# For Turing machines, if the input variables increase, will the state set Q increase ? will the tape alphabet Γ increase?

For Turing machines, if the input variables increase, will the state set Q increase ? will the tape alphabet Γ increase?

For example, for the SAT problem, the first question is whether the Boolean expression of one variable is satisfied, the second is whether the Boolean expression of three variables is satisfied. Does the latter need a larger state set Q and a larger tape alphabet Γ than the former?

The book Introduction to the Theory of Computation that I've been using for reference define a Turing machine as follows:

A Turing machine is a 7-tuple, (Q,Σ,Γ,δ,q_0 ,q_accept ,q_reject ), where Q, Σ, Γ are all finite sets and

1. Q is the set of states,
2. Σ is the input alphabet not containing the blank symbol,
3. Γ is the tape alphabet, where ☐ ∈ Γ and Σ ⊆ Γ,
4. δ: Q × Γ→Q × Γ × {L,R} is the transition function,
5. q_0 ∈ Q is the start state,
6. q_accept ∈ Q is the accept state, and
7. q_reject ∈ Q is the reject state, where q_reject ≠ q_accept .

The $$TM$$ has to be defined beforehand, and its description should not depend on the particular specific input that was given to it.