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Here we will see how to make a Turing machine for language L = {AiBjCk | i * j = k; i, j, k ≥ 1}. So this represents a kind of language where we will use only three characters A, B and C. The w is a string. So if w = AABBBBCCCCCCCC, The Turing machine will accept it.

To solve this, we will use this approach.

First replace an A with x and move right. Then skip all the A’s and move right

When the head reach to the first B then replace one B with y, then move right skipping all intermediate B’s and corresponding to the replaced B now replace one C with z and move left.

Now move towards left skipping all z and B in the way.

When pointer reach the recent y, then move right.

If the pointer is pointing at B then repeat steps 2 to 4, otherwise when pointer is pointing at z then move towards left while replacing all y to B and skipping all A’s.

When pointer comes to most recent x move to step right.

If the pointer is still pointing to A then repeat all above steps, otherwise when the head is at y then move towards right skipping all y and z.

When $ is reached then move left. String is accepted.

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