Observability of Turing Machines: a refinement of the theory of computation

The Turing machine is one of the simple abstract computational devices that can be usedto investigate the limits of computability. In this paper, they are considered from several points ofview that emphasize the importance and the relativity of mathematical languages used to describethe Turing machines. A deep investigation is performed on the interrelations between mechanicalcomputations and their mathematical descriptions emerging when a human (the researcher) starts todescribe a Turing machine (the object of the study) by different mathematical languages (the instrumentsof investigation). Together with traditional mathematical languages using such concepts as‘enumerable sets’ and ‘continuum’ a new computational methodology allowing one to measure thenumber of elements of different infinite sets is used in this paper. It is shown how mathematical languagesused to describe the machines limit our possibilities to observe them. In particular, notionsof observable deterministic and non-deterministic Turing machines are introduced and conditionsensuring that the latter can be simulated by the former are established.