Questions tagged [formal-languages]

Questions related to formal languages, grammars, and automata theory

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Is $L = \{xw^3x^{rev}\mid x, w\in\{0, 1\}^*\}$ context-free?

The title pretty much explains the question, but still: Is the language $$L = \{xw^3x^{rev}\mid x, w\in\{0, 1\}^*\}$$ context-free? I think it isn't and would motivate that suspicion by the following ...
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2answers
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Same notation/terminology for union of sets and concatenation (Kleene star)?

For the union of sets we use the union operator $\cup$ (or $\bigcup$). And for a concatenation (Kleene star) we also use the union operator. The operations are different, but why the same terminology ...
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1answer
14 views

Poping a symbol on a PDA when Input and Stack are Irrelevant

Say I had a PDA with alphabet language {0,1}, and a stack language {P,Q,\$}. In the PDA I don't really care what the inputs are at the end and I just want to clear the stack back down to the special ...
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1answer
32 views

Context-free grammar for $a^{2n} b^{2n}$

I have just started learning formal languages and here is a question I am facing a little hurdle: Construct a context-free grammar for $\{ a^{2n}b^{2n} \mid n \ge 0 \}$. This was what I got at first....
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1answer
25 views

Is it true that PRIMES are in SPARSE?

I'm wondering if PRIMES, the language of all prime numbers represented in binary, which is $\{10, 11, 101, 111, 1011, 1101, ...\}$, belongs to the SPARSE class, a set of all sparse languages, that is, ...
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1answer
20 views

Question about reduction Proof

I've recently seen a proof that the set of Turing machines $L = \{encode(M) |L(M) \text{is closed under reversal}\}$ is not decidable. The proof used following idea: Reduce from the $A_{TM}$ problem ...
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1answer
151 views

Proving $\{0^{2^n}\}$ is not regular using pumping lemma

I am currently learning the pumping lemma, and encountered the following question, which I am unable to solve: Prove that $L = \{ 0^n \mid \text{$n$ is power of 2}\}$ is not regular. I considered $w ...
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3answers
929 views

Show that $L = \{1^n w 1^n | n > 0 \text{ and } w ∈ \{0,1\}^*\}$ is regular

Show that the following language $L$ is regular by describing it using a regular expression. $$L = \{1^n w 1^n \mid n > 0 \text{ and }w ∈ \{0,1\}^*\} $$ My (apparently incorrect) answer: Given ...
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0answers
34 views

Prove that $ \{w \in \{a\}^* | \nexists n >0: |w| = n^{2}\}$ is not context-free [duplicate]

I've to prove that: $ \{w \in \{a\}^* | \nexists n >0: |w| = n^{2}\}$ is not context-free with the Pumping Lemma. Any clues?
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0answers
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How to check whether a language is regular or not? [duplicate]

I am given expressions such as \begin{align} L_2 &= \{ a^n b^{n!} \}, \\ L_3 &= \{ abcva^n \mid v \in \{a,b,c\}^*, n \in \mathbb{N}, n \text{ is even}, |v|=n/2 \}. \end{align}
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1answer
26 views

Decidability of $\{⟨G⟩ \mid \text{$G$ is CFG and $L(G) ⊈ \Sigma^+$}\}$

I want to prove that the following language is decidable: $$\mathit{SEQ}_{\mathit{CFG}} = \{⟨G⟩ \mid \text{$G$ is CFG and $L(G) ⊈ L$}\}, \text{ where } L = \Sigma^* - \{\epsilon\}$$ So, I think about ...
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1answer
750 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 $k \in \mathbb{N}$ ...
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1answer
83 views

Using the pumping theorem to show that this language is not context-free

Let $\sigma = \{a,b,c\}$ and let $L = \{s | s = a^jb^jc^k\}$ where $k=i\cdot j$ and $i,j \geq 0\}$. Using the pumping theorem, prove that $L$ is not context-free. I really don't know where to start, ...
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2answers
6k views

How to construct Context Free Grammar of words with equal number of 0's and 1's [duplicate]

i am trying to find a cfg for this cfl L = $\{ w \mid w \text{ has an equal number of 0's and 1's} \}$ is there a way to count the number of 0's or 1's in the string?
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1answer
46 views

Closure of context-sensitive languages under inverse language substitution

We define language substitution for a Context-Sensitive Language (CSL) $S$ over an alphabet $\Sigma$ is a map from $\Sigma$ into CSL's, for example: $f(abc) = L_1(a) L_2(b) L_3(c)$ such that (I guess) ...
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1answer
27 views

Proving undecidability of a language with mapping reductions

I'm referring to questions like this one: Mapping reduction to show NeverHalt is undecidable I understand with Turing reductions, you have to use oracle calls of the unknown language you're trying to ...
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1answer
45 views

Describe regular expression

I am learning about regular expression, and trying to describe a regular expression for the language L $\qquad L = \{a^i b^j c^k \mid i+j = k\}$ What is the right approach and how to describe a ...
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0answers
30 views

Unambiguous formal grammars for a specific class of languages

Suppose that $w \in \{0; 1\}^*$ is a binary word. Let's denote the number of $0$-s in $w$ as $\#_0(w)$ and the number of $1$-s in $w$ as $\#_1(w)$. Now suppose that $q \in \mathbb{Q}$ is a positive ...
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1answer
27 views

How to define the languages of the implicit set system problems?

There are implicit versions of some set system problems or matroid problems. A set system is a pair $(U, \mathcal{F})$, where $U$ is a universe of size $n$ and $\mathcal{F}$ is a collection of susbets ...
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5answers
18k views

Regular expression for a binary string containing even number of 0's

To get the regular expression I made a finite automata as the following (not sure if you can directly write regular expression without it): The regular expression for the above according to me ...
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3answers
4k 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 ...
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0answers
37 views

Designing CFG that accepts $a^n b^m c^p$ where $n=m+p+2$

I have generated the CFG of $a^n b^m c^p$ where $m = n+p+2$: $S \rightarrow ASC \mid \varepsilon$ $A \rightarrow aAb \mid \varepsilon$ $C \rightarrow bCc \mid \varepsilon$ I have been trying $a^n b^...
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1answer
30 views

Is a Turing Machine a Well-formed formula?

Today i wrote something about the bijection between turing machines and recursive functions. And i describe a Turing Machine as a Well-formed formula because it seems like a WFF to me. But is it ...
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1answer
46 views

Finding the language generated by this grammar

I'm having problems with this. Can someone help me please. Find the language generated by this grammar over the alphabet $\{0,1\}$: $S\rightarrow BAB\mid CAB$ $BA \rightarrow BC$ $CA \rightarrow AAC$ ...
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1answer
15 views

Show that a language with union is not regular by using pumping lemma

Given the language $L:= { \{ c^{2k} w \ \vert \ k \ge 1, \ w \in \{a,b,c\}^* \ and \ \vert w\vert_a \ = \ \vert w\vert_b \} \ \cup \ \{ a,b \}^* }$ I'm really unsure how to even start because of the ...
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2answers
2k views

Is intersection of regular language and context free language is “always” context free language

I have read that intersection of regular language and context-free language is always context-free. Most of the places an standard example has been used to prove this, e.g., \begin{align*} L_1 &= ...
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2answers
36 views

Proving that a language defined by a regular expression is equivalent to a right linear grammar

After thinking for a bit, I am not able to prove a double inclusion proof for the following problem. It seems very interesting to me. Consider the regular expression $r= ((1(00)^∗1 + 0)1)^∗$ and the ...
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2answers
50 views

How is CLR(1) grammar more powerful than LALR(1) grammar

I am unable to understand how Canonical LR(1) grammar is more powerful than LookAhead LR(1). Both have lookahead symbols in their items and works almost similarly, so how can CLR(1) derive a larger ...
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1answer
206 views

How to prove the language of contractible strings is context-free but not regular?

How to prove this language is context-free but not regular? I can't figure out it. A string is contractible if there is a sequence of contractions which result in the empty string, where a ...
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0answers
24 views

Prove $L =\{0^{2^n}\mid n \geqslant 0\}$ is not context free [duplicate]

Here $0^j$ means $0$ repeated $j$ times e.g. $0^2$ is $00$. So to prove this I was asked to use the pumping lemma. So let $m$ be the pumping length and assume $L$ is a CFL by contradiction. We can ...
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1answer
37 views

Proof that for every $k > 1$, there exists a language $A_k \subseteq \{0, 1\}^*$ s.t. a DFA accepting $A_k$ has $k$ states but no less

I am trying to prove that for every $k > 1$, there exists a language $A_k \subseteq \{0, 1\}^*$ such that a DFA accepting $A_k$ has $k$ states but no less. I thought about proving this in two ways: ...
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1answer
2k views

Closure properties of linear context-free languages

Under what operations are linear context-free languages closed? Suppose $L_1, L_2$ are two linear context free languages. Are there any guarantees about $L_1 \cup L_2$, $L_1 \cap L_2$, $\overline{L_1}...
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1answer
1k views

Pumping lemma for linear languages

Let $L$ be a linear language. Then there is a constant $p$ such that for all $w$ in $L$ with $|w| \ge p$, $w$ can be written as $uvxyz$ where (i) $|uvyz| \le p$ (ii) $|vy| > 0$ (iii) $uv^ixy^iz$ is ...
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1answer
47 views

CFG for $\{uvw \mid u,v,w \in\{0,1\}^*,|u|=|v|=|w| \wedge u\neq w\} $

$L=\{uvw \mid u,v,w \in\{0,1\}^*,|u|=|v|=|w| \wedge u\neq w\} $ Any help would be appreciated.
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1answer
132 views

How to design a formal grammar to convert EBNF description to a list of CFG production rules

I would like to write a grammar to convert EBNF description to a list of CFG production rules, instead of an algorithm. Can CFG production rules is generated from an EBNF description by a rewrite ...
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3answers
2k views

Proving that recursively enumerable languages are closed against taking prefixes

Define $\mathrm{Prefix} (L) = \{x\mid \exists y .xy \in L \}$. I'd love your help with proving that $\mathsf{RE}$ languages are closed under $\mathrm{Prefix}$. I know that recursively enumerable ...
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1answer
223 views

Is $\{ w_1cw_2 \mid w_1 ≠ w_2 \}$ a context-free language?

Is the language $L_1 = \{w_1cw_2 ~|~ w_1,w_2 \in \{a,b\}^{\ast} \text{ and } w_1 \neq w_2\}$ a context-free language? It certainly isn't regular, but is it context free? I'm having trouble creating ...
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1answer
31 views

PDA translating $a^{m+n} b^n$ to $x^{2m+2} y^{3n}$

On my compilation theory exam we had the following problem: Construct a PDA translator (just one stack) such that it translates the language $$ a^{m+n}b^n \rightarrow x^{2m+2}y^{3n}, \text{ where } n,...
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1answer
8k views

Why is the complement of a language that is not regular also not regular?

In my lecture notes I we were given two languages and were shown that each of the two languages were not regular. The second was the complement of the first language. To show the second was not ...
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4answers
7k views

Are modern programming languages context-free?

Which language class are today's modern programming languages like Java, JavaScript, and Python in? It appears (?) they are not context-free and not regular languages. Are these programming languages ...
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1answer
39 views

Proof that class of languages accepted by DPDA by empty stack is not closed under union

My first intuition was to take two languages $L_1$ and $L_2$ (symbol $d$ at the end is to fulfill prefix property): $$L_1 = \{ a^i b^i c^j d : i,j \ge 0 \} \mathrm{\ \ and\ \ } L_2 = \{ a^i b^j c^j d :...
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0answers
44 views

Checking correctness of grammar for $L = \{w \in \{a, b\}^* \text{ }| \text{ } w \text{ has } n_a(w) = 2n_b(w)\} $

I have written a CFG that supposedly generates $L$ below. $$L = \{w \in \{a, b\}^* \text{ }| \text{ } w \text{ has } n_a(w) = 2n_b(w)\}$$ Where $n_a(w)$ is the number of $a$'s in $w$ and similarly for ...
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1answer
2k views

Understanding application of Arden's theorem to find regular expression

I learnt Ardens theorem and its usage as follows: Ardens Theorem Let $P$ and $Q$ be two regular expressions over alphabet $Σ$. If $P$ does not contain null string, then $R = Q + RP$ has a ...
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1answer
30 views

Conditions for an Language to be infinite

given 'r' , a regular expression that does not include λ or ∅, What are the Conditions of 'r' so that L(r) would be infinite?
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19 views

Algorithmically find a formal grammar for a recursively enumerable formal language

The algorithmic problem is as follows. The input is the source code of a program accepting an integer as input and outputting a finite binary sequence. This program defines a recursively enumerable ...
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2answers
161 views

Is $L = \{ a^ib^j : 0 < j < i < 2j\}$ context free? How can it be shown?

Is $L = \{ a^ib^j : 0 < j < i < 2j\}$ context free? If so, can there be a pushdown automaton described for it? If not, does the pumping lemma apply?
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4answers
96 views

Proving that $L=\{ w \mid \lvert w \rvert$ is prime $\}$* is a regular language

I'm trying to prove that the following languague is a regular language: $L=\{ w \mid \lvert w \rvert$ is prime $\}$* What I have thought is to divide each word $w \in L$ into subwords of length 2 if ...
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1answer
34 views

Create a CFG for $L = \{ a^ib^j \mid \lvert i - j \rvert \le 2 \} $

I'm trying to find a CFG for the following language: $L = \{ a^ib^j \mid \lvert i - j \rvert \le 2 \} $ What I thought about unsuccessfully is the following: $S \rightarrow SASBS \mid SBSAS \mid \...
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1answer
21 views

Using pumping lemma to prove that $L = \{ a^ib^j \mid \lvert i - j \rvert \le 2 \} $ is irregular

Given the following language: $L = \{ a^ib^j \mid \lvert i - j \rvert \le 2 \} $ I am trying to prove that it is not regular. On the one hand my intuition tells me that the language is non-regular as ...
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2answers
24 views

Proving Irregularity of $L = \{ a^mb^nb^n \mid nm \ge 3 \} $

I'm trying to prove the irregularity of the following language: $$L = \{ a^mb^nb^n \mid nm \ge 3 \} $$ I tried to demonstrate that it doesn't verifies the Pumping Lemma but for all words I tried it ...

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