Single-tape and Multi-tape Turing machines through the lens of the Grossone methodology

TL;DR

This paper explores how different mathematical languages, inspired by physics, affect the accuracy of describing and observing Single and Multi-tape Turing machines, highlighting the impact of the observation tools on results.

Contribution

It introduces a new mathematical language based on the Grossone methodology to analyze Turing machines, emphasizing the role of observation tools in computational descriptions.

Findings

Traditional and new languages yield different observational results.

The new language improves the precision of describing computational processes.

Comparison shows the influence of language choice on computational analysis.

Abstract

The paper investigates how the mathematical languages used to describe and to observe automatic computations influence the accuracy of the obtained results. In particular, we focus our attention on Single and Multi-tape Turing machines which are described and observed through the lens of a new mathematical language which is strongly based on three methodological ideas borrowed from Physics and applied to Mathematics, namely: the distinction between the object (we speak here about a mathematical object) of an observation and the instrument used for this observation; interrelations holding between the object and the tool used for the observation; the accuracy of the observation determined by the tool. Results of the observation executed by the traditional and new languages are compared and discussed.