Question

Please draw the Turing machine with JFLAP format in paper by the following instruction.

To construct a linear bounded automaton (LBA) that accepts the language L = {aⁿ : n = m², m ≥ 1}, we need to make sure that the LBA accepts all strings that are in the language L and rejects all strings that are not in the language L.
One way to construct such an LBA is as follows:
1. The LBA reads the input string from the input tape and moves the head to the right end of the tape.
2. The LBA then marks the first symbol of the input string with a special symbol, say B.
3. The LBA then repeatedly checks if the string can be divided into two equal parts such that the first part consists of a^m symbols and the second part consists of a^n-m symbols, where n = m².
4. If the string cannot be divided into two such parts, then the LBA rejects the input string and halts.
5. If the string can be divided into two such parts, the LBA continues to repeat step 3 for each part.
6. For each part, the LBA marks the first symbol with a special symbol, say C.
7. The LBA then moves the head to the right end of the string and starts scanning the string from right to left.
8. For each a symbol encountered, the LBA moves one step to the left and changes the marked symbol from C to D.
9. The LBA continues this process until it has scanned all the symbols of the string.
10. If the LBA has marked all the symbols with D, then it accepts the input string. Otherwise, it rejects the input string.
The idea behind this LBA is to check if the input string can be divided into two parts such that the first part consists of aᵐ symbols and the second part consists of aⁿ-m symbols, where n = m².
If the string can be divided into such parts, then the LBA checks that each part contains a square number of a symbols. To do this, the LBA marks the first symbol of each part with the symbol C and scans the part from right to left,
marking each symbol with the symbol D as it goes. If the LBA can mark all the symbols with D, then the part contains a square number of a symbols. If the LBA can mark both parts with D,
then the entire string contains a square number of a symbols and the LBA accepts it.
To construct a linear bounded automaton that accepts L = {aⁿ : n = m², m ≥ 1}, we need to check if the string can be divided into two parts such that each part contains a square number of a symbols.
This can be done by marking the first symbol of each part with the symbol C and scanning the part from right to left, marking each symbol with the symbol D as it goes. If both parts can be marked with D,
then the entire string contains a square number of a symbols and the LBA accepts it.

Answer 1

Creating a Linear Bounded Automaton for language L accepts strings where the number of 'a's is a perfect square. It uses a marking and scanning technique to divide and check lengths of parts of strings. The Turing Machine will accept only if all parts of the string are properly marked and divided, indicating a perfect square length.

Step-by-step explanation:

Constructing a Linear Bounded Automaton for Specific Language

Creating a Linear Bounded Automaton (LBA) to accept the language L = {aⁿ : n = m², m ≥ 1} involves verifying whether a given string of 'a's is of length that is a perfect square. The process starts by marking the first symbol with a special symbol, B, and moving the tape head to check if the remaining string can be equally divided into two parts, satisfying the required conditions. If a division is viable, the LBA marks the first symbol of each part with C and scans from right to left, marking with D until all the symbols are processed. The Turing Machine constructed in accordance with these instructions will only accept strings with lengths that are perfect squares, rejecting all others upon failing the division or marking test.

To break the string into two halves, the automaton might need to count and compare lengths iteratively, which is a simplified strategy hinted to by the marking with C and D. If all symbols transform to D, indicating successful division and marking, the LBA accepts the input; otherwise, it rejects it. This LBA is an abstraction and theoretical model for computation, demonstrating how machines can be structured to decide whether strings belong in certain formal languages.