tructions using 15 states numbered 1, 15, plus the halt state (H), and six symbols noted 1, X, Y, Z, W plus the underscore _, which stands for blank. The instruction nomenclature is "input state, read symbol, output state, write symbol, move right (>) or left (<).' The numbers m, n to be multiplied are entered in the input tape under the string form _M_N_, where M and N are strings of m + 1 and n + 1 successive 1s, respectively. The head is initially located to the leftmost (blank) symbol. The output tape takes the form _M_N_P_, where P is the unary representation of mn = p (P is a string of p + 1 successive 1s). 1, 1, -, > 10, 11, < 1, 1, 2, W, > 10, 1, 12, Z, > 2, 1, 2, 1, > 12,1,12,1, > 2, 3, -, > 12, 13, > 3, 1, 4, Y, > 13, 1, 13, 1, > 4, 1, 4, 1, > 13, 14, 1, < 14, 1, 14, 1, < 5, 6, 1, < 14, 14, < 6, 6, < 14, Z, 10, Z, > 6, 1, 6, 1, < 14, Y, 10, Y, > 6, Z, 6, Z, < 11, 1, 11, 1, < 6, Y, 6, Y, < 11, -, 11, -, < 6, X, 7, X, > 11, Z, 11, 1, < 6, W, 7, W, > 11, Y, 6, Y, < 7, 8, > 8, Y, 8, 1, < 7, 1, 9, X, > 8, 8, -, < 9, 1, 9, 1, > X, 8, 1, < 9, 9, -, > W, 15, 1, < 9, Y, 10, Y, > 15, , H, , > an erase/blank character). The table also shows that 38 different instructions, or TM transitions, are necessary to complete the multiplication of the two input numbers. To recall, such an operation can be performed with any numbers of any size, without limitation in size, since the TM tape is theoretically infinite in length. I will come back to this point later. Example 7.5 This concerns the operation of division, which is also relatively simple to implement with a TM. Given two integers m, n, the task is to find the quotient [m/n] and the remainder r = m - [m/n], where the brackets indicate the integer part of the ratio m/n. A first test consists in verifying that n > 0, i.e., that the field containing n has at least two 1s (with the convention 1 decimal = 110unary). This test can be performed through the above-described Comp program, which is also derived in an exercise. If n = 0, a special character meaning "zero divide