arXiv subjects
Publications and source records attributed to Bjørn Kjos-Hanssen.
Let $π$ be an automorphism of the Turing degrees induces by a homeomorphism $φ$ of the Cantor space $2^ω$ such that $φ$ preserves all Bernoulli measures. It is proved that $π$ must be trivial. In particular, a permutation of $ω$ can only induce the trivial automorphism of the Turing degrees.
Cilibrasi and Vitanyi have demonstrated that it is possible to extract the meaning of words from the world-wide web. To achieve this, they rely on the number of webpages that are found through a Google search containing a given word and they associate the page count to the probability that the word appears on a webpage. Thus, conditional probabilities allow them to correlate one word with another word's meaning. Furthermore, they have developed a similarity distance function that gauges how closely related a pair of words is. We present a specific counterexample to the triangle inequality for this similarity distance function.
We study Doob's martingale convergence theorem for computable continuous time martingales on Brownian motion, in the context of algorithmic randomness. A characterization of the class of sample points for which the theorem holds is given. Such points are given the name of Doob random points. It is shown that a point is Doob random if its tail is computably random in a certain sense. Moreover, Doob randomness is strictly weaker than computable randomness and is incomparable with Schnorr randomness.
A real $x$ is $Δ^1_1$-Kurtz random ($Π^1_1$-Kurtz random) if it is in no closed null $Δ^1_1$ set ($Π^1_1$ set). We show that there is a cone of $Π^1_1$-Kurtz random hyperdegrees. We characterize lowness for $Δ^1_1$-Kurtz randomness as being $Δ^1_1$-dominated and $Δ^1_1$-semi-traceable.
We consider the individual points on a Martin-Löf random path of Brownian motion. We show (1) that Khintchine's law of the iterated logarithm holds at almost all points; and (2) there exist points (besides the trivial example of the origin) having effective dimension $<1$. The proof of (1) shows that for almost all times $t$, the path $f$ is Martin-Löf random relative to $t$ and so the effective dimension of $(t,f(t))$ is 2.
We show that positive measure domination implies uniform almost everywhere domination and that this proof translates into a proof in the subsystem WWKL$_0$ (but not in RCA$_0$) of the equivalence of various Lebesgue measure regularity statements introduced by Dobrinen and Simpson. This work also allows us to prove that low for weak $2$-randomness is the same as low for Martin-Löf randomness (a result independently obtained by Nies). Using the same technique, we show that $\leq_{LR}$ implies $\leq_{LK}$, generalizing the fact that low for Martin-Löf randomness implies low for $K$.
We answer a question of Ambos-Spies and Kučera in the affirmative. They asked whether, when a real is low for Schnorr randomness, it is already low for Schnorr tests.
We study the reverse mathematics and computability-the\-o\-re\-tic strength of (stable) Ramsey's Theorem for pairs and the related principles COH and DNR. We show that SRT$^2_2$ implies DNR over RCA$_0$ but COH does not, and answer a question of Mileti by showing that every computable stable $2$-coloring of pairs has an incomplete $Δ^0_2$ infinite homogeneous set. We also give some extensions of the latter result, and relate it to potential approaches to showing that SRT$^2_2$ does not imply RT$^2_2$.
We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable locally finite lattice) is isomorphic to an initial segment of $\mathcal{D}_{h}$. Corollaries include the decidability of the two quantifier theory of $% \mathcal{D}_{h}$ and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of $ω_{1}^{CK}$. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve $ω_{1}$. On the other hand, we construct countable lattices that are not isomorphic to an initial segment of $\mathcal{D}_{h}$.
When testing a set of data for randomness according to a probability distribution that depends on a parameter, access to this parameter can be considered as a computational resource. We call a randomness test Hippocratic if it is not permitted to access this resource. In these terms, we show that for Bernoulli measures $μ_p$, $0\le p\le 1$ and the Martin-Löf randomness model, Hippocratic randomness of a set of data is the same as ordinary randomness. The main idea of the proof is to first show that from Hippocrates-random data one can Turing compute the parameter $p$. However, we show that there is no single Hippocratic randomness test such that passing the test implies computing $p$, and in particular there is no universal Hippocratic randomness test.
The members of Martin-Löf random closed sets under a distribution studied by Barmpalias et al. are exactly the infinite paths through Martin-Löf random Galton--Watson trees with survival parameter $\frac{2}{3}$. To be such a member, a sufficient condition is to have effective Hausdorff dimension strictly greater than $γ=\log_2 \frac{3}{2}$, and a necessary condition is to have effective Hausdorff dimension greater than or equal to $γ$.
In algorithmic randomness, when one wants to define a randomness notion with respect to some non-computable measure $λ$, a choice needs to be made. One approach is to allow randomness tests to access the measure $λ$ as an oracle (which we call the "classical approach"). The other approach is the opposite one, where the randomness tests are completely effective and do not have access to the information contained in $λ$ (we call this approach "Hippocratic"). While the Hippocratic approach is in general much more restrictive, there are cases where the two coincide. The first author showed in 2010 that in the particular case where the notion of randomness considered is Martin-Löf randomness and the measure $λ$ is a Bernoulli measure, classical randomness and Hippocratic randomness coincide. In this paper, we prove that this result no longer holds for other notions of randomness, namely computable randomness and stochasticity.
We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-Löf, computably, Schnorr, and Kurtz random sets, weakly 1-generics and their complementary classes, we find that there exist characterizations of the third and fourth levels of the arithmetic hierarchy purely in terms of these notions. More generally, there exists an equivalence between arithmetic complexity and existence of numberings for classes of left-r.e. sets with shift-persistent elements. While some classes (such as Martin-Löf randoms and Kurtz non-randoms) have left-r.e. numberings, there is no canonical, or acceptable, left-r.e. numbering for any class of left-r.e. randoms. Finally, we note some fundamental differences between left-r.e. numberings for sets and reals.
We prove that superhigh sets can be jump traceable, answering a question of Cole and Simpson. On the other hand, we show that such sets cannot be weakly 2-random. We also study the class superhigh$^\Diamond$, and show that it contains some, but not all, of the noncomputable $K$-trivial sets.
We consider two axioms of second-order arithmetic. These axioms assert, in two different ways, that infinite but narrow binary trees always have infinite paths. We show that both axioms are strictly weaker than Weak König's Lemma, and incomparable in strength to the dual statement (WWKL) that wide binary trees have paths.
There is an infinite subset of a Martin-Löf random set of integers that does not compute any Martin-Löf random set of integers. To prove this, we show that each real of positive effective Hausdorff dimension computes an infinite subset of a Martin-Löf random set of integers, and apply a result of Miller.
We divide the class of infinite computable trees into three types. For the first and second types, $0'$ computes a nontrivial self-embedding while for the third type $0''$ computes a nontrivial self-embedding. These results are optimal and we obtain partial results concerning the complexity of nontrivial self-embeddings of infinite computable trees considered up to isomorphism. We show that every infinite computable tree must have either an infinite computable chain or an infinite $Π^0_1$ antichain. This result is optimal and has connections to the program of reverse mathematics.
We prove various results on effective numberings and Friedberg numberings of families related to algorithmic randomness. The family of all Martin-Löf random left-computably enumerable reals has a Friedberg numbering, as does the family of all $Π^0_1$ classes of positive measure. On the other hand, the $Π^0_1$ classes contained in the Martin-Löf random reals do not even have an effective numbering, nor do the left-c.e. reals satisfying a fixed randomness constant. For $Π^0_1$ classes contained in the class of reals satisfying a fixed randomness constant, we prove that at least an effective numbering exists.