Can the tape of a Turing machine be represented as a discrete function?

Question

I'm just thinking aloud here, and to expand on this, each square of the tape could lie between integers, and discrete y values of the function could map to symbols. That way a given discrete/step function could describe a given program, and vice versa. I'm playing with ideas of somehow using integration on some analytic analogue of such a function to give some indication on whether the program halts.

Answer 1

What you are talking about sounds vaguely close to a form of the Church-Turing thesis, that there is a relation between "recursively enumerable" functions (which are a subset of functions applied to the natural numbers) and Turing machines.

When working with Turing machines and decidability, it's important to note that there's a difference between the contents of the tape at any point in time and the machine itself, but also that:

1. Every combination of Turing Machine + initial tape can be reduced to a TM with a blank initial tape;

2. Every TM with an alphabet of $N$ symbols can be reduced to one with only two symbols; and

3. It is generally impossible to prove whether any given TM and input will halt, and the set of TMs that can be proven to halt or not is infinitesimal compared to the set of all TMs.