What is a normalized Turing Machine

Question

Could you explain to me the difference between a Turing machine and a normalized turing machine?

How does for example a standard operation (e.g., addition, subtraction, multiplication) differ at each case?

Answer 1

Normalized turing machine means that there exists only 1 accepted state and 1 rejected state.

Answer 2

The definition is:

A Turing machine $M$ over alphabet $\{a_1,\dots,a_k \}$ is called normalized if its program satisfies the following condition: from every non-final state $q_i$ of $M$ we can only reach states $q_j$, where $j ≤ k (i + 1)$.

This can be achieved by just reordering the states of any Turing Machine by swapping them to change their order. So the normalized machine looks just the same as the original one, you just look at the states in a different order. But since the states are in a set, this order is not really part of the machine; it can, however, be convenient for some proofs.

Arithmetic operations are not really standard operations for TMs. They just move left or right and rewrite the symbol their head is on. Anything more complex is already a program for a TM.