Discuss the following:
Q: Consider the following Turing-machine model (which is used in one of the standard textbooks in recursion theory, a branch of mathematical logic: Recursively Enumerable Sets and Degrees, by Robert I. Soare, Springer-Verlag, New York, 1987):
The Turing machine is equipped with the following:
(i) a tape that is infinitely long in both directions and is divided into cells; at any given step, each cell either is blank or contains a 1 (we will refer to the latter type of cell as a non-blank cell)
(ii) a read head (which reads one cell of the tape at each step)
(iii) a finite set Q of states (Q = {q_0, q_1, q_2, ..., q_k} for some nonnegative integer k)
At any given step of a Turing computation,
(a.) all but at most finitely many cells of the tape are blank (the rest, if any, contain a 1)
(b.) the machine can either remain in the current state or change to a different state
(c.) the contents of the cell being read can either change (from 1 to blank, if it currently contains a 1; from blank to 1, if it currently contains a blank) or remain as is
(d.) the read head must move by one unit, either to the right (R) or to the left (L), before the next step
The program for a Turing machine consists of a finite set of 5-tuples (q, s, q', s', m), where q and q' are the current and subsequent states of the machine, and s and s' indicate the current and subsequent contents of the cell being read; if m = L, the machine is to move one unit to the left of the current cell; if m = R, the machine is to move one unit to the right of the current cell.
The input to the machine, n, is represented on the tape by a string of n + 1 consecutive 1's, with all the other cells blank.
Examples:
If n = 0, only one cell of the tape contains a 1 (since n + 1 = 0 + 1 = 1).
If n = 1, there are two consecutive cells containing a 1 (since n + 1 = 1 + 1 = 2), and all the other cells are blank.
If n = 2, there is a string of three consecutive cells containing a 1 (since n + 1 = 2 + 1 = 3), and all the other cells are blank.
The machine begins in state q_1, with the read head positioned at the leftmost cell of the tape that contains a 1.
The halting state of the machine is q_0; if the machine ever reaches state q_0, the machine halts and outputs the total number of 1's on the tape, which is the value of the function being computed.
Using this Turing-machine model, create a program for a Turing machine that computes the function f: N -> N defined by f(n) = 2n + 3, where N is the set of all natural numbers (i.e., N is the set of all nonnegative integers).