Final answer:
A Turing machine transducer to compare unary numbers follows a process of alternately erasing symbols from both numbers and checking if they complete at the same time. If they are equal, the machine writes 1; otherwise, it writes 0 before entering a halt state.
Step-by-step explanation:
To design a standard Turing machine (TM) as a transducer capable of comparing two unary numbers and determining whether they are equal, we need to define a set of states and transitions that process the input and produce the correct output. You start in the initial state, scanning the first symbol of the first number. If it is a 1, you replace it with an empty symbol (like a space ' '), move to the right, and enter a new state which indicates you have removed one symbol from the first number. In this new state, if you encounter 1s, you continue moving right until you reach the 0 separator. Once you hit the 0, you change to another state and move right until you find the first 1 of the second number. Replace that 1 with a space, indicating you have also removed a symbol from the second number, and move back left to find the separator 0 again.
When you reach the 0 this time, change to a state that moves you left back towards the first number. Repeat this process of erasing one 1 from both numbers and alternating sides until you reach a space where a 1 should be in either number. If you finish one number (i.e., encounter a space instead of a 1) and the other number still has 1s left, then the numbers are not equal. You would then move to the end of the tape and write 0 to indicate inequality. However, if both numbers run out of symbols (only spaces left) simultaneously, they are equal; you transition to the end of the tape and write 1 to indicate equality. You should also define a halt state. This process ultimately checks if the two unary numbers separated by a 0 are equal or not.