arXiv subjects

Achilles A. Beros

Publications and source records attributed to Achilles A. Beros.

9 recordsLinked to original sources

arXiv2020

From eventually different functions to pandemic numberings

A function is strongly non-recursive (SNR) if it is eventually different from each recursive function. We obtain hierarchy results for the mass problems associated with computing such functions with varying growth bounds. In particular, there is no least and no greatest Muchnik degree among those of the form SNR$_f$ consisting of SNR functions bounded by varying recursive bounds $f$. We show that the connection between SNR functions and canonically immune sets is, in a sense, as strong as that between DNR (diagonally non-recursive) functions and effectively immune sets. Finally, we introduce pandemic numberings, a set-theoretic dual to immunity.

math.LO↗

arXiv2019

Planar digraphs for automatic complexity

We show that the digraph of a nondeterministic finite automaton witnessing the automatic complexity of a word can always be taken to be planar. In the case of total transition functions studied by Shallit and Wang, planarity can fail. Let $s_q(n)$ be the number of binary words $x$ of length $n$ having nondeterministic automatic complexity $A_N(x)=q$. We show that $s_q$ is eventually constant for each $q$ and that the eventual constant value of $s_q$ is computable.

cs.FL↗

arXiv2017

Effective bi-immunity and randomness

We study the relationship between randomness and effective bi-immunity. Greenberg and Miller have shown that for any oracle X, there are arbitrarily slow-growing DNR functions relative to X that compute no ML random set. We show that the same holds when ML randomness is replaced with effective bi-immunity. It follows that there are sequences of effective Hausdorff dimension 1 that compute no effectively bi-immune set. We also establish an important difference between the two properties. The class Low(MLR, EBI) of oracles relative to which every ML random is effectively bi-immune contains the jump-traceable sets, and is therefore of cardinality continuum.

math.LO↗

arXiv2017

Canonical immunity and genericity

Whereas the usual notions of immunity -- e.g., immunity, hyperimmunity, etc. -- are associated with Cohen genericity, canonical immunity, as introduced by Beros, Khan and Kjos-Hanssen, is associated instead with Mathias genericity. Specifically, every Mathias generic is canonically immune and no Cohen 2-generic computes a canonically immune set.

math.LO↗

arXiv2016

Classifying the Arithmetical Complexity of Teaching Models

This paper classifies the complexity of various teaching models by their position in the arithmetical hierarchy. In particular, we determine the arithmetical complexity of the index sets of the following classes: (1) the class of uniformly r.e. families with finite teaching dimension, and (2) the class of uniformly r.e. families with finite positive recursive teaching dimension witnessed by a uniformly r.e. teaching sequence. We also derive the arithmetical complexity of several other decision problems in teaching, such as the problem of deciding, given an effective coding $\{\mathcal L_0,\mathcal L_1,\mathcal L_2,\ldots\}$ of all uniformly r.e. families, any $e$ such that $\mathcal L_e = \{L^e_0,L^e_1,\ldots,\}$, any $i$ and $d$, whether or not the teaching dimension of $L^e_i$ with respect to $\mathcal L_e$ is upper bounded by $d$.

math.LO↗

arXiv2016

Index Sets of Universal Codes

We examine sets of codes such that certain properties are invariant under the choice of oracle from a range of possible oracles and establish a connection between such codes and Medvedev reductions. In examing the complexity of such sets of \emph{universal codes}, we prove completeness results at various levels of the arithmetic hierarchy as well as two general theorems for obtaining $Π_1^1$-completeness for sets of universal codes. Among other corollaries, we show that the set of codes for Medvedev reductions of bi-immune sets to DNC functions is $Π_1^1$-complete.

math.LO↗

arXiv2016

Teachers, Learners and Oracles

We exhibit a family of computably enumerable sets which can be learned within polynomial resource bounds given access only to a teacher, but which requires exponential resources to be learned given access only to a membership oracle. In general, we compare the families that can be learned with and without teachers and oracles for four measures of efficient learning.

math.LO↗

arXiv2014

Normal numbers and limit computable Cantor series

Given any oracle, A, we construct a basic sequence Q, computable in the jump of A, such that no A-computable real is Q-distribution-normal. A corollary to this is that there is a Delta^0_{n+1} basic sequence with respect to which no Delta^0_n real is distribution-normal. As a special case, there is a limit computable sequence relative to which no computable real is distribution-normal.

math.LO↗

arXiv2014

A DNC function that computes no effectively bi-immune set

In Diagonally Non-Computable Functions and Bi-Immunity, Carl Jockusch and Andrew Lewis proved that every DNC function computes a bi-immune set. They asked whether every DNC function computes an effectively bi-immune set. We construct a DNC function that computes no effectively bi-immune set, thereby answering their question in the negative.

math.LO↗