# Recent questions tagged context-free-languages

1 vote
1
Which of the following definitions generates the same languages as $L,$ where $L = \{x^{n}y^{n},n \geq 1\}$ $E \rightarrow xEy \mid xy$ $xy \mid x^{+}xyy^{+}$ $x^{+}y^{+}$ $(i)$ Only $(i)$ and $(ii)$ only $(ii)$ and $(iii)$ only $(ii)$ only
2
Which of the following definitions generates the same languages as $L,$ where $L = \{x^{n}y^{n},n \geq 1\}$ $E \rightarrow xEy \mid xy$ $xy \mid x^{+}xyy^{+}$ $x^{+}y^{+}$ $(i)$ $(i)$ and $(ii)$ only $(ii)$ and $(iii)$ only $(ii)$ only
1 vote
3
If $L_1$ and $L_2$ are context free language and $R$ a regular set, then which one of the languages below is not necessarily a context free language? $L_1L_2$ $L_1\cap L_2$ $L_1\cap R$ $L_1\cup L_2$
4
If $L$ be a language recognizable by a finite automaton, then language from $\{L\} = \{w$ such that $w$ is a prefix of $v$ where $v\in L\}$, is a regular language. context-free language. context-sensitive language. recursive enumeration language
1 vote
5
If $L1$ is CFL and $L2$ is regular language which of the following is false? $L1-L2$ is not Context free $L1$ intersection $L2$ is Context free $\sim L1$ is Context free Both (A) and (C)
6
Which of the following statement is true? Deterministic context free language are closed under complement. Deterministic context free language are not closed under Union. Deterministic context free language are closed under intersection with regular set. All of the options
7
Given the following two languages : $L_{1}= \{a^{n}b^{n}, \mid n\geq 0,n \neq 100 \}$ $L_{2}= \{w \in \{a,b,c \} ^{\ast} \mid n_{a}(w)=n_{b}(w)=n_{c}(w) \}$ Which of the following options is correct? Both $L_{1}$ and $L_{2}$ ... are context free language $L_{1}$ is context free language, $L_{2}$ is not context free language $L_{1}$ is not context free language, $L_{2}$ is context free language
1 vote
8
Consider the following statements. The intersection of two context-free languages is always context-free The super-set of a context-free languages is never regular The subset of a decidable language is always decidable Let $\Sigma = \{a,b,c\}.$ Let $L\subseteq \Sigma$ be the language of all strings in which either ... $(2)$ Only $(1),(2)$ and $(3)$ Only $(4)$ None of $(1),(2),(3),(4)$ are true.
1 vote
9
Context free languages are closed under union, intersection union, kleene closure intersection, complement complement, kleene closure
10
Say that a variable $A$ in $CFL\: G$ is usable if it appears in some derivation of some string $w \in G$. Given a $CFG\: G$ and a variable $A$, consider the problem of testing whether $A$ is usable. Formulate this problem as a language and show that it is decidable.
11
Let $C = \{ww^{R} \mid w in \{0,1\}^{\ast}\}$. Prove that $C$ is not a DCFL. (Hint: Suppose that when some DPDA $P$ is started in state $q$ with symbol $x$ on the top of its stack, $P$ never pops its stack below $x$, no matter what input ... that case, the contents of $P's$ stack at that point cannot affect its subsequent behavior, so $P's$ subsequent behavior can depend only on $q$ and $x$.)
12
Let $B = \{a^{i}b^{j}c^{k}\mid i,j,k \geq 0\: \text{and}\: i = j\: \text{or}\: i = k \}$. Prove that $B$ is not a DCFL.
13
Let $A = L(G_{1})$ where $G_{1}$ is defined in Question $2.55$. Show that $A$ is not a DCFL.(Hint: Assume that $A$ is a DCFL and consider its DPDA $P$. Modify $P$ so that its input alphabet is $\{a, b, c\}$. When it first enters an accept state, it pretends that $c’s$ are $b’s$ in the input from that point on. What language would the modified $P$ accept?)
14
Show that the class of DCFLs is not closed under the following operations: Union Intersection Concatenation Star Reversal
15
We defined the $CUT$ of language $A$ to be $CUT(A) = \{yxz| xyz \in A\}$. Show that the class of $CFLs$ is not closed under $CUT$.
16
We defined the rotational closure of language $A$ to be $RC(A) = \{yx \mid xy \in A\}$.Show that the class of CFLs is closed under rotational closure.
17
Let $\Sigma = \{0,1\}$. Let $C_{1}$ be the language of all strings that contain a $1$ in their middle third. Let $C_{2}$ be the language of all strings that contain two $1s$ ... $C_{1}$ is a CFL. Show that $C_{2}$ is not a CFL.
18
Let $A = \{wtw^{R}\mid w,t\in\{0,1\}^{\ast}\:\text{and} \mid w \mid = \mid t \mid \}$. Prove that $A$ is not a CFL.
19
For strings $w4 and$t,4 write $w \circeq t$ if the symbols of $w$ are a permutation of the symbols of $t.$ In other words, $w \circeq t$ if $t$ ... then the SCRAMBLE of a regular language is context free. What happens in part $(a)$ if $\Sigma$ contains three or more symbols? Prove your answer.
20
Let $Y = \{w\mid w = t_{1}\#t_{2}\#\dots t_{k} \:\text{for}\: k\geq 0,\text{each}\: t_{i}\in 1^{\ast}, \text{and}\: t_{i}\neq t_{j} \text{whenever}\: i\neq j\}$.Here $\Sigma = \{1,\#\}$. Prove that $Y$ is not context free.
21
Read the definitions of NOPREFIX(A) and NOEXTEND(A) in Question $1.40$. Show that the class of CFLs is not closed under NOPREFIX. Show that the class of CFLs is not closed under NOEXTEND.
22
Say that a language is prefix-closed if all prefixes of every string in the language are also in the language. Let $C$ be an infinite, prefix-closed, context-free language. Show that $C$ contains an infinite regular subset.
23
For the definition of the shuffle operation. For languages $A$ and $B,$ let the $\text{shuffle}$ of $A$ and $B$ be the language $\{w| w = a_{1}b_{1} \ldots a_{k}b_{k},$ where $a_{1} · · · a_{k} ∈ A$ and $b_{1} · · · b_{k} ∈ B,$ each $a_{i}, b_{i} ∈ Σ^{*}\}.$ Show that the class of context-free languages is not closed under shuffle.
24
For the definition of the perfect shuffle operation. For languages $A$ and $B,$ let the $\text{perfect shuffle}$ of $A$ and $B$ be the language $\text{{$w| w = a_{1}b_{1} · · · a_{k}b_{k},$where$a_{1} · · · a_{k} ∈ A$and$b_{1} · · · b_{k} ∈ B,$each$a_{i}, b_{i} ∈ Σ$}}.$ Show that the class of context-free languages is not closed under perfect shuffle.
1 vote
25
Let $L_{1}:=\{a^{n}b^{m}\mid m,n\geq 0\: \text{and}\: m\geq n\}$ and $L_{2}:=\{a^{n}b^{m}\mid m,n\geq 0\: \text{and}\: m < n\}.$ The language $L_{1}\cup L_{2}$ is: regular, but not context-free context-free, but not regular both regular and context-free neither regular nor context-free
1 vote
26
It is undecidable whether the complement of a CFL is also a CFL. Exercise $9.5.2$ can be used to show it is undecidable whether the complement of a CFL is regular, but that is not the same thing. To prove our initial claim, we need to define a different language that ... $7.2.5(b)$.
How can the decision algorithm be constructed for deciding whether context-free language $L$ is finite? By constructing redundant CFG G in CNF generating language $\underline{L}$ By constructing non-redundant CFG G in CNF generating language $L$ By constructing non-redundant CFG G in CNF ... $\wedge$ stands for null) Which of the following is correct? a only b only c only None of a, b and c
Determine whether or not the following languages are context-free. (a) $L=$ {$a^nww^Ra^n : n ≥ 0, w ∈$ {$a,b$}*} (b) $L=$ {$a^nb^ja^nb^j : n ≥ 0, j ≥ 0$}. (C) $L=$ {$a^nb^ja^jb^n : n ≥ 0, j ≥ 0$}. (d) $L=$ {$a^nb^ja^kb^l : n + j ≤ k + l$}. (e)$L=$ {$a^nb^ja^kb^l : n ≤ k, j ≤ l$}. (f) $L=$ {$a^nb^nc^j : n ≤j$}. (g) $L=$ {$w ∈$ {$a, b, c$}* $: n_a(w)= n_b (w)=2n_c(w)$}.
Is the language L = {$a^nb^m : n = 2^m$} context-free?