The Distance Function on a Computable Graph

TL;DR

This paper explores the computability and reducibility properties of the distance function in graphs using computable model theory, introducing new reducibility notions and characterizing their spectra.

Contribution

It adapts truth-table reducibilities to functions, develops new theorems, and characterizes the spectra of the distance function in computable graphs.

Findings

Spectra can be a single btt-degree approximable from above.

Spectra can include all such btt-degrees.

Spectra can be the bT-degrees of functions approximable in at most n steps.

Abstract

We apply the techniques of computable model theory to the distance function of a graph. This task leads us to adapt the definitions of several truth-table reducibilities so that they apply to functions as well as to sets, and we prove assorted theorems about the new reducibilities and about functions which have nonincreasing computable approximations. Finally, we show that the spectrum of the distance function can consist of an arbitrary single btt-degree which is approximable from above, or of all such btt-degrees at once, or of the bT-degrees of exactly those functions approximable from above in at most n steps.