightforwardly effectivized. A set of reals is said to be of effective strong measure zero if for any computable sequence {εn}n∈N{εn}n∈N of positive rationals, a sequence of intervals InIn of diameter εnεn covers the set. We observe that a set is of effective strong measure zero if and only if it is of (...) measure zero with respect to any outer measure constructed by Monroe's Method from a computable atomless outer premeasure defined on all open balls. This measure-theoretic restatement permits many characterizations of strong measure zero in terms of semimeasures as well as martingales. We show that for closed subsets of Cantor space, effective strong nullness is equivalent to another well-studied notion called diminutiveness , the property of not having a computably perfect subset. Further, we prove that if P is a nonempty effective strong measure zero Π01 set consisting only of noncomputable elements, then some Martin-Löf random reals compute no element in P , and P has an element that computes no autocomplex real. Finally, we construct two different special Π01 sets, one of which is not of effective strong measure zero, but consists only of infinitely-often K-trivial reals, and the other is perfect and of effective strong measure zero, but contains no anti-complex reals. (shrink)

The famous Gödel incompleteness theorem states that for every consistent, recursive, and sufficiently rich formal theory T there exist true statements that are unprovable in T . Such statements would be natural candidates for being added as axioms, but how can we obtain them? One classical approach is to add to some theory T an axiom that claims the consistency of T . In this paper we discuss another approach motivated by Chaitin's version of Gödel's theorem where axioms claiming the (...) randomness of some strings are probabilistically added, and show that it is not really useful, in the sense that this does not help us prove new interesting theorems. This result answers a question recently asked by Lipton. The situation changes if we take into account the size of the proofs: randomly chosen axioms may help making proofs much shorter .We then study the axiomatic power of the statements of type “the Kolmogorov complexity of x exceeds n ” in general. They are Π1Π1 statements of Peano arithmetic. We show that by adding all true statements of this type, we obtain a theory that proves all true Π1Π1-statements, and also provide a more detailed classification. In particular, as Theorem 7 shows, to derive all true Π1Π1-statements it is enough to add one statement of this type for each n if strings are chosen in a special way. On the other hand, one may add statements of this type for most x of length n and still obtain a weak theory. We also study other logical questions related to “random axioms”.Finally, we consider a theory that claims Martin-Löf randomness of a given infinite binary sequence. This claim can be formalized in different ways. We show that different formalizations are closely related but not equivalent, and study their properties. (shrink)