turing-machine | Turing Machine Compiler | Machine Learning library

There are two ways to solve the addition problem. Assume your input tape is in the form ^a+b$, where ^ and $ are symbols telling you you've reached the front and back of the input.

1. You can increment b and decrement a by 1 each step until a is 0, at which point b will be your answer. This is assuming you're comfortable writing a TM that can increment and decrement.
2. You can implement a full adding TM, using carries as you would if you were adding binary numbers on paper.

For either option, you need code to find the least significant bit of both a and b. The problem specifies that the most significant bit is first, so you'll want to start at + for a and $ for b.

For example, let's say we want to increment 1011$. The algorithm we'll use is find the least significant unmarked digit. If it's a 0, replace it with a 1. If it's a 1, move left.

1. Start by finding $, moving the read head there. Move the read head to the left.
2. You see a 1. Move the read head to the left.
3. You see a 1. Move the read head to the left.
4. You see a 0. write 1.
5. Return the read head to $. The binary number is now 1111$.

To compare two numbers, you need to keep track of which values you've already looked at. This is done by extending the alphabet with "marked" characters. 0 could be marked as X, 1 as Y, for example. X means "there's a 0 here, but I've seen it already.

So, for equality, we can start at ^ for a and = for b. (Assuming the input looks like ^a=b$.) The algorithm is to find the start of a and b, comparing the first unmarked bit of each. The first time you get to a different value, halt and reject. If you get to = and $, halt and reject.

Let's look at input ^11=10$:

1. Read head starts at ^.
2. Move the head right until we find an unmarked bit.
3. Read a 1. Write Y. Tape reads ^Y1=10$. We're in a state that represents having read a 1.
4. Move the head right until we find =.
5. Move the head right until we find an unmarked bit.
6. Read a 1. This matches the bit we read before. Write a Y.
7. Move the head left until we find ^.
8. Go to step 2.
9. This time, we'll read a 1 in a and read the 0 in b. We'll halt and reject.

Hope this helps to get you started.