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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2
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1answer
983 views

every DCFL has an LR(1), an LALR(1) and even an SLR(1) grammar

Here is one discussion that says, "every DCFL has an LR(1), an LALR(1) and even an SLR(1) grammar." And wiki says, "a language can be generated by an LR(k) grammar if and only if it is deterministic [...
13
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1answer
5k views

Is there a context free, non-regular language $L$, for which $L^*$ is regular?

I know that there are non-regular languages, so that $L^*$ is regular, but all examples I can find are context-sensitive but not context free. In case there are none how do you prove it?
10
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1answer
410 views

Constructing all context-free languages from a set of base languages and closure properties?

One way of looking at regular expressions is as a constructive proof of the following fact: it's possible to construct the regular languages by starting with a small set of languages and combining ...
9
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1answer
392 views

How powerful are CFGs that allow an infinite number of rules?

I was wondering recently what would happen if we'd allow context-free grammars to have an infinite number of rules. Clearly, if we'd allow arbitrary such infinite sets of rules, every language $L$ ...
7
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1answer
182 views

Smallest class of automata model whose corresponding language class contains CFL and is closed against (dis)allowing nondeterminism in the model

From a comment, an interesting question popped up. The class of CFLs (the languages recognized by PDAs) are obviously not closed under nondeterminism - what I mean by this is that deterministic PDAs ...
6
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2answers
3k views

Is Python a context-free language?

From Wikipedia: Off-side_rule#Implementation, there is a statement: ...This requires that the lexer hold state, namely the current indentation level, and thus can detect changes in indentation ...
6
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2answers
319 views

Grammatical characterization of deterministic context-free languages

Deterministic context-free languages are commonly defined using an automaton concept, the (restricted, deterministic) pushdown automaton. To some that is confusing, as the name context-free refers to ...
6
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1answer
95 views

When is ε-removal from a CFG idempotent?

For which context-free grammars is it idempotent to remove $\varepsilon$-productions? Given that there are multiple rewriting algorithms which preserve language and leave the grammar without $\...
6
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2answers
7k views

Is it decidable whether a given context free grammar generates an infinite number of strings?

Is the decision problem "Does a given context free grammar generate an infinite number of strings" decidable? In order to test whether a context free grammar generates an infinite number of strings or ...
4
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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\...
4
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1answer
418 views

Undecidable problem intersection of two DCFL languages is DCFL?

We have two deterministic pushdown automatas A and B, which languages are deterministic context-free. The problem is to decide if there exists a deterministic pushdown automata, which language is an ...
3
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1answer
333 views

CFG Equivalent of regular expressions

So I was wondering something about the Chomsky hierarchy. DFAs (and NFAs) accept regular languages, while NPDAs accept context-free languages. Right-regular or left-regular grammars produce regular ...
3
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3answers
5k views

Closure of CFL against right-quotient with regular languages

Let $A/B$ = $\{ w \mid wx \in A$ for some $x \in B \}$. Show that if A is context free and B is regular, then $A/B$ is context free. My interpretation of this is is that we need to show that if a ...
3
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3answers
3k views

Is this language LL(1) parseable?

I tried to find a simple example for a language that is not parseable with an LL(1) parser. I finally found this language. $$L=\{\,a^nb^m\mid n,m\in\mathbb N,\>n\ge m\,\}$$ Is my hypothesis true ...
3
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1answer
2k views

Are permutations of context-free languages context-free?

Given a context-free language $L$, define the language $p(L)$ as containing all permutations of strings in $L$ (i.e. all strings in $L$ such that the order of symbols is not important). Is $p(L)$ ...
3
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2answers
344 views

Is $L=\{ xyx^Ry^R \mid x,y \text{ is an element of }\{0,1\}^*\}$ context-free?

Is the language $L=\{ xyx^Ry^R \mid x,y \text{ is an element of }\{0,1\}^*\}$ context-free? Note: $x^R$ is the reverse of $x$. My Work: I think this is a context free language. Since a pushdown ...
2
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1answer
406 views

Is converting an ambiguous grammar to an unambiguous grammar computable?

Is the problem of converting ambiguous grammar into unambiguous grammar computable? (Consider Domain as all context free languages).
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0answers
2k views

Converting Chomsky Normal Forms to Greibach Normal Form

Here is a passage from Kozen's Automata and Computability (pages 145-146) that I'm confused about: Now we show how to convert an arbitrary grammar to an equivalent one (except possibly for $ \...
2
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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 ...
2
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1answer
76 views

Lower bound for number of nonterminals in a CFG

Let's say we have a context-free grammar for the language $a\mbox{*}b\mbox{*}c\mbox{*}$. Is there a way to determine a lower bound for the number of nonterminals in this grammar? I'm pretty sure you ...
2
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2answers
244 views

Can this language be defined by a Context Free Grammer?

I was solving one of my practice questions, defining a language with Context Free Grammar Productions , but I am stuck on one question , Here are my attempt: Question: $L = \{a^n b^m c^p \mid n = m + ...
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2answers
6k views

How to show that given language is unambiguous

Given following grammar: $$ \begin{align} S \rightarrow &A1B \\ A \rightarrow & 0A \mid \varepsilon \\ B \rightarrow & 0B \mid 1B \mid \varepsilon \\ \end{align} $$ How can I show that ...
2
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1answer
134 views

Is the language of words with as many a's in the first as b's in the second part context-free?

Is $L = \{ W_1W_2 \mid W_1,W_2 \in (a+b)^* , N_a(W_1) = N_b(W_2)\}$ context free? Can we construct an NPDA for the language? There is a book here that claims $L$ is not CF (without any elaboration), ...
2
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1answer
366 views

How to use a CFG to restrict a subset of a*b*c*d* so that there are at most as many a's and b's as d's?

Give Context-free Grammar for the language $\{a^i b^j c^k d^h \mid i,j,h \ge 0, k>0, i+j \le h\}$ This is a training exercise, for which we don't get any answers, in a course I'm taking. I have ...
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2answers
397 views

Ambiguous context free

Is there any technique to prove that a given language L is not ambiguous context-free? Here I don't know that whether L is CFL or not.
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1answer
349 views

How can I show context free grammars are strictly more expressive than regular expressions with an example?

I need to show a CFG can express everything that can be expressed by a regular expression, and something that cannot.. I have no idea what example is traditionally used for this.
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2answers
884 views

Context Free Grammar for language

The language is $L = \{a^{i} b^{j} c^{k} \;|\; k \neq 2j\}$. I'm trying to write a grammar for this language, what I have so far is: $S \rightarrow AT_{1} \;|\; AT_{2} \;|\; AT_{3} \;|\; AB \;|\; AC$ ...
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2answers
186 views

Develop the context free grammar to match this language (puzzle)

This is a puzzle type question which asks to create a context-free grammar to match this language: ...
1
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1answer
151 views

Find a CFG for the language $\{ x\$y \mid x,y\in\{a,b\}^* \wedge |x| \ne |y| \}$?

Consider the language below, on the alphabet $\Sigma = \{a,b,\$\}$: $$L = \left\{ x$y \mid x,y\in\{a,b\}^* \land \left|x\right| \ne \left|y\right| \right\}$$ I need to define a CFG for this language. ...
1
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1answer
681 views

Converting to Chomsky normal form

Im having some problems with a qeuestion regarding converting a context free grammar to chomsky normal form. I have S -> abC | babS | de C -> aCa |b I know what to do with the case ...
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2answers
125 views

Prove/ Disprove: If $L$ is a CFL then $A(L)$ is a CFL too

Consider the operation $A(L)$: $$A(L) = \{ w: w\in L \land w_R \notin L \}$$ where $w_R$ is the reverse of $w$. Prove/ Disprove: if $L$ is a CFL language so does $A(L)$. I am almost certain ...
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1answer
3k views

Is Context Free Language closed under perfect shuffle?

Note that this is not shuffle but perfect shuffle, defined as follows: Let $w = a_{1}a_{2} \ldots a_{n}$ and $x = b_{1}b_{2} \ldots b_{n}$ be two strings of the same length. Then the perfect shuffle ...
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2answers
229 views

Is the language of words that contain a square regular or context-free? [duplicate]

$ L = \{w \in\{a,b\}^{*} : \exists_{x,y,z} , w=xyyz \wedge y \neq \epsilon \}$ I have a problem with this exercise. I need to determine if this language is regular, context-free or not both and ...
0
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1answer
165 views

Prove $ L = \{ww^{R} \in \{a, b\}^{*} : |w|_{a} \equiv |w|_{b} \equiv 0$ $ (mod$ $13) \} $ is regular or context-free or neither

$ L = \{ww^{R} \in \{a, b\}^{*} : |w|_{a} \equiv |w|_{b} \equiv 0$ $ (mod$ $13) \} $ Exercises: If the language L is regular (build a DFA or regular expression) else if the language L is context-...
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0answers
4k views

How to prove that the language { ww | w ∈ {a,b}* } is / isn't context free? [duplicate]

Is the language { ww | w ∈ {a,b}* } context free? I have tried to create a pushdown automaton but I didn't find any solution. I think you need a queue and not a stack. Is there a way to prove this ...
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2answers
167 views

Grammar for a language with 1/3 of a's

I have this language: $$ L = \left\{ w \in \{a,b,c\}^* \;\big|\; |w| / |w|_a = 3 \right\} $$ where $|w|_a$ is the number of occurrences of $a$. How can I find a grammar that generates it?
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3answers
15k views

What is complement of Context-free languages?

I need to know what class of CFL is closed under i.e. what set is complement of CFL. I know CFL is not closed under complement, and I know that P is closed under complement. Since CFL $\subsetneq$ P I ...
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1answer
1k views

Arithmetic expressions grammar transformation

In the article Parsing Expressions by Recursive Descent by Theodore Norvell (1999) the author starts with the following grammar for arithmetic expressions: ...
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1answer
1k views

Techniques to prove a language is not DCFL

I know that DCFL is closed under complementation and intersection with regular languages. By using these we can prove that a language is not DCFL. Are there any other techniques that will help me to ...
6
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1answer
336 views

Determining whether a CFG is $LL(k)$ for any $k$?

In Knuth's original paper on $LR(k)$ grammars, he proved that the decision problem "Given a CFG $G$, is there a $k$ such that $G$ is an $LR(k)$ grammar?" is undecidable. Is there a similar result ...
5
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1answer
699 views

Deterministic context-free languages are closed under regular right-product

I am looking for a proof for the following problem: For languages $L$ and $R$, if $L$ is deterministic context-free and $R$ is regular, then $LR$ is a deterministic context-free language. Note:...
4
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1answer
620 views

Are LR(k) languages and DCFLs equivalent?

In the familiar book of Theory of Computation by M. Sipser, the author proved that for endmarked context-free languages, the set of languages having a LR(k) grammar for a predefined $k \in \mathbb{N}$ ...
3
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1answer
306 views

Is this a Context Free Language?

I got this question on my final exam: Is the following language context-free? $$ L = \{w\bar w^R \mid w\in \{0,1\}^* \}$$ Notation: The string $\bar w$ is obtained from $w$ by replacing all 0s ...
3
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1answer
525 views

Solving the emptiness problem for a CFG in Chomsky normal form (linear)

Given a CFG in Chomsky normal form, is there an algorithm that solves the emptiness problem in linear runtime? I thought about using depth search here, but I think it's a little bit above linear ...
3
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1answer
77 views

Language whose intersection with a CFL is always a CFL (2)

This is a follow-up to this question, which asks for an example of a non-regular language $L$ which satisfies the following condition, intersection resilience: If $L'$ is context-free then so is $L ...
3
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1answer
2k views

PDA for { xy : |x| = |y|, x ≠ y} from its grammar, and intuition behind it

I know the grammar for the language $\{ xy : |x| = |y|, x ≠ y \}$ if $\Sigma=\{a,b\}$: $$ \begin{align*} &S→AB∣BA \\ &A→a∣aAa∣aAb∣bAa∣bAb \\ &B→b∣aBa∣aBb∣bBa∣bBb \end{align*} $$ I ...
3
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1answer
743 views

Is Myhill-Nerode equivalence class of a language which contains all palindrome pairwise distinct?

In my formal language class, we define a language called PAL, which is on a alphabet set $\Sigma = \{0,1\}$. $PAL = \{w \in \{0,1\}^* : w = w^R\}$. We have proved that every string in this language ...
2
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1answer
1k views

Prove that regular languages and context-free languages aren't closed under $Perm(L)$

Let the operation $$Perm(L) = \{ w | \exists u \in L \text{ such that } u \text{ is a permutation of } w \}$$ Prove that both regular languages and CFLs aren't closed under $Perm(L)$. I've tried ...
2
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1answer
103 views

Are Context-Free Grammars with only one Production Rule always Unambiguous?

Consider the following (Context-Free) Grammars with only one production rule (not including the epsilon production): $S \rightarrow aSb\;|\;\epsilon$ $\require{cancel} \cancel{S \rightarrow aSSb\;|\;\...
2
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0answers
184 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 ...