Adding Two Unary Number

AI Summary

An AI-generated summary of this video lecture.

The video presents a lecture on designing a Turing machine to perform addition of two numbers represented in unary notation. The instructor begins by stating the problem: given two unary numbers separated by a blank symbol, the machine must output their sum in unary. He illustrates the concept using a specific example where the input represents the numbers 3 and 4. The tape is initialized with a blank symbol, followed by three '1's, a separator 'B', four '1's, and trailing blanks. The goal is to transform this input into a sequence of seven '1's, representing the sum 7. The instructor then proceeds to design the state transition diagram step-by-step, explaining the logic behind each state and transition to achieve the desired result.

Chapters

1. 0:00 – 2:00 00:00-02:00

   The video opens with the question "Design a Turing machine for addition of two number in unary?" displayed at the top. The instructor introduces the problem and begins drawing a tape diagram. He marks the start with 'B'. He writes three vertical lines '111' for the number 3. He places a 'B' as a separator. Next, he writes four vertical lines '1111' for the number 4. He labels the groups '3' and '4' above the tape. He explains the input is 111B1111. He states the output should be 1111111 (seven '1's). He emphasizes the machine must produce the correct unary sum.

2. 2:00 – 5:00 02:00-05:00

   The instructor starts constructing the state diagram. He draws the initial state q0. He adds a self-loop on q0 labeled "1,1,R", indicating that the machine reads '1', writes '1', and moves right, staying in q0. This allows the machine to scan through the first number without changing it. He then draws a transition from q0 to a new state q1 labeled "B,1,R". He explains that when the machine encounters the separator 'B', it writes '1' and moves right. This effectively replaces the separator with a '1', merging the two numbers into a single block of '1's. Now the tape conceptually looks like 11111111. He draws state q1 and adds a self-loop "1,1,R" to scan through the second number. When it hits the end blank 'B', he draws a transition to state q2 labeled "B,B,L". This moves the head left to the last '1'. In state q2, he draws a transition to q3 labeled "1,B,R". He explains that this writes 'B' (deleting the '1') and moves right. This step removes the extra '1' that was added when the separator was replaced. Finally, from q3, he draws a transition to the final state q4 labeled "B,B,R".

3. 5:00 – 5:31 05:00-05:31

   The instructor completes the state diagram by finalizing the transition to the final state q4. He briefly reviews the entire logic, confirming that the machine successfully adds the two numbers. He mentions that this approach works by effectively concatenating the numbers and then subtracting one to correct the count. He concludes the lecture segment, having successfully designed the Turing machine for unary addition. The video ends with the completed diagram visible on the screen, showing the states q0, q1, q2, q3, and q4 connected by the defined transitions.

The lecture demonstrates a systematic approach to designing a Turing machine for a basic arithmetic operation. By breaking down the problem into scanning, merging, and correcting steps, the instructor shows how simple state transitions can achieve complex tasks like addition. The key insight is treating the separator as a bridge to merge the unary strings and then adjusting the count to ensure accuracy. This method highlights the power of Turing machines in manipulating symbols on a tape to perform logical operations.