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$$\small{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline \textbf{Year}&\textbf{2019}&\textbf{2018}&\textbf{2017-1}&\textbf{2017-2}&\textbf{2016-1}&\textbf{2016-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum} \\\hline\textbf{1 Mark Count}&2&2&2&3&3&3&2&2.5&3 \\\hline\textbf{2 Marks Count}&3&3&5&3&3&3&3&3.3&5 \\\hline\textbf{Total Marks}&8&8&12&9&9&9&\bf{8}&\bf{9.2}&\bf{12}\\\hline \end{array}}}$$

# Recent questions in Theory of Computation

1 vote
1
Consider the following languages: L1 ={a^nb^nc^m } U {a^nb^mc^m} n, m ≥ 0 L2={wwR|w {a,b}*} Where R represents reversible operation Which one of the following is (are) inherently ambiguous language(s)? 1) Only L1 (2) only L2 3) Both L1 and L2 (4) neither L1 nor L2
1 vote
2
$\mathbf{Q57}$ Let $\mathrm{A={001,0011,11,101}}$ and $\mathrm{B={01,111,111,010}}$. Similarly , Let $\mathrm{C={00,001,1000}}$ and $\mathrm{B={0,11,011}}$. Which of the following pairs have a post correspondence solution? $\mathrm{1)\; Only \;pair \;(A,B) }$ $\mathrm{ 2) \;Only\; pair \;(C,D) }$ $\mathrm{ 3) \; Both (A,B)\; and (C,D) \; }$ $\mathrm{\; 4) \;Neither \;(A,B) \;nor\; (C,D) }$
3
Say that a $CFG$ is minimal if none of its rules can be removed without changing the language generated. Let $MIN_{CFG} = \{\langle G \rangle \mid \text{G is a minimal CFG}\}$. Show that $MIN_{CFG}$ is $T-$recognizable. Show that $MIN_{CFG}$ is undecidable.
1 vote
4
Say that a variable $A$ in $CFG \:G$ is necessary if it appears in every derivation of some string $w \in G$. Let $NECESSARY_{CFG} = \{\langle G, A\rangle \mid \text{A is a necessary variable in G}\}$. Show that $NECESSARY_{CFG}$ is Turing-recognizable. Show that $NECESSARY_{CFG}$is undecidable.
5
Let $X = \{\langle M, w \rangle \mid \text{M is a single-tape TM that never modifies the portion of the tape that contains the input$w$} \}$ Is $X$ decidable? Prove your answer.
1 vote
6
Consider the problem of determining whether a $PDA$ accepts some string of the form $\{ww \mid w \in \{0,1\}^{\ast} \}$ . Use the computation history method to show that this problem is undecidable.
7
Prove that the following two languages are undecidable. $OVERLAP_{CFG} = \{\langle G, H\rangle \mid \text{G and H are CFGs where}\: L(G) \cap L(H) \neq \emptyset\}$. $PREFIX-FREE_{CFG} = \{\langle G \rangle \mid \text{G is a CFG where L(G) is prefix-free}\}$.
8
Let $f(x)=\left\{\begin{matrix}3x+1 & \text{for odd}\: x& \\ \dfrac{x}{2} & \text{for even}\: x & \end{matrix}\right.$ for any natural number $x$. If you start with an integer $x$ and iterate $f$, you obtain a sequence, $x, f(x), f(f(x)), \dots$ ... problem. Suppose that $ATM$ were decidable by a $TM\: H$. Use $H$ to describe a $TM$ that is guaranteed to state the answer to the $3x + 1$ problem.
1 vote
9
Use Rice’s theorem, to prove the undecidability of each of the following languages. $INFINITE_{TM} = \{\langle M \rangle \mid \text{M is a TM and L(M) is an infinite language}\}$. $\{\langle M \rangle \mid \text{M is a TM and }\:1011 \in L(M)\}$. $ALL_{TM} = \{\langle M \rangle \mid \text{ M is a TM and}\: L(M) = Σ^{\ast} \}$.
10
Rice's theorem. Let $P$ be any nontrivial property of the language of a Turing machine. Prove that the problem of determining whether a given Turing machine's language has property $P$ is undecidable. In more formal terms, let $P$ be a language consisting of Turing ... any $TMs$. Prove that $P$ is an undecidable language. Show that both conditions are necessary for proving that $P$ is undecidable.
11
Rice's theorem. Let $P$ be any nontrivial property of the language of a Turing machine. Prove that the problem of determining whether a given Turing machine's language has property $P$ is undecidable. In more formal terms, let $P$ ... $M_{1}$ and $M_{2}$ are any $TMs$. Prove that $P$ is an undecidable language.
1 vote
12
A two-dimensional finite automaton $(2DIM-DFA)$ is defined as follows. The input is an $m \times n$ rectangle, for any $m, n \geq 2$. The squares along the boundary of the rectangle contain the symbol $\#$ and the internal squares contain ... . Consider the problem of determining whether two of these machines are equivalent. Formulate this problem as a language and show that it is undecidable.
13
Define a two-headed finite automaton $(2DFA)$ to be a deterministic finite automaton that has two read-only, bidirectional heads that start at the left-hand end of the input tape and can be independently controlled to move in either direction. The tape of a $2DFA$ is finite and is just large ... $E_{2DFA}$ is not decidable.
14
Give an example of an undecidable language $B$, where $B \leq_{m} \overline{B}$.
15
Let $J = \{w \mid \text{either$w = 0x$for some$x \in A_{TM},$or$w = 1y\:$for some$y \in \overline{A_{TM}}\:\:$}\}$. Show that neither $J$ nor $\overline{J}$ is Turing-recognizable.
16
Show that $A$ is decidable iff $A \leq_{m} 0 ^{\ast} 1^{\ast}$ .
17
Show that $A$ is Turing-recognizable iff $A \leq_{m} A_{TM}$.
Let $AMBIG_{CFG} = \{\langle G \rangle \mid \text{G is an ambiguous CFG}\}$. Show that $AMBIG_{CFG}$ is undecidable. (Hint: Use a reduction from $PCP$ ... $a_{1},\dots,a_{k}$ are new terminal symbols. Prove that this reduction works.)
Prove that there exists an undecidable subset of $\{1\}^{\ast}$ .
In the silly Post Correspondence Problem, $SPCP$, the top string in each pair has the same length as the bottom string. Show that the $SPCP$ is decidable.