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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93
votes
5answers
80k views

How to prove that a language is not context-free?

We learned about the class of context-free languages $\mathrm{CFL}$. It is characterised by both context-free grammars and pushdown automata so it is easy to show that a given language is context-free....
26
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2answers
34k views

How to prove that a language is context-free?

There are many techniques to prove that a language is not context-free, but how do I prove that a language is context-free? What techniques are there to prove this? Obviously, one way is to exhibit ...
48
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1answer
17k views

Show that { xy ∣ |x| = |y|, x ≠ y } is context-free

I remember coming across the following question about a language that supposedly is context-free, but I was unable to find a proof of the fact. Have I perhaps misremembered the question? Anyway, here'...
29
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4answers
25k views

How to prove that a grammar is unambiguous?

My problem is how can I prove that a grammar is unambiguous? I have the following grammar: $$S → statement ∣ \mbox{if } expression \mbox{ then } S ∣ \mbox{if } expression \mbox{ then } S \mbox{ else } ...
7
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2answers
17k views

Context Free Grammar for {a^ib^j | i,j ≥ 0; i ≠ 2j}

Can someone help with this: $L=\{a^ib^j \mid i,j \ge 0 \text{ and } i \ne 2j\}$ I'm trying to write a grammar for this language? I don't know how to do this. I tried this: $S \rightarrow aaAb \...
6
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2answers
7k views

Converting to CFG from a CFL? [duplicate]

I am trying to learn CFG. Now to make a CFG from a CFL it is really difficult for me. Is there any simple rule or steps so that I can easily find a CFG for a CFL. I am trying to solve one problem ...
5
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2answers
30k views

How to convert PDA to CFG

I learned how to convert context-free grammar to pushdown automata but how can I do the opposite? to convert PDA to CFG? For example: to write CFG for the automata My attempt: $S=A_{03}$ because $...
15
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3answers
7k views

Example of a non-context free language that nonetheless CAN be pumped?

So basically L satisfies the conditions of the pumping lemma for CFL's but is not a CFL (that is possible according to the definition of the lemma).
10
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3answers
2k views

If $L$ is context-free and $R$ is regular, then $L / R$ is context-free?

I'm am stuck solving the next exercise: Argue that if $L$ is context-free and $R$ is regular, then $L / R = \{ w \mid \exists x \in R \;\text{s.t}\; wx \in L\} $ (i.e. the right quotient) is context-...
9
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1answer
443 views

How can ws with |w| = |s| and w ≠ s be context-free while w#s is not?

Why does (if so) the seperator $\#$ is making a difference between the two languages ? Let say: $L=\{ws : |w|=|s|\, w,s\in \{0,1\}^{*}, w \neq s \}$ $L_{\#}=\{w\#s : |w|=|s|\, w,s\in \{0,1\}^{*}, ...
7
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1answer
3k views

prove no DPDA accepts language of even-lengthed palindromes

How do you prove that the language of even-lengthed palindromes, i.e., $L=\left\{ ww^R \mid w\in \left\lbrace 0,1 \right\}^* \right\}$, can not be accepted by a determinsitc Push-Down-Automaton? Is ...
10
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5answers
1k views

Language of the values of an affine function

Write $\bar n$ for the decimal expansion of $n$ (with no leading 0). Let $a$ and $b$ be integers, with $a > 0$. Consider the language of the decimal expansions ...
19
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1answer
28k views

Intersection of context free with regular languages

The intersection of a context free language L with a regular language M, is said to be always context free. I understood the cross product construction proof, but I still don't get why it is context ...
16
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1answer
9k views

Construct a PDA for the complement of $a^nb^nc^n$

I am wondering if this is even possible, since $\{a^n b^n c^n \mid n \geq 0\} \not\in \mathrm{CFL}$. Therefore a PDA that can distinguish a word $w\in\{a^n b^n c^n \mid n \geq 0\}$ from the rest of $...
5
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2answers
2k views

Pumping lemma: if you can keep pumping, what does this tell you?

Hypothetically, let's say you are using the pumping lemma for either regular or context free languages. Now using either, you come across a case that remains true despite pumping it. In this situation,...
7
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3answers
2k views

Context-free grammar for $L = \{a^{2^k}, k \in\mathbb{N}\}$

In an exercise, I am asked to find a context free grammar for language $L = \{a^{2^k}, k \in \mathbb{N}\}$. I have been trying to use a "doubling" variable. If $a^{2n} \in L, n\in\mathbb{N}$ then use ...
3
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2answers
6k views

Find a pushdown automaton for { x#y ∣ x ≠ y }

I was told to built a PDA that recognizes the following language: $$L = \{x\#y \mid x,y \in \{0,1\}^{\ast} \wedge x \neq y\}$$ My attempt is basically to push $x$ to the stack for every $1$ and $0$ ...
12
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3answers
6k views

Easy proof for context-free languages being closed under cyclic shift

The cyclic shift (also called rotation or conjugation) of a language $L$ is defined as $\{ yx \mid xy \in L \}$. According to wikipedia (and here) the context-free languages are closed under this ...
7
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3answers
5k views

Proving that any CF language over a 1 letter alphabet is regular

I would like to prove that any context free language over a 1 letter alphabet is regular. I understand there is Parikh's theorem but I want to prove this using the work I have done so far: Let $L$ be ...
2
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1answer
1k views

Context Free Grammar for language L

Can someone help with this: $L=\{a^ib^j \mid i,j \ge 1 \text{ and } i \ne j \text{ and } i<2j\}$ I'm trying to write a grammar for this language? I tried this: $S \to S_1 \mid S_2 \\ S_1 \to aXb ...
1
vote
2answers
386 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 ...
20
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2answers
20k views

Is the complement of { ww | … } context-free?

Define the language $L$ as $L = \{a, b\}^* - \{ww\mid w \in \{a, b\}^*\}$. In other words, $L$ contains the words that cannot be expressed as some word repeated twice. Is $L$ context-free or not? I'...
7
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3answers
19k views

Regular Expression to Context-Free Grammar

Anyone knows if there is an algorithm for directly write the context-free grammar that generates a given regular expression?
20
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2answers
1k views

Are context-free languages in $a^*b^*$ closed under complement?

The context-free languages are not closed under complement, we know that. As far as I understand, context-free languages that are a subset of $a^*b^*$ for some letters $a,b$ are closed under ...
14
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2answers
11k views

Are all context-free and regular languages efficiently decidable?

I came across this figure which shows that context-free and regular languages are (proper) subsets of efficient problems (supposedly $\mathrm{P}$). I perfectly understand that efficient problems are a ...
19
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1answer
2k views

Is language equality for linear context-free grammars decidable?

Let's consider two context-free grammars $G_1$ and $G_2$ and ask the following question: Is $L(G_1) = L(G_2)$, that is, are the two grammars equivalent? In general, this problem is undecidable. ...
8
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1answer
4k views

Are DCFLs closed under reversal?

According to this chart, DCFLs are closed under reversal. However, I am not convinced as the intuitive proof (reversing the arrows of the controlling finite state machine and switching the pushes and ...
5
votes
2answers
506 views

What's the reason for the second condition of the pumping lemma(s)?

For a language $L$ with pumping length $p$, and a string $s\in L$, the pumping lemmas are as follows: Regular version: If $|s| \geq p$, then $s$ can be written as $xyz$, satisfying the following ...
3
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3answers
2k views

Why DCFL is not closed under kleene star?

I have read somewhere that DCFL is not closed under kleene star. but I haven't found any example
2
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1answer
7k 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 ...
12
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2answers
17k views

Examples of context-free languages with a non-context-free complements

Context-free languages are not closed under complementation. In the lectures we have been given the same argument as here on Wikipedia: For $$A = \{\mathtt a^n \mathtt b^n \mathtt c^m;~m, n ∈ ℕ_0\}\...
11
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1answer
2k views

A pumping lemma for deterministic context-free languages?

The pumping lemma for regular languages can be used to prove that certain languages are not regular, and the pumping lemma for context-free languages (along with Ogden's lemma) can be used to prove ...
5
votes
1answer
3k views

A non-regular language satisfying the pumping lemma

I got a problem to solve, which is to demostrate that the language $L$, given by: $L = \{ab^nc^n\mid n \geq 0\} \cup \{a^kw \mid k\geq 2 \wedge w \in \Sigma^*\}$ Satisfies the pumping lemma. Is not ...
3
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1answer
495 views

Inducing a context free grammar [closed]

I have a file containing a subset of possible strings from a context free language. I am looking for a mechanism to induce the grammar from this information. Is that possible?
0
votes
1answer
263 views

Example Context free grammar

Is there a nice way to give context free grammar for $$\{a^nb^ma^kb^l:n+m=k+l\}?$$ From PDA point of view it seems we just push + on stack if we see a, push + on stack if we see b, pop + from stack ...
16
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2answers
2k views

Decidablity of Languages of Grammars and Automata

Note this is a question related to study in a CS course at a university, it is NOT homework and can be found here under Fall 2011 exam2. Here are the two questions I'm looking at from a past exam. ...
11
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2answers
5k views

How can I prove this language is not context-free?

I have the following language $\qquad \{0^i 1^j 2^k \mid 0 \leq i \leq j \leq k\}$ I am trying to determine which Chomsky language class it fits into. I can see how it could be made using a context-...
7
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2answers
956 views

Do NPDA work in parallel?

Assume my language is $$ L= ww^{r}\ $$ Now when we use NPDA for this,we will guess middle every time. It may be actual middle or it may not, so a new branch is created every time as I have a choice ...
5
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2answers
2k views

Designing a PDA w/o $\epsilon$-moves and $\leq 2$ states to accept an $\epsilon$-free CFL by final state

I understand that any CFL can be accepted by a PDA by final state or empty store but I have been rather stumped by this question. The question states that the PDA has at most 2 states. Clearly 1 will ...
3
votes
2answers
141 views

Using pumping lemma to show a language is not context free(Complicated)

How can i show that the following long language is not context free using the pumping lemma? $L=\left\{abc^{i_1}bc^{i_2}...bc^{i_{2m}}def^{j_1}ef^{j_2}..ef^{j_{2n}}ghq^{k_1}hq^{k_2}...hq^{k_o}\right\}...
2
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1answer
2k views

Finding a unambiguous grammar

As an exercise we were supposed to find a grammar $G$ that generates language $L(G) = \{w \in \{a,b\}^* \mid |w|_a = |w|_b\}$. That was not so hard, I found a grammar which I think is correct: $S \...
2
votes
1answer
423 views

Is the language of all ucv with u ≠ v context-free?

Is $L = \{w_1cw_2 : w_1, w_2 \in \{a, b\}^* , w_1 \neq w_2 \}$ a CFL? In my opinion it is not since if we want to know the inequality of $w_1$ and $w_2$ we must be aware of their equality and that is ...
2
votes
1answer
737 views

Is this unambiguous grammar equivalent?

I have a simple context free grammar $(\{A\}, \{(,)\}, R, A)$, which consists of this one production rule: $A \rightarrow AA\, \vert\, (A)\, \vert\, \epsilon$ I believe this is ambiguous! For ...
1
vote
1answer
356 views

Language whose intersection with a CFL is always a CFL

Prove or disprove: If the language $L$ is such that for every context-free language $L_0$, the language $L \cap L_0$ is context-free, then $L$ is regular. I haven't managed to prove this, but I'm ...
35
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2answers
4k views

What does “context” in “context-free grammar” refer to?

There are lots of definitions online about what a Context-Free Grammar is, but nothing I find is satisfying my primary trouble: What context is it free of? To investigate, I Googled "context ...
6
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2answers
4k views

In context-free grammar (CFG), what is the importance of doing both leftmost and rightmost derivations?

I am a beginner learning theoretical computer science. While learning about CFG, I found that doing both leftmost and rightmost derivations gave me the same parse tree. So, my question is: Why is it ...
23
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1answer
3k views

Decide whether a context-free languages can be accepted by a deterministic pushdown automaton

Given a context-free grammar G, there exists a Nondeterministic Pushdown Automaton N that accepts exactly the language G accepts. (and visa versa) There may also exist a Deterministic Pushdown ...
13
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3answers
2k views

What would you get if you add parameters to context free grammars?

I was thinking of grammars for indendation-sensitive languages and it looks like CF grammars would do the trick if combined with parameters. As an example, consider this fragment for simplified Python ...
8
votes
1answer
500 views

Is this language Context-Free?

Is the language $$L = \{a,b\}^* \setminus \{(a^nb^n)^n\mid n \geq1 \}$$ context-free? I believe that the answer is that it is not a CFL, but I can't prove it by Ogden's lemma or Pumping lemma.
8
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2answers
3k views

How do I show that whether a PDA accepts some string $\{ w!w \mid w \in \{ 0, 1 \}^*\}$ is undecidable?

How do I show that the problem of deciding whether a PDA accepts some string of the form $\{ w!w \mid w \in \{ 0, 1 \}^*\}$ is undecidable? I have tried to reduce this problem to another undecidable ...

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