Turing Machines on Graphs and Inescapable Groups

TL;DR

This paper generalizes Turing machines through unconventional tapes, classifies these tapes under specific constraints, and explores implications for computability and paths in Cayley graphs of finitely generated groups.

Contribution

It introduces a novel tape-based generalization of Turing machines that offers an alternative perspective on recursively enumerable degrees without using oracles.

Findings

Classified all tapes conforming to certain geometric constraints

Provided an alternative definition of recursively enumerable degrees

Addressed questions about computable paths in Cayley graphs

Abstract

We present a generalization of standard Turing machines based on allowing unusual tapes. We present a set of reasonable constraints on tape geometry and classify all tapes conforming to these constraints. Surprisingly, this generalization does not lead to yet another equivalent formulation of the notion of computable function. Rather, it gives an alternative definition of the recursively enumerable Turing degrees that does not rely on oracles. The definitions give rise to a number of questions about computable paths inside Cayley graphs of finitely generated groups, and several of these questions are answered.