Questions tagged [context-free]

Questions about the set of languages (equivalently) described by context-free grammars or accepted by (non-deterministic) pushdown automata.

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2answers
99 views

Can the String, $0^p 0^p 0^p$, be Used with the Pumping Lemma to Show that $w^r w w^r$ is Not Context Free?

I'm trying to show that $L=\left\{w^rww^r:w \in \{0,1\}^*\right\}$ is not context free using the pumping lemma. I thought picking the string, $0^p0^p0^p$, would be a good candidate for this, but ...
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1answer
311 views

conversion of this grammar to CNF

My task is to convert the following grammar to CNF: $S \to SS \mid (S) \mid \lambda$ after removing lambda productions: $S\to SS, S\to (S), S\to(), S\to S$ after removing unit productions: $S\to ...
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1answer
2k views

Is my proof for a context free language correct? Same number of a's as b's

I have the following grammar G: $$ \begin{align*} &S \to aB|bA \\ &A \to a|aS|bAA \\ &B \to b|bS|aBB \end{align*} $$ I am going to prove that this language L(G) consists of words with the ...
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1answer
6k views

Pushdown automaton for complement of { ww | … }

I want to be able to describe the idea behind the pushdown automaton (no tables or diagrams). So, I already know that $L = \{ ww \mid w \text{ in } (0,1)^*\}$ is not context free. Since CFL are not ...
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1answer
995 views

Constructing Context Free Grammar [duplicate]

I am stuck and having a hard time with this question. I want to construct a CFG for the language $$L = \{{a^lb^mc^n | l,m\in N, n=|l-m|\}}$$ I know that the language consists of strings where: 1. ...
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1answer
1k views

Determining whether a context-free language (CFL) described by a given grammar is regular (RL)

In my homework we're given the following problem: Determine whether the context-free language described by the following grammar is regular, showing all the reasoning steps: S -> T T | U T -> 0 T | ...
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4answers
3k views

What information do we get from a compiler's parse tree?

In the compiler course by Alex Aiken on Coursera, more specifically lecture 05-02 Context Free Grammars, the professor says that CFGs give answers of the type yes/no, i.e. whether the given string of ...
3
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5answers
14k views

Context-free grammar for language with unequal numbers of a and b

I've been trying to get a CFG for the language of all words with unequal numbers of a and b, i.e. $$\{u \in \{a, b\}^* \mid \text{number of occurrences of $a$ and $b$ in $u$ are unequal} \},$$ but ...
4
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3answers
240 views

Designing a CFG that produces as many c's as the difference of numbers of a's and b's

The question is to design a CFG for the language of words that have as many c's as the difference of numbers of a's and b's, that is $\qquad\displaystyle L = \{(a^l)(b^m)(c^n) \mid l, m \in \mathbb{N}...
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1answer
1k views

Does Reverse Polish Notation have an LL grammar?

Let L be the language of all arithmetic expressions written in Reverse Polish Notation, containing only binary operators. $\Sigma(L) = \{n, o\}$, n := number, o := operator. Is there an LL grammar G ...
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2answers
90 views

Grammar generating specific language

Construct a context-sensitive grammar that generates L: L = {a^n b^m c^k|k>n, k>m} I believe my productions should go along this lines: ...
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1answer
55 views

Which grammar is this?

Having the grammar G = (V,P,S) with variable V = {S,A} over the alphabet {a,b} with the ...
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1answer
1k views

Is the Syntax of C Language completely defined by CFGs?

I think the Question is self sufficient. Is the syntax of C Language completely defined through Context Free Grammars or do we have Language Constructs which may require non-Context Free definitions ...
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2answers
236 views

What does this context-free grammar generate?

The grammar is $$ S\to aSb\ |\ bSa\ |\ SS\ |\ \epsilon. $$ I think this generates the set of strings with equal numbers of $a$'s and $b$'s based on examples I've done. Is this correct?
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3answers
2k views

How to prove every context-free language over a unary alphabet is regular?

How can I show that every context-free language over a unary alphabet is regular?
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2answers
515 views

The complement of $\{w: w\text{ has equal numbers of $a$'s, $b$s' and $c$'s}\}$ is context-free

Let $L$ be the language $\{w \mid w \text{ has equal numbers of \(a\)'s, \(b\)'s and \(c\)'s}\}$. Prove that its complement $\overline{L}$ is context free.
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1answer
2k views

Prove that {0^{n^3} | n≥0} is not context free

I'm not very comfortable with pumping lemma for context-free grammar. I understand the sufficient conditions that must hold but proving it gets me everytime. For example, I need to prove whether $L = \...
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1answer
74 views

Deciding the class of certain languages [closed]

I am preparing for my exam in Formal languages and Automata theory and I'm looking at some old exam questions right now. I need help with the following question: For each of the following ...
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2answers
95 views

Is $\{s_0 w s_1 : s_0s_1\in L_1, w\in L_2 \}$ context free if $L_1$ and $L_2$ are?

In class, it was alluded to that a language: \begin{equation*} \{s_0 w s_1 : s_0s_1\in L_1, w\in L_2 \} \end{equation*} would be context free, if $L_1$ and $L_2$ are context free. Intuitively, that ...
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1answer
652 views

Non context free language that is pumpable? [duplicate]

For my homework assignment, I have to come up with a non-cfl that is pumpable. I came up with the following: $$ C = \{a^n b^n c^n d^m \mid n \ge 1 \text{ and } m \ge 1 \} $$ I'm not sure whether ...
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1answer
2k views

why does ${a^nb^n}$ fit the pumping lemma for context-free languages?

I am writing somthing about Ppumping Lemma. I know that the language $L = \{ a^nb^n| n ≥ 0 \}$ is context-free. But I don't understand how this language satisfies the conditions of pumping lemma (for ...
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2answers
2k views

Context Free Grammar for $\{0^n1^n \mid n \geq 0\}^*$

Give a context free grammar for the following language over $\Sigma = \{0,1\}$: $ L = L_1^* $ where $ L_1 = \{0^n1^n : n \geq 0\}$. Not really sure where to start with this one. Any help is ...
2
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1answer
658 views

Would it be possible that the intersection of two non-context-free languages is context-free?

Assume two languages $L_1$ and $L_2$, both of which are non-context-free. Let $L = L_1 \cap L_2$. Could $L$ be context-free?
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1answer
1k views

Derive a Context Free Grammar from a language

I am having challenges (in two phases) with creating a CFG. Derive the CFG for the following language Show parse trees for the strings cacab and aacabbb obtained from the grammar designed above. I ...
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5answers
4k views

How is non-ambuiguity different from determinism?

I am trying to understand what is meant by "deterministic" in expressions such as "deterministic context-free grammar". (There are more deterministic "things" in this field). I would appreciate an ...
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2answers
669 views

Prove L to not context free using pumping lemma on language L

$L = \{0^i1^j0^i1^j|i,j \geq 0\}$ I've tried letting $s = 0^p1^p0^p1^p$. But not sure where to go from here. Help would be appreciated.
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3answers
11k views

Language of balanced parentheses; Biconditional proof about parentheses

Let L be language of balanced parentheses. (a) Prove If there are equal number of ('s and )'s and every prefix of w contains at least as many ('s as )'s, then w is in L. (b) Prove If w is in L, then ...
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3answers
157 views

Determine whether two languages are context free

(1) $L_1 = \{a^ib^{i+j}c^j|i,j\geq 0\} $ (2) $L_2 = \{xy | x,y \in \{0,1\}^*, x \neq y, |x| = |y| \}$ I doubt that $L_1$ is CFL. I've been trying to go with the string $s$ = $a^pb^{2p}c^p$. Thus, we ...
2
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1answer
2k views

Use pumping lemma to show L is not context free

Show that L = $\{0^{2^n}| n\geq 0\}$ is not a context free language. Let string $s = 0^{2^p}$. Then we know we can write $s$ as $s = uvxyz$. I know that |vy| > 0 and $|vxy| \leq p$. So how do I ...
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2answers
168 views

Equivalence of Context-Free-Grammar and Context-Free-Grammar in CNF

Given any Context-Free-Grammar, $G$, and another in Chomsky Normal Form, $G_c$, how can we check if both $G$ and $G_c$ generate the same language? One of the trivial ways I know of is to convert $G$ ...
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1answer
200 views

Prove that X/Y/Z is context-free

Given languages X, Y and Z, each with alphabet, define X/Y/Z as: X/Y/Z = { w ∈ Σ* | ∃u ∈ Y and ∃v ∈ Z; such that wuv ∈ X }. Prove that if X is context-...
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0answers
25 views

Given languages A,B and C, each with alphabet Σ, define A/B/C as: [duplicate]

Given languages A,B and C, each with alphabet Σ, define A/B/C as A/B/C = { w∈Σ* | ∃u ∈ B and ∃v ∈ C; such that wuv ∈ A }. Prove that if A is context-free, and B and C are regular, then A/B/C is ...
2
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2answers
296 views

Does this language have a context-free grammar?

Here is a question that I encountered in one of my exams: Find one context-free grammar that recognizes the language: $\qquad L = \{a^n(b^mc^m)^pd^n \mid m, n, p \geq 0\} $ Can you find such a ...
0
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1answer
514 views

How do I prove that Context Free languages have more memory than FSM [closed]

This is very much clear to me that an FSM has limited memory (sufficient to store present state). How do I prove that (intutively or otherwise) that a CFL has more memory than a DFA or NFA (thus ...
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1answer
69 views

Concatenation among different language types

I am trying to figure out the result of the concatenation among different language types (regular, context free, ...). I think the result strongly depends on the nature of the languages which will be ...
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3answers
208 views

Deciding if language is Context-Free

I need help with deciding if $L$ is context-free. $$L = \{a^pb^{q+r}c^sd^{q+t}e^{p+r} \mid p, q, r, s \ge 0\ , s > t\}$$ Can be rewritten into: $$L = \{a^pb^qb^rc^sd^qd^te^pe^r \mid p, q, r, s \...
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2answers
107 views

Grammar to Language

Having: $\qquad \begin{align} S &\to aT \\ T &\to a \mid UTV \\ U &\to ab \mid ba \\ V &\to ac \mid ca \end{align}$ What language would be generated by this? How ...
2
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0answers
180 views

Constructing a Context Free Grammar for checking non-equality of strings [duplicate]

I have been studying the book Introduction to Computation by Michael Sipser on my own, and I'm stuck on this exercise from the chapter on Pushdown Automato and Context-Free Languages. The exercise is ...
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3answers
4k views

Is the language that accepts strings concatenated with their reverse regular?

If the set of regular languages is closed under the concatenation operation and is also closed under the reverse operation ($x^R$ is the reverse of $x$) then is the language generated by $$\{ww^R|w\in\...
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1answer
373 views

Determine if two grammars for the same language are ambiguous

I'm reading the book: Formal Syntax and Semantics of Programming Languages. I don't understand this exercise: Consider the following two grammars, each of which generates strings of correctly ...
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2answers
5k views

Prove Context Free languages not closed under difference?

Given two context-free languages $L_1$ and $L_2$, the language given by the difference of the two languages, $L_1 - L_2$, is (in general) not context-free. Is it possible to prove this without using ...
3
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3answers
2k views

Are regular and context free languages closed against making them prefix-free?

For a language L we define: $\qquad A(L) = \{ x \in L \mid \text{ no proper prefix of x is in L} \} $ Are regular / context free languages closed under this operation ? For regular languages I ...
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0answers
5k views

Context Free Grammar for $\{A^nB^nC^n | n \in \mathbb{N}\}$ [duplicate]

Is $L = \{A^n B^n C^n \mid n \in \mathbb{N}\}$ a context-free language, e.g. $AAAABBBBCCCC \in L$ If so, what's that context-free grammar that produces it?
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3answers
169 views

How can I produce a summation function from this set of production rules for a grammar?

I am not entirely sure if the title is the correct way to phrase what is occurring. There is a recurring process which I decided to attempt to model using production rules similar to those used in a ...
2
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1answer
4k views

Does the empty language have a CFG in CNF?

I just not sure does empty set have a context-free grammar in Chomsky normal form? That is, for $B=\emptyset$, then a context-free grammar is $S \to S$, I think which doesn't have a Chomsky normal ...
2
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2answers
107 views

I'm lookin for a method to construct a particular grammar?

I'm looking for an algorithm to construct a grammar which, given a set of words which can have multiple identical symbols, represents a compressed version of this set, that is, I can generate only the ...
0
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1answer
751 views

Removing identical variables in CFG Unit Productions

Productions of the form A-> A are removed immediately, but what if the production is of the form A -> AA? example: A -> AA | a
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2answers
588 views

Myhill-Nerode style characterization of CFL?

Define the Nerode equivalence over a language $L \subseteq \Sigma^{*}$ as $u \sim_L v$ iff $uw \in L \Leftrightarrow vw \in L$ for every $w \in \Sigma^{*}$. The Nerode equivalence ${\sim}_L$ has ...
1
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1answer
78 views

Is CFL closed against exchanging complementation and reversal?

Let $L$ be a language such that $\overline{L}^R$ (the reversal of the complement of $L$) is context-free. Is then also $\overline{(L^R)} \in \mathrm{CFL}$?
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2answers
206 views

Proving that CFLs are closed under even-ness using grammars

This is a question from a 2007 exam paper for a course I'm studying, question 2 on page 2. Theorem: Let $L$ be a context-free language. Let $L_{even}$ be the subset of $L$ consisting of all the ...