Number of Turing Machines?

Question

Do you have any idea about this question? I do not know what it asks.

-What is the number of Turing machines with with the state set of (Q-start, Q2, Q3, Q4, Q5, Q6, Q-accept, Q-reject),input alphabet(0,1)and tape alphabet (0,1,x,U) where U is the blank symbol? The start , accept and reject states are the ones with the appropriate names. Show your work.

Answer 1

We can assume single-tape deterministic Turing machines are the model intended. Also assume all Turing machines begin in Q-start with the tape head pointed at the left-most non-blank symbol (or any blank symbol, if the tape is entirely blank).

At each stage, the TM:

1. Reads a symbol from the tape.
2. Chooses the next state to visit.
3. Chooses what to write on the tape.
4. Chooses whether to move the head left/still/right.

We have 8 states, 4 tape symbols (to read/write), and 3 movement options (note: your model may require the tape head move left or right; then use 2 instead).

There are 6 states in which we must specify behavior. Assuming crashes aren't allowed (i.e., all behaviors are accounted for and handled somehow) then we have 6 x 4 x 8 x 4 x 3 = 2,304 Turing machines. If we allow crashing, we can alter our calculation to 6 x 4 x (1 + 8 x 4 x 3) = 2,328 Turing machines. The +1 allows each state and read tape symbol either to crash or to define a response (state x write x move).

I might have missed some considerations in there, but I take the question to be asking for this kind of analysis.