Some questions about the Computability of Turing Machines

Question

I'm learning for a test and I have some important questions about Computability of deterministic and non deterministic Turing Machines.

Consider we have the partial functions $f,g,h,t: \mathbb{N} \rightarrow \mathbb{N}$ with $f$ is Turing Machine computable, $g$ not Turing Machine computable, $h$ is not While solvable and $t$ is While computable. Are the following answers correct? And is there anything changing if we had an non deterministic TM?

There is no proof to do words are ok :)

1. Is $f \circ g$ Turing Machine computable?
2. Is $g \circ f$ Turing Machine computable?
3. Is $t \circ h$ While computable?
4. Is $h \circ t$ While computable?

My answers:

First of all, we know that Turing Machine = While Computability. (#)

1. I would say, that we do not know if it is or is not TM computable, because there may a TM who can handle the output of an not TM computable function.

2. I would say no, because if whatever $g$ takes, it won't be TM computable.

3. and 4. Because of (#) it is the same like 1. and 2.

Could that be right? It is for an multiple choice test and those questions are tricky.

Answer 1

The answer is "we don't know without more information" for all 4.

Suppose $f$ is the identity function $f(x) = x$. Then $f \circ g = g \circ f = g$, which is non-computable.

On the other hand, suppose $g$ is a total non-computable function (which is a special case of $g$ being a partial non-computable function), and suppose $f$ is the constant zero function $f(x) = 0$. In this case:

• $f \circ g$ is the constant zero function, which is obviously computable.
• $g \circ f$ is also a constant function. We don't necessarily know what the constant is, but that doesn't matter; every constant function is computable. See How can it be decidable whether $\pi$ has some sequence of digits?

Deterministic vs. non-deterministic Turing machine doesn't affect the answer, because a deterministic Turing machine can simulate the execution of a non-deterministic Turing machine (potentially with an exponential slowdown) by dovetailing.