A Turing Machine for the language \(\mathrm{L}=\left\{\mathrm{a}^{\mathrm{n}}…
2022
A Turing Machine for the language \(\mathrm{L}=\left\{\mathrm{a}^{\mathrm{n}} \mathrm{b}^{\mathrm{m}} \mathrm{c}^{\mathrm{n}} \mathrm{d}^{\mathrm{m}} \mid \mathrm{n} \geq 1, \mathrm{~m} \geq 1\right\}\) is designed. The resultant model is \(M =\)\(\left(\left\{\mathrm{q}_{0}, \mathrm{q}_{1}, \mathrm{q}_{2}, \mathrm{q}_{3}, \mathrm{q}_{4}, \mathrm{q}_{5}, \mathrm{q}_{6}, \mathrm{q}_{7}, \mathrm{q}_{\mathrm{f}}\right\},\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\},\left\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{X}_{1}, \mathrm{X}_{2}, \mathrm{Y}_{1}, \mathrm{Y}_{2}\right\}, \delta, \mathrm{q}_{0}, \mathrm{~B},\left\{\mathrm{q}_{\mathrm{f}}\right\}\right)\) and part of \(' 𝛿 '\) is given in the transition table. You need to write the following questions based on design of Turing Machine for the given language. Note that, while designing the Turing Machine \(X_1\) and \(X_2\) are used to work with \(′𝑎′𝑠\) and \(′𝑐′𝑠\) and \(Y_1\) and \(Y_2\) are used to handle \(′𝑏′𝑠\) and \(′𝑑′𝑠\) of the given string.
\(\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline & \mathrm{a} & \mathrm{b} & \mathrm{c} & \mathrm{d} & \mathrm{X}_{1} & \mathrm{X}_{2} & \mathrm{Y}_{1} & \mathrm{Y}_{2} & \mathrm{~B} \\ \hline \mathrm{q}_{0} & \left(\mathrm{q}_{1}, \mathrm{X}_{1}, \mathrm{R}\right) & & & & \mathrm{M} 2 & & & & \\ \hline \mathrm{q}_{1} & \left(\mathrm{q}_{1}, \mathrm{a}, \mathrm{R}\right) & \left(\mathrm{q}_{1}, \mathrm{~b}, \mathrm{R}\right) & \mathrm{M} 1 & & & \left(\mathrm{q}_{1}, \mathrm{X}_{2}, \mathrm{R}\right) & & & \\ \hline \mathrm{q}_{2} & \left(\mathrm{q}_{2}, \mathrm{a}, \mathrm{L}\right) & \left(\mathrm{q}_{2}, \mathrm{~b}, \mathrm{~L}\right) & & & \left(\mathrm{q}_{2}, \mathrm{X}_{1}, \mathrm{R}\right) & \left(\mathrm{q}_{2}, \mathrm{X}_{2}, \mathrm{~L}\right) & & & \\ \hline \mathrm{q}_{3} & \mathrm{M} 3 & \left(\mathrm{q}_{4}, \mathrm{Y}_{1}, \mathrm{R}\right) & & & & \left(\mathrm{q}_{6} \mathrm{X}_{2}, \mathrm{R}\right) & & & \\ \hline \mathrm{q}_{4} & & \left(\mathrm{q}_{4}, \mathrm{~b}, \mathrm{R}\right) & & \left(\mathrm{q}_{5}, \mathrm{Y}_{2}, \mathrm{~L}\right) & & \mathrm{M} 5 & & \left(\mathrm{q}_{4}, \mathrm{Y}_{2}, \mathrm{R}\right) & \\ \hline \mathrm{q}_{5} & & \left(\mathrm{q}_{5}, \mathrm{~b}, \mathrm{~L}\right) & & & & \left(\mathrm{q}_{5}, \mathrm{X}_{2}, \mathrm{~L}\right) & \mathrm{M} 4 & \left(\mathrm{q}_{5}, \mathrm{Y}_{2}, \mathrm{~L}\right) & \\ \hline \mathrm{q}_{6} & & & & & & \left(\mathrm{q}_{6}, \mathrm{X}_{2}, \mathrm{R}\right) & & \left(\mathrm{q}_{7}, \mathrm{Y}_{2}, \mathrm{R}\right) & \\ \hline \mathrm{q}_{7} & & & & & & & & \left(\mathrm{q}_{7}, \mathrm{Y}_{2}, \mathrm{R}\right) & \left(\mathrm{q}_{6}, \mathrm{~B}, \mathrm{R}\right) \\ \hline \end{array}\)
What is the Move in the cell with number \('M2'\) of the resultant Table?
- A.
\(\left(\mathrm{q}_{1}, \mathrm{X}_{1}, \mathrm{R}\right)\) - B.
\(\left(\mathrm{q}_{1}, \mathrm{a}, \mathrm{R}\right)\) - C.
\(\left(\mathrm{q}_{2}, \mathrm{X}_{1}, \mathrm{R}\right)\) - D.
Error Entry
Attempted by 26 students.
Show answer & explanation
Correct answer: D
Answer: The cell labeled M2 is an error entry (no valid transition).
Reasoning:
Role of the initial state: The start state marks the first unprocessed 'a' by replacing it with the marker X1 and then moves right to continue processing. It is intended to act only when it sees an unmarked 'a'.
Meaning of X1: X1 denotes an already-processed 'a'. If the head sees X1, that indicates that 'a' was previously handled and other states (not the initial marking state) are responsible for scanning over markers.
Why the table cell is erroneous: Defining a transition in the initial state for symbol X1 would either re-mark an already-marked symbol or overwrite markers, breaking the intended marking scheme used to match a's with c's. Therefore no valid transition belongs in that cell as part of the correct design.
Conclusion: The correct entry for M2 is that the cell is an error entry (i.e., no transition should be defined there).