Questions related to formal languages, grammars, and automata theory

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3
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0answers
43 views

Automatic tool for resolving left-recursion within CFG [closed]

Though facing the fear that someone might not like my question but does somebody know a useful tool to either resolve left recursion or to simplify a context-free grammar automatically ? I need ...
3
votes
1answer
125 views

Generators of families of langauges?

From Wikipedia's definition of regular langauges The collection of regular languages over an alphabet $Σ$ is defined recursively as follows: The empty language $Ø$ is a regular language. ...
1
vote
3answers
111 views

What happens with trios, full trio, (full) semi-AFL, (full) AFL if we require closure under intersection?

Wikipedia says: A trio is a family of languages closed under e-free homomorphism, inverse homomorphism, and intersection with regular language. A full trio, also called a cone, is a trio ...
0
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2answers
47 views

Can a language be the one recognized by more than one automatons?

The language recognized by an automaton is defined as the set of strings that are accepted by the automaton. I wonder if it is possible that the languages recognized by two automatons are the same? ...
0
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1answer
54 views

Definition of prefixes of a string

From Wikipedia: The prefixes of a string is the set of all prefixes to a string, with respect to a given language: $$ \operatorname{Pref}_L(s) = \{t \mid s=tu \mbox { for } t,u\in ...
3
votes
1answer
44 views

Definition of the cyclic shift of a formal language?

Wikipedia says, the context-free languages are closed under the cyclic shift of $L$ (the language $\{vu : uv \in L \}$) So I am looking for the definition of the cyclic shift operation on formal ...
2
votes
1answer
29 views

Differences between substitution and rewriting?

I am continuing with my self-study of formal languages. Given two alphabets $\Sigma$ and $\Delta$, a string substitution is a mapping from $\Sigma$ to $\mathcal P(\Delta^*)$, which induces a mapping ...
5
votes
2answers
149 views

Which language families admit inductive definitions?

I am self-learning about formal languages. I learned that the family of the regular languages can be defined inductively, in terms of the operations they are closed under (namely the smallest ...
1
vote
1answer
65 views

Formal language inverse

How can you specify the "inverse" of a word, so: let's say a word consists of a's and b's the language is: $ww^{-1}$ the second word is the same as the first but every a is replaced by b and every b ...
1
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2answers
121 views

Syntax and formal grammar of a formal language

For a formal language, I wonder what differences and relations are between its syntax and its formal grammar. A formal grammar is a set of formation rules that describe how to generate the strings ...
1
vote
1answer
187 views

Why is this language over {a,b,c} regular?

The language of all words over the alphabet {a,b,c} such that the number of as in the word minus the number of cs in the word is divisible by three. How is this language regular? Lecturer ...
-1
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2answers
144 views

Using the Pumping Lemma to show that the language $a^n b a^n$ is not regular

I've seen a lot couple of questions regarding the pumping lemma that are pretty similar to each other and this one is unfortunately not the exception. Most likely will be this question marked as a ...
10
votes
1answer
341 views

Computational complexity vs. Chomsky hierarchy

I'm wondering about the relationship between computational complexity and the Chomsky hierarchy, in general. In particular, if I know that some problem is NP-complete, does it follow that the ...
-1
votes
1answer
31 views

Construct grammar given the following language [duplicate]

Construct grammar given the following language! $ L = \{(ab)^{n+1}u(ba)^n|n>0, l_c(u) = 1, u\in\{a,c,d\}^* \}$ My interpretation in a less accurate way: $(ab)^{n+1}$ says we need to concatenate ...
1
vote
1answer
134 views

Draw a graph of DFA for a regular language

I'm trying to draw a DFA graph for the regular language where every chain: ...
1
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0answers
30 views

Notable decidable operations on context-sensitive languages [closed]

It is not always so easy to determine which basic questions on languages are (un)decidable. Also due to Rice's theorem, many nontrivial questions on languages are undecidable. What are notable or ...
1
vote
2answers
45 views

Proving a language is not decideable using a reduction from Busy Beaver?

I was given this function: $F(n)$ returns the smallest TM (measured in number of states) such that on input $\epsilon$, the TM makes at least $n$ steps before eventually halting ($n$ is a natural ...
0
votes
1answer
35 views

Is my grammar correct for this context-free language?

$\{a^nb^2a^n \mid n \ge0\}$ I'm studying for my final and I came across this language. I haven't dealt with characters of the same length on opposite ends with something in between. I came up with ...
2
votes
3answers
79 views

What is the regular expression to the given language?

I can't really find out, how can the following given Language be written down with regular expressions $ L = \{ a^{3k-1} b^n a^{2t} \mid n > 0; k, t\ge1 \} $ I had some guesses, but I don't know ...
3
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0answers
66 views

What are appropriate isomorphisms between formal languages?

A formal language $L$ over an alphabet $\Sigma$ is a subset of $\Sigma^*$, that is, a set of words over that alphabet. Two formal languages $L$ and $L'$ are equal, if the corresponding sets are ...
11
votes
1answer
161 views

The number of different regular languages

My question is: Given an alphabet $\Sigma = \{ a,b \}$, how many different regular languages are there that can be accepted by an $n$-state nondeterministic finite automaton? As an example, let us ...
1
vote
1answer
37 views

why recursive languages does not come under Chomsky hierarchy? [duplicate]

why recursive languages does not come under Chomsky hierarchy ? why Chomsky did not classified it? any real life resemblance of recursive language something that i can connect it with like regular ...
7
votes
1answer
217 views

Is the reversal of a minimal DFA also minimal?

The question is pretty much in the title. Is there ever a time where some language $L$ can be accepted by a minimal DFA with $n$ states, but $L^R$, the reversal of $L$, can be accepted by a DFA with ...
3
votes
0answers
55 views

Prove or disprove that every $L$ in this class is a CFL iff $L$ is equivalent to a substitution

Let $L$ be a language with every string of the form $(w_i\#)^*$ with $w_i\in\{0,1\}^*$. Set $w'\sim w$ if there is a permutation $\pi_1$ such that $w_i=w'_{\pi_1(i)}$ for all $i$. If additionally ...
1
vote
2answers
123 views

Is $a^n b^m$ never regular if n and m have some relation between them?

I know what regular and context free language are and how regular language needs finite memory and other stuff. What concerns me is that I think if $a^nb^m$ such that $n$ and $m$ have some relation ...
0
votes
2answers
90 views

regular expression in license plates

I'm trying to write a regular expression for some particular license plates. They consist of one up to three capital letters, a hyphen, one up to two capitol letters and one up to four numbers. The ...
0
votes
1answer
36 views

How does one figure out where a class of languages falls under some complexity class? [closed]

I was wondering how can someone prove that one class of languages is of a certain complexity? For example, how could I show the Turing-recognizable languages are in P? Would I have to come up with ...
1
vote
1answer
61 views

Constructing an unrestricted grammar for a^n b^m c^n d^m [closed]

I've been trying to construct an unrestricted grammar which has the language: L = {a^n b^m c^n d^m | n>0, m>0} But I can't seem to figure it out without ...
5
votes
1answer
114 views

Where/when did Stephen Kleene first define the Kleene closure/star?

I'm working on a paper and would like to review the origins of Kleene's closure. I am unable to find any article of Kleene's that has the original definition of the Kleene closure. Is there a paper ...
1
vote
1answer
62 views

regular expression: sum of positive fixed point decimal numbers [closed]

I need help with this exercise. Indicate the regular expression for the following Languages. Explain your expression in one sentence and indicate the basis form of the alphabet. Indicate also every ...
1
vote
1answer
64 views

Turing machine with repeated strings

How would I go about making a Turing machine to accept the following language L? $$L = \{ www \mid w = \{0,1\}^* \text{ and } w > 0\}$$ I was thinking counting the number of symbols in the input ...
1
vote
1answer
68 views

Proving that a language does not belong to a language class by using more specific instances of that language

You have a description of a language that you have to prove is regular, context free, or other. In order to prove that it does not belong to a certain class of languages, you might think that it will ...
3
votes
1answer
100 views

A Question from Introduction to Formal Languages by Gyorgy E. Revesz; Exercise 1.1

The exercise says "Show that the grammar $G = \langle\{S\}, \{a, b\}, S, \{S \to \lambda, S \to aSb\}\rangle$ generates the language $L = \{a^i b^i \mid i = 0, 1, 2, \ldots\}$." Now, I'm new to ...
2
votes
1answer
62 views

Is there a Context-free grammar for this language?

Is there a Context-free grammar for the following language: $L=\{ x\#1^m|x \in \{0,1\}^* \space and \space the \space m^{th} \space char \space in \space x \space ...
1
vote
1answer
59 views

Prove not context free

How can we prove that: $$ L = \{ w_1\#w_2 \mid w_1 \in w_2;\; |w_2| > |w_1|;\; w_1 , w_2 \in \{0, 1\}^*\} $$ is not context-free? The language defines $w_1$ as a sub-string of $w_2$, and they ...
4
votes
1answer
79 views

What does $\{$ a set $\}^{+}$ mean in the context of languages?

I came across this notation and I don't know the meaning of it, or if it's a typo: $\{$ some set $\}^{+}$ What does the + mean, i.e., the plus operator applied to a set?
5
votes
0answers
82 views

Expressiveness of modern regular expressions

I recently discussed with a friend about a website that proposed regex challenges, mainly matching a group a of words with a special property. He was looking for a regex that matches strings like ...
1
vote
1answer
47 views

Small-step semantics: for-loops

I'm trying to construct the small-step semantic rules involving the for-loops, but I can't find anything about it in the literature (only about while-loops). I was wondering if anyone could help me ...
7
votes
1answer
119 views

Proving a language (ir)regular (standard methods have failed)

I'm currently trying to prove a language regular (for personal amusement). The language is: The language containing all numbers in ternary that have even bit-parity when encoded in binary. Now, I've ...
2
votes
2answers
160 views

Intersection/Union of recursively enumerable languages that aren't decidable?

For $L_1, L_2 $ and $L_1 \in RE $ and $ L_1\notin R$ and $L_2 \in RE $ and $ L_2\notin R$ I was asked to prove/disprove if the following can occur: $L_1 \cap L_2 \in R$ $L_1 \cup L_2 \in R$ $L_1 ...
9
votes
3answers
741 views

Why use languages in Complexity theory

I'm just starting to get into the theory of computation, which studies what can be computed, how quickly, using how much memory and with which computational model. I have a pretty basic question, but ...
5
votes
3answers
460 views

Is it compulsory that every infinite set be non regular?

I am confused regarding the statements provided by one of our faculty regarding "Is it compulsory that every infinite set is non regular though every finite set is a regular set". Providing ...
3
votes
1answer
65 views

Regular expressions and semi-linear sets

In proving Parikh's Theorem, my Theory of Computer Science textbook defines a linear set as: $u_0 + \langle u_1, \dots, u_m \rangle = \{u_0 + a_1u_1 + \dots + a_mu_m \mid a_1, \dots, a_m \in ...
1
vote
1answer
51 views

How to check ambiguity of a specific grammar

Giving the following Grammar: S → ^ | SaSMSM | SMSaSM | SMSMSa M → b | c ^ means eopsilon. How can i check whether its ambgious or not? My intuition is ...
0
votes
2answers
78 views

Find a CFG for a language

In an assignment I've been asked to find a CFG for $a^x b^y a^z b^w$, where, $x,y,z,w \in \mathbb{N}^+$, $y > x$, $z > w$, and $x+z = y+w$. A hint was given, think of the language as $(a^p ...
3
votes
2answers
264 views

Proving that a word is *not* generated by a context-free grammar

I saw the answer in one of the solutions and I cannot figure out how they got the answer. The question is asked if the word is in the language or not for CNF... How did they get the answer so that ab ...
2
votes
0answers
93 views

Good introductions to Formal Language Theory and Formal Grammars

Does anyone know any good introductions to Formal Language theory and Formal Grammar, that cover the mathematical basis of Syntax and things like context free grammars and pushdown automata. In ...
0
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1answer
21 views

Canonical infinitely ambiguous languages

In an article I am currently reading the grammar S → SS | a | ε is being described as canonical infinitely ambiguous. The infinitely ambiguous part I have no problem recognizing, but does ...
3
votes
3answers
86 views

Compression of non-adjacent structure using grammar

I'm working with compression algorithms that use context-free grammars (e.g. RE-PAIR and SEQUITUR). These grammars look for frequently occurring digrams (pairs of adjacent symbols) in an input string ...
0
votes
1answer
45 views

a regular language so that $unary(L) \notin $Context Free Languages [closed]

I need a regular language $ L\subseteq \{0,1\}^{*} $ so that $unary(L)$ is not context free. unary of $L$ is defined by: $$unary(L) = \{0^{1x} : x \in L \}$$ Example $L = \{0, 11\} $ $\rightarrow ...