Recent questions tagged theory-of-computation

1
Let $L \subseteq \{0,1\}^*$ be an arbitrary regular language accepted by a minimal $\text{DFA}$ with $k$ states. Which one of the following languages must necessarily be accepted by a minimal $\text{DFA}$ with $k$ states? $L-\{01\}$ $L \cup \{01\}$ $\{0,1\}^* – L$ $L \cdot L$
2
Let $L_1$ be a regular language and $L_2$ be a context-free language. Which of the following languages is/are context-free? $L_1 \cap \overline{L_2} \\$ $\overline{\overline{L_1} \cup \overline{L_2}} \\$ $L_1 \cup (L_2 \cup \overline{L_2}) \\$ $(L_1 \cap L_2) \cup (\overline{L_1} \cap L_2)$
1 vote
3
Consider the following deterministic finite automaton $\text{(DFA)}$ The number of strings of length $8$ accepted by the above automaton is ___________
4
Suppose we want to design a synchronous circuit that processes a string of $0$'s and $1$'s. Given a string, it produces another string by replacing the first $1$ in any subsequence of consecutive $1$'s by a $0$ ... $\begin{array}{l} t=s+b \\ y=s \overline{b} \end{array}$
5
​​​​​​Consider the following two statements about regular languages: $S_1$: Every infinite regular language contains an undecidable language as a subset. $S_2$: Every finite language is regular. Which one of the following choices is correct? Only $S_1$ is true Only $S_2$ is true Both $S_1$ and $S_2$ are true Neither $S_1$ nor $S_2$ is true
6
For a string $w$, we define $w^R$ to be the reverse of $w$. For example, if $w=01101$ then $w^R=10110$. Which of the following languages is/are context-free? $\{ wxw^Rx^R \mid w,x \in \{0,1\} ^* \}$ $\{ ww^Rxx^R \mid w,x \in \{0,1\} ^* \}$ $\{ wxw^R \mid w,x \in \{0,1\} ^* \}$ $\{ wxx^Rw^R \mid w,x \in \{0,1\} ^* \}$
7
​​​​​​​Which of the following regular expressions represent(s) the set of all binary numbers that are divisible by three? Assume that the string $\epsilon$ is divisible by three. $(0+1(01^*0)^*1)^*$ $(0+11+10(1+00)^*01)^*$ $(0^*(1(01^*0)^*1)^*)^*$ $(0+11+11(1+00)^*00)^*$
1 vote
8
Suppose that $L_1$ is a regular language and $L_2$ is a context-free language. Which one of the following languages is $\text{NOT}$ necessarily context-free? $L_1 \cap L_2$ $L_1 \cdot L_2$ $L_1- L_2$ $L_1\cup L_2$
9
Let $\langle M \rangle$ denote an encoding of an automaton $M$. Suppose that $\Sigma = \{0,1\}$. Which of the following languages is/are $\text{NOT}$ recursive? $L= \{ \langle M \rangle \mid M$ is a $\text{DFA}$ such that $L(M)=\emptyset \}$ $L= \{ \langle M \rangle \mid M$ is ... $L(M)=\emptyset \}$ $L= \{ \langle M \rangle \mid M$ is a $\text{PDA}$ such that $L(M)=\Sigma ^* \}$
10
Consider the following language: $L= \{ w \in \{0,1\}^* \mid w \text{ ends with the substring } 011 \}$ Which one of the following deterministic finite automata accepts $L?$
11
For a Turing machine $M$, $\langle M \rangle$ denotes an encoding of $M$ ... and $L_2$ are decidable $L_1$ is decidable and $L_2$ is undecidable $L_1$ is undecidable and $L_2$ is decidable Both $L_1$ and $L_2$ are undecidable
12
In a pushdown automaton $P=(Q, \Sigma, \Gamma, \delta, q_0, F)$, a transition of the form, where $p,q \in Q$, $a \in \Sigma \cup \{ \epsilon \}$, and $X,Y \in \Gamma \cup \{ \epsilon \}$, represents $(q,Y) \in \delta(p,a,X).$ Consider the ... $\Gamma = \{ \#, A\}$. The number of strings of length $100$ accepted by the above pushdown automaton is ___________
1 vote
13
Consider $L=L_1 \cap L_2$ where $L_1 = \{ 0^m 1^m 20^n 1^n \mid m,n \geq 0 \}$ $L_2 = \{0^m1^n2^k \mid m,n,k \geq 0 \}$ Then, the language $L$ is Recursively enumerable but not context free Regular Context free but not regular Not recursive
Let $L_1$ and $L_2$ be languages over $\Sigma = \{a,b\}$ represented by the regular expressions $(a^* +b)^*$ and $(a+b)^*$ respectively. Which of the following is true with respect to the two languages? $L_1 \subset L_2$ $L_2 \subset L_1$ $L_1 = L_2$ $L_1 \cap L_2 = \phi$
Consider the following languages: $L_1=\{a^{\grave{z}^z} \mid \grave{Z} \text{ is an integer} \}$ $L_2=\{a^{z\grave{z}} \mid \grave{Z} \geq 0\}$ $L_3=\{ \omega \omega \mid \omega \epsilon \{a,b\}^*\}$ Which of the languages is(are) regular? Choose the correct answer from the options given below: $L_1$ and $L_2$ only $L_1$ and $L_3$ only $L_1$ only $L_2$ only