Let's solve **each part step by step**, explaining the logic and the construction of the Turing Machines (TMs). --- --- ## **Step 1: Introduction to the Concept** A **Turing Machine (TM)** is a theoretical computational model that manipulates symbols on a tape according to a set of rules. It can read and write symbols, move left or right, and change states. TMs can be designed to perform specific tasks such as arithmetic operations or pattern recognition by encoding the problem and designing appropriate transitions. --- ## **Step 2: (a) TM to Replace Unary k with 2k** ### **Problem** Given integer \( k \) in unary (as a sequence of \( k \) 1's), design a TM to replace it with \( 2k \) (i.e., double the number of 1's). ### **Algorithm** 1. **Scan the tape from left to right.** 2. **For every 1 encountered, append another 1 at the end of the block.** 3. **After all original 1's are processed, erase any marks used (if any).** ### **Explanation** - Mark each 1 as processed (say, by changing it to X). - For every X, move to the end and write a 1. - After processing all Xs, convert Xs back to 1s. --- ## **Step 3: (b) TM to Replace Unary \( i \) and \( j \) (separated by blank) with \( i - j \)** ### **Problem** Given \( i \) and \( j \) in unary separated by blank, output \( i-j \) in unary. ### **Algorithm** 1. **Scan for the first 1 (from the left block).** 2. **For each 1 in the left block, cross out a 1 from the right block.** 3. **After all 1's in the right block are crossed out, the remaining 1's in the left block represent \( i-j \).** 4. **Clean up the tape to leave only the result.** ### **Explanation** - Cross off one 1 from each side, left and right, repeatedly. - When the right block is empty, the left block has the answer. --- ## **Step 4: (c) TM to Recognize Strings with 1 a, 2 b's, 3 c's (Length 6)** ### **Problem** Recognize strings (length 6) with exactly 1 'a', 2 'b's, 3 'c's. ### **Algorithm** 1. **Mark one 'a', two 'b's, and three 'c's one by one.** 2. **If at any time more or fewer than the required number is found, reject.** 3. **If all six positions are marked correctly, accept.** ### **Explanation** - Mark each required character (say, by changing to X). - If you can't find the required count, or if extra symbols are left, reject. --- ## **Step 5: Whole Final Answer (in Very Few Sentences)** (a) The TM scans each 1, marks it, and appends another 1 at the end, thus doubling the unary input. (b) The TM repeatedly matches and erases one 1 from each block, leaving \( i-j \) 1's in unary. (c) The TM marks exactly one 'a', two 'b's, and three 'c's, and accepts only if the string is of length 6 and all counts match. --- ## **Step 6: Brief Explanation of Each Step** - **Step 1:** Introduced the concept of TMs. - **Step 2:** Designed a TM to double a unary number by marking and copying. - **Step 3:** Designed a TM to subtract two unary numbers by matching and canceling. - **Step 4:** Designed a TM to recognize a specific pattern by counting and marking symbols. - **Step 5:** Summarized the solutions concisely. Let me know if you want state diagrams or pseudo-code for any part!