Classifying the Arithmetical Complexity of Teaching Models

TL;DR

This paper analyzes the arithmetical complexity of various teaching models and decision problems, classifying their position in the arithmetical hierarchy and providing insights into their computational difficulty.

Contribution

It determines the arithmetical complexity of key classes and decision problems in teaching models, advancing understanding of their computational properties.

Findings

Class of uniformly r.e. families with finite teaching dimension has a specific arithmetical complexity.

Class with finite positive recursive teaching dimension has a determined arithmetical complexity.

Decidability of teaching dimension bounds in effective coding schemes is characterized.

Abstract

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 $d$.