The Grossone methodology perspective on Turing machines

This chapter discusses how the mathematical language used to describe and to observe automatic computations influences the accuracy of the obtained results. The chapter presents results obtained by describing and observing different kinds ofTuringmachines (single andmulti-tape, deterministic and non-deterministic)through the lens of a new mathematical language named Grossone. This emerging language is strongly based on three methodological ideas borrowed from Physics and applied toMathematics: the distinction between the object (indeed 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. In the chapter, the new results are compared to those achievable by using traditional languages. It is shown that both languages do not contradict each other but observe and describe the same object (Turing machines) but with different accuracies.