The probability distribution as a computational resource for randomness testing

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 $\mu_p$, $0\le p\le 1$ and the Martin-L\"of 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

Kolmogorov structure functions for automatic complexity

For a finite word $w$ we define and study the Kolmogorov structure function $h_w$ for nondeterministic automatic complexity. We prove upper bounds on $h_w$ that appear to be quite sharp, based on numerical evidence.Comment: Preliminary version: "Kolmogorov structure functions for automatic complexity in computational statistics", Lecture Notes in Comput. Sci., vol. 8881, Springer, Cham, 2014, 652--665, 8th International Conference on Combinatorial Optimization and Applications (COCOA 2014

Models of the Chisholm set

We give a counter-example showing that Carmo and Jones' condition 5(e) may conflict with other conditions on the models in their paper \emph{A new approach to contrary-to-duty obligations}.Comment: Paper for Filosofi hovedfag spesialomr{\aa}de 1 exam, University of Oslo, Fall 1996. First cited in Carmo and Jones, Deontic logic and contrary-to-duties, Handbook of Philosophical Logic, 2002, footnote 2

On the complexity of automatic complexity

Generalizing the notion of automatic complexity of individual strings due to Shallit and Wang, we define the automatic complexity $A(E)$ of an equivalence relation $E$ on a finite set $S$ of strings. We prove that the problem of determining whether $A(E)$ equals the number $|E|$ of equivalence classes of $E$ is $\mathsf{NP}$-complete. The problem of determining whether $A(E) = |E| + k$ for a fixed $k\ge 1$ is complete for the second level of the Boolean hierarchy for $\mathsf{NP}$, i.e., $\mathsf{BH}_2$-complete. Let $L$ be the language consisting of all strings of maximal nondeterministic automatic complexity. We characterize the complexity of infinite subsets of $L$ by showing that they can be co-context-free but not context-free, i.e., $L$ is $\mathsf{CFL}$-immune, but not $\mathsf{coCFL}$-immune. We show that for each $\epsilon>0$, $L_\epsilon\not\in\mathsf{coCFL}$, where $L_\epsilon$ is the set of all strings whose deterministic automatic complexity $A(x)$ satisfies $A(x)\ge |x|^{1/2-\epsilon}$

Automatic complexity of shift register sequences

Let $x$ be an $m$-sequence, a maximal length sequence produced by a linear feedback shift register. We show that $x$ has maximal subword complexity function in the sense of Allouche and Shallit. We show that this implies that the nondeterministic automatic complexity $A_N(x)$ is close to maximal: $n/2-A_N(x)=O(\log^2n)$, where $n$ is the length of $x$. In contrast, Hyde has shown $A_N(y)\le n/2+1$ for all sequences $y$ of length $n$.Comment: Preliminary version: "Shift registers fool finite automata", Lecture Notes in Computer Science 10388 (2017), 170-181, Workshop on Logic, Language, Information and Computation (WoLLIC) 201

Local initial segments of the Turing degrees

Recent results on initial segments of the Turing degrees are presented, and some conjectures about initial segments that have implications for the existence of non-trivial automorphisms of the Turing degrees are indicated

A strong law of computationally weak subsets

We show that in the setting of fair-coin measure on the power set of the natural numbers, each sufficiently random set has an infinite subset that computes no random set. That is, there is an almost sure event $\mathcal A$ such that if $X\in\mathcal A$ then $X$ has an infinite subset $Y$ such that no element of $\mathcal A$ is Turing computable from $Y$

Infinite subsets of random sets of integers

There is an infinite subset of a Martin-L\"of random set of integers that does not compute any Martin-L\"of random set of integers. To prove this, we show that each real of positive effective Hausdorff dimension computes an infinite subset of a Martin-L\"of random set of integers, and apply a result of Miller

Permutations of the integers induce only the trivial automorphism of the Turing degrees

Let $\pi$ be an automorphism of the Turing degrees induces by a homeomorphism $\varphi$ of the Cantor space $2^\omega$ such that $\varphi$ preserves all Bernoulli measures. It is proved that $\pi$ must be trivial. In particular, a permutation of $\omega$ can only induce the trivial automorphism of the Turing degrees

Effective Banach spaces

This thesis addresses Pour-El and Richards' fourth question from their book "Computability in analysis and physics", concerning the relation between higher order recursion theory and computability in analysis. Among other things it is shown that there is a computability structure that is uncountable. The example given is a structure on the Banach space of bounded linear operators on the set of almost periodic functions.Comment: Master's thesis, University of Oslo, 1997. Adviser: Dag Normann. Translated from Norwegian. Original title: "Effektive Banach-rom