Questions related to formal languages, grammars, and automata theory

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1answer
58 views

Show that language generated by grammar is regular

We have grammar with nonterminals $ X_1,...X_n$ terminals $V_t$ and rewriting rules of form: $X_i \rightarrow a \in V_t $ $X_i \rightarrow X_jX_k, \ i \ge j , \ i > k $ How can I show that ...
10
votes
2answers
580 views

Why are regular expressions defined with union, concatenation and star operations?

A regular expresssion is defined recursively as $a$ for some $a \in \Sigma$ is a regular expression, $\varepsilon$ is a regular expression, $\emptyset$ is a regular expression, $(R_1 \cup R_2)$ ...
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0answers
19 views

inclusion relation or equality of languages

Draw a bull's eye diagram that shows inclusion relation or equality between the following sets-Give an example in each case:  The set of regular expressions REX  The set of all regular languages ...
-3
votes
0answers
34 views

Describing the language of a CFG

Let G be a CFG such that: S -> aSb | bY | Ya Y -> bY | aY | lambda Give a simple description of L(G) in English. From what I can tell from this ...
1
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1answer
42 views

Of which Chomsky-type is the language $L = \{a^jb^ic^{2i} | i,j \in \mathbb{N}^0\}$?

At first I thought the language would be context sensitive because it seems that it can be shown with the pumping lemma for regular languages, that it's not a regular language and analogously with the ...
14
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1answer
242 views

The number of different regular languages

Given an alphabet $\Sigma = \{ a,b \}$, how many different regular languages are there that can be accepted by an $n$-state non-deterministic finite automaton? As an example, let us consider $n=3$. ...
2
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1answer
32 views

How to use the Pumping Lemma to prove that a restricted subset of 0*1*2*3*, where there are as many 3's as 0's and 1's, is not a CFL?

Use the pumping lemma for context-free languages to show that the following language is not context-free: $ L = \{0^i 1^j 2^i 3^k \mid k=i+j \} $ So I have started like this: Let us assume ...
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1answer
67 views

Order classic notions of computability by power

I need some help with a question. I'm currently studying for an exam and I could therefore use some help with this following question: Order the following formalisms (but one) according to their ...
2
votes
1answer
28 views

Prove that language of possible stack content is regular

So, here's the problem: Suppose that $A=(Q,\Sigma,\Gamma,\delta,s,\bot, F)$ is a PDA, let $$L = \{ \gamma \in \Gamma^* \hspace{5pt}|\hspace{5pt} \exists_{x,y\in \Sigma^*} \exists_{q\in Q}: ...
2
votes
3answers
48 views

Is there a standard (common) notation for the following operation on binary strings?

Typically, we use the notation $S = \{0, 1\}^n$ to denote the set of all $n$-bit strings. Suppose that I wanted to extract a subset of the strings where certain bits have some fixed values. For ...
0
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0answers
6 views

OCL constraints are used to validate/verify instances of meta models. Which (v/v) is true?

I have a meta model of which valid instances are defined by OCL invariants. I'm not sure whether to say that I am validating or verifying instances of that model when I check whether they conform to ...
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0answers
15 views

LL(1) grammar for reverse polish notation [closed]

Could anyone provide me an example of LL(1) grammar for reverse polish notation? Exactly LL(1).
0
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1answer
35 views

The Chomsky–Schützenberger representation theorem

I've been trying to proof The Chomsky–Schützenberger, but I stuck on creating regular language from that theorem. I mean reagular language, which is intersected with Duck language. Could anyone give ...
1
vote
1answer
69 views

Can every context free grammar be transformed into equivalent grammar of this form?

Show, that every context free grammar can be transformed into equivalent context free grammar ( with possible loss of $\lambda $ ) where $a \in V_t$ and $A,B,C \in V_n $ with rewriting rules of ...
3
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0answers
45 views

Context free grammar as minimal solution of a system of equations

It is a well-known fact that language generated by a context-free grammar is the minimal solution of a particular system of equations, for example: $$\begin{align*} X &=\{{\epsilon}\} \cup Y\\ X ...
5
votes
1answer
3k views

Are all languages in P?

Are all languages in $\mathbf{P}$? Note: The definitions of all the symbols and functions here follow the document [1]. The following is my attempt to answer the question. Assume that we design a ...
13
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1answer
308 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. ...
3
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1answer
32 views

How to recursively infer a word/string from a context-free grammar?

Give the recursive inference of the word $abcddd$ from the Context-free Grammar: $A\rightarrow aAd\mid B$ $B\rightarrow bBd\mid C$ $C\rightarrow cC\mid cD$ $D\rightarrow Dd\mid ϵ$ This ...
4
votes
1answer
54 views

How to use Parikh's Theorem to show language is not context free

Parikh's Theorem is quite complicated, I understand intuition of that theorem but I don't see how to use that to prove that language is not context free. I kindly ask you to show me how to do, ...
0
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0answers
33 views

Pumping Lemma to prove that L is not context free

I have the language and I want to prove that is not context-free. So I started like this: is variable. Choose w = Case 1: vxy has no c. Choose i = 2 has more a than c or more b than c. Case 2: ...
0
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1answer
27 views

What do we mean when we say an edge (u,v) connects some component to other component in forest G = (V,A)

Let H = (V,E) be a connected, undirected graph. Let A be a subset of E. Let C = (W , F) be a connected component (tree) in the forest G = (V,A). Let (u,v) be an edge connecting C to some other ...
0
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1answer
57 views

How to prove {a^(n^2) | n>0} is not context-free?

So I have a language: $$ L = \{a^{n^2} \mid n > 0\} $$ I need to prove that this language isn't context-free using the pumping lemma. I have a vague thought process as to how to do the proof but ...
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1answer
61 views

Is every countably infinite language recursive?

We'll say the alphabet for the languages is finite, say {0,1}.
2
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1answer
92 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 ...
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2answers
580 views

Using pumping lemma to show $L = \{a^i b^j a^k \ | \ k > i + j\}$ cannot be accepted by an FA

$L = \{a^i b^j a^k \ | \ k > i + j\}$ Use the pumping lemma to show that this language cannot be accepted by an FA. Proof: Suppose $L$ can be accepted by an FA. Suppose a string $s = ...
1
vote
1answer
34 views

CFL that runs in NP-time

What is an example of a context-free language that runs in NP-time? I've done searches but cant find one. Frankly, I do not know how to determine when a CFL is P or NP. Can someone tell me, please?
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0answers
22 views

How to prove a language is not context free using pumping lemma? [duplicate]

So I have a question in particular here, I need to prove that the following is not context free: $\{0^m1^n0^m1^n | m,n \in \mathbb{N} \}$ I am fully aware that I need to use the pumping lemma for ...
8
votes
2answers
177 views

Method for measuring the 'similarity' between FSA grammars?

I'm working with a pattern matching algorithm that generates an acyclic finite state automaton that accepts a given text string and all its substrings. The FSA algorithm is being run on a symbolic ...
0
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0answers
41 views

Proof of completeness for CFG having twice as many zeroes as ones [duplicate]

One possible CFG containing twice as many zeros as ones can be, S -> 0S0S1S | 0S1S0S | 1S0S0S | ϵ (This CFG is redundant but it will do the job. So I am not interested in the redundancy. Other ...
1
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3answers
207 views

Regularity of “middles” of words from regular language

I need some help with the following problem. Let $L \subseteq \Sigma^*$ be a regular language. I have to prove that the language $P = \{\alpha \mid \beta\alpha\gamma \in L, \beta,\gamma \in ...
0
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1answer
38 views

Showing a language is a subset of another language?

I'm actually trying to give an example of a language being context-free and its superset that isn't context-free. I came up with this, but I'm not sure if this particular language is a superset of the ...
0
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1answer
38 views

Prove using pumping free lemma for context-free languages

One of the exercises I tried to make I failed miserably. The question was as follows: Show that the language $L = \{ w \,|\, n_a(w) \cdot n_b(w) = n_c(w) \}$ is not context-free. (with $n_a(w)$ ...
7
votes
1answer
458 views

Is it possible to build DFA for odd-length words with 1 in the middle?

$L := \{w \in \{0,1\}^* | $the length of $w$ is odd $ \wedge $ 1 is in the middle of $w\}$ So the alphabet is $\{0,1\}^*$. My problem is that I can't keep track of the equality of chars before and ...
0
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1answer
44 views

If the strings of a language can be enumerated in lexicographic order, is it recursive?

If the strings of a language L can be effectively enumerated in lexicographic order then is the statement "L is recursive but not necessarily context free" is true?
7
votes
2answers
1k views

Will $L = \{a^* b^*\}$ be classified as a regular language?

Will $L = \{a^* b^*\}$ be classified as a regular language? I am confused because I know that $L = \{a^n b^n\}$ is not regular. What difference does the kleene star make?
0
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2answers
2k views

Is $a^n b^n c^n$ context-free? [duplicate]

I am new to grammars and I want to learn context free grammars which are the base of programming languages. After solving some problems, I encountered the language $$\{a^nb^nc^n\mid n\geq 1\}\,.$$ ...
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1answer
66 views

Need to give a CFG for this language?

I have the language: $$ L = \{0^m1^n \mid 0 ≤ m ≤ n\text{ or }0 ≤ n ≤ 2m\}. $$ My goal is to give an equivalent context-free grammar for this language, but I am unsure if I am going about it the ...
0
votes
3answers
84 views

What is the language generated by a given grammar

Given the grammar $s \to aSb \mid bSb \mid a \mid b$; what is the language generated by the grammar over the alphabet $\{a,b\}$? When I was solving this question I was a bit confused about ...
0
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2answers
69 views

If L is a regular language, how to prove that L' is also regular?

I've been trying to construct a proof of the following statement the whole day but I got stuck: If $L$ is a regular language, the language $L_{}{'}$ consisting of all words in $L$ containing the ...
1
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2answers
61 views

Can a regular language have uncountably many strings?

Obviously it can have a countably infinite number of strings. (Take the language descibed by the regular expression 0* as an example.) But can a RL have uncountably many strings? I'm leaning toward ...
1
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1answer
103 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 ...
1
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1answer
16 views

What is the complement of the language with all ucv with u ≠ v?

If $L = \{w_1cw_2: w_1,w_2 \in \{a,b\}^* , w_1 \neq w_2\}$ what is the complement of language L? one of my friend said that it is $\overline{L} = \{w_1cw_2: w_1,w_2 \in \{a,b\}^* , w_1 = w_2\}$ and he ...
3
votes
1answer
38 views

Is there an example of a recursive language which is not context sensitive?

I have been looking for a prototypical language for recursive languages (decidible) which is no context sensitive without success. For instance $a^*$ is prototypical of regular languages, $a^nb^n$ for ...
0
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2answers
32 views

Is the given language finite or infinite?

I have an idea regarding whether this language is finite or not, but for some reason I am still having some issues regarding exactly grasping what makes a language finite or infinite. I know that ...
0
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0answers
28 views

Is the language of all DFAs that accept the empty language regular?

Is $E_{DFA}$ in the class of regular languages? $\qquad E_{DFA} = \{ \langle D \rangle \mid D \text{ is a DFA }, L(D) = \emptyset\}$ My argument is that it is because all of the DFAs in $E_{DFA}$ ...
0
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1answer
33 views

Convert C language code to problem specification by computing the invariant of a program

Suppose that you need to give a problem specification of some problem P and you have an implementation of P, in C. I have 2 questions: Can you obtain the formal specification of the problem if you ...
1
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2answers
58 views

Show that a language cannot be generated by linear grammar

I have a language $ L= \{ w \in \{a,b\}^* ; |w|_b=2i, i \ge 0 \}$ that is a language with even number of b's. I found a grammar for it with these rules: $S \rightarrow aS \ | \ bL \ | \ \lambda ...
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1answer
50 views

Prove that TM does not decide this language

So my problem is how can I show that this TM does not decides this language. $$L = \{a^nb^nc^n\ |\ n \geq 0\} $$ It might be a basic problem and seem silly to you but still I do not know how to ...
3
votes
2answers
47 views

Meta-grammar for context-free grammars

Formal grammars like regular expressions (REs) or context-free grammars (CFGs) specify languages, i.e. sets of strings over an alphabet. Grammars themselves can be seen as languages, e.g. the set of ...
1
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1answer
84 views

Regular and Non-Regular Language

My friends and I are taking a formal languages class and for one of our homework questions we have to prove if these languages are regular: 1) L = {apaqi : p and q are fixed integer values, i >= 0} ...