Approximation of subsets of natural numbers by c.e. sets

TL;DR

This paper explores a new approximation method for subsets of natural numbers using computably enumerable sets, focusing on maximal and irregular sets, and analyzes their relationships within computability theory.

Contribution

It introduces a novel approximation approach for non-c.e. sets based on maximal subset content and categorizes sets into regular and irregular types.

Findings

Defined A.maximal sets within non-c.e. sets.

Characterized c.regular and c.irregular non-c.e. sets.

Analyzed the relationships between c.e. subsets of irregular sets.

Abstract

The approximation of natural numbers subsets has always been one of the fundamental issues in computability theory. Computable approximation, $\Delta_2$-approximation, as well as introducing the generically computable sets have been some efforts for this purpose. In this paper, a type of approximation for natural numbers subsets by computably enumerable sets will be examined. For an infinite and non-c.e set, $W_i$ will be an $A$.maximal (maximal inside $A$) if $W_i \subseteq A$, is infinite and $\forall j (W_i \subseteq W_j \subseteq A) \to \Delta (W_i, W_j )< \infty$, where $\Delta$ is the symmetric difference of the two sets. In this study, the natural numbers subsets will be examined from the maximal subset contents point of view, and we will categorize them on this basis. We will study c.regular sets that are non-c.e. and include a maximal set inside themselves, and c.irregular sets that are non-c.e. and non-immune sets which do not include maximal sets. Finally, we study the graph of relationship between c.e. subsets of c.irregular sets.