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Questions tagged [regular-languages]

Questions about properties of the class of regular languages and individual languages.

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8answers
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How to prove that a language is not regular?

We learned about the class of regular languages $\mathrm{REG}$. It is characterised by any one concept among regular expressions, finite automata and left-linear grammars, so it is easy to show that a ...
47
votes
8answers
62k views

How to prove a language is regular?

There are many methods to prove that a language is not regular, but what do I need to do to prove that some language is regular? For instance, if I am given that $L$ is regular, how can I prove that ...
14
votes
3answers
1k views

What are the possible sets of word lengths in a regular language?

Given a language $L$, define the length set of $L$ as the set of lengths of words in $L$: $$\mathrm{LS}(L) = \{|u| \mid u \in L \}$$ Which sets of integers can be the length set of a regular language?...
3
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2answers
4k views

How do I find a regular expression for a particular language?

I have a language, and I want to find a regular expression for the language. How do I do that? Is there a step-by-step, systematic procedure for that? Pretend I am just learning this topic; what ...
11
votes
2answers
4k views

Single-tape Turing Machines with write-protected input recognize only Regular Languages

Here is the problem: Prove that single-tape Turing Machines that cannot write on the portion of the tape containing the input string recognize only regular languages. My idea is to prove that this ...
13
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2answers
1k views

Closure against right quotient with a fixed language

I'd really love your help with the following: For any fixed $L_2$ I need to decide whether there is closure under the following operators: $A_r(L)=\{x \mid \exists y \in L_2 : xy \in L\}$ $A_l(L)=\{...
10
votes
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 ...
5
votes
3answers
923 views

Is {wxw^r} a regular language?

Is $\{ WxW^{\mathrm{R}} \mid W,x\in\{0,1\}^+\}$ a regular language? If so, why? The notation $W^{\mathrm{R}}$ means the reverse string of $W$? If we consider the best answer in this solution, ...
16
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4answers
47k views

How to show that a “reversed” regular language is regular

I'm stuck on the following question: "Regular languages are precisely those accepted by finite automata. Given this fact, show that if the language $L$ is accepted by some finite automaton, then $L^{...
9
votes
6answers
972 views

Is $A$ regular if $A^{2}$ is regular?

If $A^2$ is regular, does it follow that $A$ is regular? My attempt on a proof: Yes, for contradiction assume that $A$ is not regular. Then $A^2 = A \cdot A$. Since concatenation of two non-...
2
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5answers
36k views

Regular expression for the strings without a particular substring

How can we design a regular expressions without particular substrings. The goal of this is to create language L which won't contain a particular substring (i.e. 110)...
8
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1answer
674 views

Proving that language is regular or not regular

Let $L$ be a regular language. Prove that: $L_{+--}=\left\{w: \exists_u |u|=2|w| \wedge wu\in L\right\}$ $L_{++-}=\left\{w: \exists_u 2|u|=|w| \wedge wu\in L \right\}$ $L_{-+-}=\left\{w:\...
15
votes
1answer
17k 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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3answers
6k views

Is there any uncountable Turing decidable language?

There are many(and I mean many) countable languages which are Turing-decidable. Can any uncountable language be Turing decidable?
17
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3answers
7k views

Pumping lemma for simple finite regular languages

Wikipedia has the following definition of the pumping lemma for regular langauges... Let $L$ be a regular language. Then there exists an integer $p$ ≥ 1 depending only on $L$ such that every ...
11
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1answer
1k views

Algorithm to test whether a language is regular

Is there an algorithm/systematic procedure to test whether a language is regular? In other words, given a language specified in algebraic form (think of something like $L=\{a^n b^n : n \in \mathbb{N}\...
7
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1answer
5k views

Prove that regular languages are closed under the cycle operator

I've got in a few days an exam and have problems to solve this task. Let $L$ be a regular language over the alphabet $\Sigma$. We have the operation $\operatorname{cycle}(L) = \{ xy \mid x,y\in \...
5
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1answer
2k views

If $L$ is a regular language then so is $\sqrt{L}=\{w:ww\in L\}$

I am interested in proving that $\sqrt{L}=\{w:ww\in L\}$ is regular if $L$ is regular but I don't seem to be getting anywhere. If possible I was hoping for a hint to get me going in the right ...
15
votes
2answers
1k views

Number of words of a given length in a regular language

Is there an algebraic characterization of the number of words of a given length in a regular language? Wikipedia states a result somewhat imprecisely: For any regular language $L$ there exist ...
9
votes
2answers
1k views

Smallest DFA that accepts given strings and rejects other given strings

Given two sets $A,B$ of strings over alphabet $\Sigma$, can we compute the smallest deterministic finite-state automaton (DFA) $M$ such that $A \subseteq L(M)$ and $L(M) \subseteq \Sigma^*\setminus B$?...
4
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2answers
2k views

Determine if this language is regular

Let $L = \{xyx \mid \text{ for some }x,y \in \{0,1\}^+\}$. Is this language regular? So I was trying to construct a DFA, but I don't how to do this with this language.
2
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1answer
2k views

are regular languages closed under division

I am trying to solve this question which appeared in previous exam paper Can someone help me what i am failing to understand For languages $A$ and $B$ define $A \div B = \{x \in \Sigma^{\ast} : xy ...
8
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3answers
4k views

Deriving the regular expression for C-style /**/ comments

I'm working on a parser for a C-style language, and for that parser I need the regular expression that matches C-style /**/ comments. Now, I've found this expression on the web: ...
3
votes
1answer
1k views

Is $L_{half} = \{w : \text{for some } z \in L, x \in \Sigma^*, z = wx \wedge |w| = |x| \} $ regular? [duplicate]

Suppose we have some regular language $L$, then can we say that $$L_{half} = \{w : \text{for some } z \in L, x \in \Sigma^*, z = wx \wedge |w| = |x| \} $$ is also regular? I have a 'feeling' that ...
0
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1answer
578 views

Regular expression of a language over {a,b,c} which does not contain substring bbb

I'm trying to figure out how to build a regular expression for a language that doesn't contain substring bbb. The alphabet is {a,b,c}. I'm trying to construct a DFA and convert to help me get the ...
11
votes
2answers
8k 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 ...
5
votes
2answers
614 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
votes
1answer
1k views

Kleene star of an infinite unary language always yields a regular language

Let $L = \{a^n \mid n \ge 0\}$, where $a^0 = \epsilon$ and $a^n = a^{n-1}a$ for all $n \ge 1$. Thus $L$ consists of sequences of $a$ of all lengths, including a sequence of length $0$. Let $L_2$ be ...
12
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2answers
2k views

If $L$ is a subset of $\{0\}^*$, then how can we show that $L^*$ is regular?

Say, $L \subseteq \{0\}^*$. Then how can we prove that $L^*$ is regular? If $L$ is regular, then of course $L^*$ is also regular. If $L$ is finite, then it is regular and again $L^*$ is regular. Also ...
8
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5answers
1k views

Is there a known method for constructing a grammar given a finite set of finite strings?

From my reading it seems that most grammars are concerned with generating an infinite number of strings. What if you worked the other way around? If given n strings of m length, it should be possible ...
4
votes
1answer
682 views

Time Complexity of Regular Languages

I wonder how I can go about proving that if a language L is decidable in o(nlog(n)) then L must be regular. I should probably mention that by "decidable" I mean "being decidable by single-tape ...
3
votes
6answers
15k views

Designing a DFA that accepts strings such that nth character from last satisfies condition

This is a homework question, so I am only looking for hints. I got a question in an assignment which states : Design a DFA that accepts strings having 1 as the 4th character from the end, on the ...
5
votes
1answer
152 views

Closure of regular languages are closed under certain cut operations

Let $f : \mathbb{N} \to \mathbb{N}$ be an integer function. For a language $L$, define $$f(L) = \{w \mid \exists x : |x| = f(|w|) \text{ and } wx \in L\}$$ For example, if $f(n) = n$ this is just ...
5
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1answer
2k views

Prove that the language of non-prime numbers written in unary is not regular

Im trying to prove that the following language is not regular. $$\text{Notprime} = \{a^n \text{where \(n\) isn't prime}\} = \{\epsilon, a, aaaa, aaaaaa, aaaaaaaa, \ldots\}$$ Heres what I have: "If ...
4
votes
2answers
2k 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 a ...
3
votes
5answers
3k views

Show that every infinite language has a non-regular subset

I'm trying to solve this problem: Let $L$ be some infinite language, show that there exists a sub-language of $L$ that is not regular But can this be correct? If I have the language $\{a\}^*$ for ...
2
votes
3answers
2k views

If L is regular, show that even(L) is also regular

I am stuck on the following question. If $L$ is regular show that $\mathrm{even}(L)$ is also regular, where $\mathrm{even}(L) = \{ even(w) : w \in L \}$, $w$ is a string in $L$ and $\mathrm{even}(w)...
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. ...
6
votes
1answer
661 views

What's wrong with my pumping lemma proof?

The language $L = \{0^{2n} \space |\space n \ge 0 \}$ is obviously regular – for example, it matches the regular expression $(00)^*$. But the following pumping lemma argument seems to show it's ...
5
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1answer
3k views

DFA for a strings whose every subsequence of length five has at least two zeroes

I have a regular language consisting of such {0,1}^k sequences, in which every subsequence of length 5 has at least two 0's in ...
3
votes
3answers
2k views

Show that regular languages are closed under Mix operations

Let $L_1, L_2$, two regular languages and the operations: $$Mix_1(L_1, L_2) =\{ a_1b_1a_2b_2\ldots a_nb_n | n\ge 0 \land a_1,a_2,\ldots ,a_n,b_1,b_2,\ldots ,b_n\in\Sigma\\ \land a_1a_2\ldots a_n\in ...
2
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1answer
3k views

Regular expression for a string not containing a set of substrings

I'm trying to figure out how to build a regular expression for a language that doesn't contain strings that contain $101$ or $001$. The alphabet is defined as $\{0, 1\}$. I'm stuck on trying to figure ...
2
votes
3answers
299 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 $P = \{\alpha \mid \beta\alpha\gamma \in L, \beta,\gamma \in \Sigma^*\}$...
2
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1answer
164 views

When using the Pumping lemma, how do I deal with different cases of y?

I want to prove L is not regular:$$L={\{www|w \in \Sigma^*\}}$$ $$\Sigma=\{a,b\}$$ I am sure I can do so using pumping lemma. I used $$ab^pab^pab^p$$as my chosen string but I am stuck. I do not know ...
2
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1answer
186 views

Can the definition of regular languages be simplified?

Wikipedia says The collection of regular languages over an alphabet Σ is defined recursively as follows: The empty language Ø is a regular language. For each a ∈ Σ (a belongs to Σ), the ...
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vote
2answers
582 views

Proving Regularity of Languages that are 1/k of an already known regular language

There is this question in Kozen, that states if a language is regular then the first half would also be regular. Also I found a material on the internet that extends the thinking saying a language ...
0
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2answers
2k views

Are supersets of non-regular languages also non-regular?

I have to proof that if $L_1 \subset L_2$ and $L_1$ is not regular then $L_2$ it not regular. This is my proof. Is it valid? Since $L_1$ is not regular, there does not exists a finite automata $M_1$ ...
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2answers
2k views

Are all irregular languages infinite? [duplicate]

How can I prove whether irregular languages are infinite? I thought about proving it by the definition of regular language but got stuck.
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2answers
2k views

Why is a regular language called 'regular'?

I have just completed the first chapter of the Introduction to the Theory of Computation by Michael Sipser which explains the basics of finite automata. He defines a regular language as anything ...
17
votes
4answers
7k views

Using Pumping Lemma to prove language $L = \{(01)^m 2^m \mid m \ge0\}$ is not regular

I'm trying to use pumping lemma to prove that $L = \{(01)^m 2^m \mid m \ge0\}$ is not regular. This is what I have so far: Assume $L$ is regular and let $p$ be the pumping length, so $w = (01)^p 2^p$....