Noncomputable Spectral Sets

TL;DR

This paper explores the complexity of spectral sets derived from computability measures, generalizing shortest program sets, and investigates their properties and relationships to classical Turing degrees.

Contribution

It introduces a generalized notion of spectral sets, analyzes their complexity and properties, and establishes connections to canonical Turing degrees under certain conditions.

Findings

Spectral sets can be exactly the canonical sets 0', 0'', 0''' under specific Godel numberings.

Spectral sets always contain some useful computability information.

Constructed a set that is neither disjoint from nor contains any infinite arithmetic set, yet is 0-majorized.

Abstract

It is possible to enumerate all computer programs. In particular, for every partial computable function, there is a shortest program which computes that function. f-MIN is the set of indices for shortest programs. In 1972, Meyer showed that f-MIN is Turing equivalent to 0'', the halting set with halting set oracle. This paper generalizes the notion of shortest programs, and we use various measures from computability theory to describe the complexity of the resulting "spectral sets." We show that under certain Godel numberings, the spectral sets are exactly the canonical sets 0', 0'', 0''', ... up to Turing equivalence. This is probably not true in general, however we show that spectral sets always contain some useful information. We show that immunity, or "thinness" is a useful characteristic for distinguishing between spectral sets. In the final chapter, we construct a set which neither contains nor is disjoint from any infinite arithmetic set, yet it is 0-majorized and contains a natural spectral set. Thus a pathological set becomes a bit more friendly. Finally, a number of interesting open problems are left for the inspired reader.