Questions tagged [closure-properties]

Questions about operations on objects of some kind that result in objects of the same kind.

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Why L' is not regular?

$$L'=\{ww|w\in L\}$$ I need to give an example of regular language L for which the concatenation of 2's $w$ gives $L'$ which is not regular. How can I give such an example if according to closure ...
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1answer
105 views

If $L$ is a regular language then so is $L/a =\{w | wa ∈ L\}$, where $L$ is a language over $\Sigma$ and $a \in \Sigma$

I'm trying to work out a proof by construction that $L/a$ would be regular. $a$ is any final symbol at the end of an accepted string, so I figured the only part of the machine that would have to be ...
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2answers
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Myhill-Nerode and closure properties

It is well known that regular languages are characterized by the Myhill-Nerode equivalence. For language $L$ over $\Sigma^*$ define the equivalence $x\sim_L y$ over $\Sigma^*$ iff for all $z\in\Sigma^*...
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3answers
3k views

Why are DCFL not closed under concatenation or Union whereas CFL is?

I understand that DCFL they are not closed under concatenation or Union. As without non determinism, PDA cannot decide when to jump to the next one in case of concatenation and without epsilon moves ...
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3answers
102 views

Show that if A is regular, then the subset containing only even language strings, is also regular

A language A, even(A) is the subset of A consisting of those strings in A of even length: even(A) = { x∈A | |x| is even} I need to use closure properties show ...
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2answers
129 views

If L is a regular language then the language replace(L,σ,τ) is also regular

I am stuck at the following problem: Prove that if $L$ is a regular language over some alphabet $\Sigma$ and that $\sigma, \tau \in \Sigma$, Then the language $replace(L,\sigma,\tau)$ is regular. ...
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1answer
27 views

Policies for handling symbols leaking out of a lexical scope

Suppose it's possible for a symbol to escape the scope in which it is defined. What are considered the possible policies for handling that? I mean possible in the sense of what choices of language ...
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3answers
2k views

Are regular and context free languages closed against making them prefix-free?

For a language L we define: $\qquad A(L) = \{ x \in L \mid \text{ no proper prefix of x is in L} \} $ Are regular / context free languages closed under this operation ? For regular languages I ...
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1answer
99 views

Prove/disprove: If $𝐿_1$ is a finite language but not empty and $𝐿_2$ is NOT regular then $𝐿_1 \circ 𝐿_2$ is NOT regular

That what I have so far, but I am not sure at all. Assume toward contradiction that $𝐿_1 \circ 𝐿_2$ is regular. Define $\Sigma' = \{\sigma'|\sigma\in\Sigma\} $. Define a regular substitution $\...
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2answers
59 views

Is the difference of two context-free languages still context-free?

i am asking myself the following question: Assuming: A and B are context-free languages, then A - B (difference) must also be context-free language, right? but I do not know how to prove it.
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1answer
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Have any natural complexity classes been proven not to be closed under complement?

Many important (non-deterministic) complexity classes like NP are believed not to be closed under complement. But have any of them been proven not to be? I'm sure one could construct some contrived ...
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1answer
49 views

Quotient of languages, regular quotient and their closedness

Left quotient is defined as below at this link: Left quotient of $L1$ by $L2$: $L1\backslash L2:= \{u\in \Sigma^*|vu\in L1$ for some $v\in L2 \}$ Wikipedia defines it as follows: $L_1\...
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3answers
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Show that P is closed against the Kleene star

I have that question that looks kinda easy at first but it is quite hard. Let $L\in P$. Prove that $L^*\in P$ my approach: I tried to generate a Turing machine but I got stuck with the thing of ...
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2answers
1k 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
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3answers
4k views

Closure of CFL against right-quotient with regular languages

Let $A/B$ = $\{ w \mid wx \in A$ for some $x \in B \}$. Show that if A is context free and B is regular, then $A/B$ is context free. My interpretation of this is is that we need to show that if a ...
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1answer
41 views

Are models of computation closed under composition?

It's common to ask whether a particular class of languages $\mathcal{C} \subseteq \mathcal{P}(\Sigma^*)$, for some alphabet $\Sigma$, is closed under complement, or union, or intersection, or ...
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2answers
7k views

Is every subset of a decidable set, also decidable?

Is it true that if A is a subset of B, and B is decidable, than A is guaranteed to be decidable? I believe it would be true because all the subsets of B should also be decidable making A decidable. I'...
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7answers
1k 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-...
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4answers
380 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 ...
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1answer
2k views

Difference between substitution, morphism, and homomorphism

In closure properties, I got confused between substitution and morphism. 1) According to Wikipedia, string substitution means to map letters in a set of alphabets to languages (possibly in a ...
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1answer
55 views

How can the union of two 'context-free but not regular' languages be regular?

I cannot understand how the union of two languages which are context-free but not regular, can result in a regular language: If $L_1$ and $L_2$ are 'context-free but not regular' languages, defined ...
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1answer
614 views

Complexity classes that are closed under subtraction

Are NP or P closed under subtraction? Im having a hard time deciding whether they are or aren't. Question was edited Original question: Im having some hard time figuring out what languages are closed ...
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45 views

Intersection of decision problems?

Say we have two problems $\Pi_1\in NP$ and $\Pi_2\in coNP$. Where does $\Pi_1\cap\Pi_2$ live?
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91 views

Can the regular image of a context-free language be undecidable?

I just need to know the truth or falsity of a simple statement. Let $L_1$ be a context-free language over an alphabet which contains some number of characters $\Sigma$, as well as a single, special ...
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1answer
23 views

Getting from one language to the other using closure properties(automata) [duplicate]

I am trying to deduct how i can, using closure properties, deduct that since the following language is not context free $$L=\left\{abc^{i_1}bc^{i_2}...bc^{i_{2m}}def^{j_1}ef^{j_2}..ef^{j_{2n}}ghq^{k_1}...
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2answers
79 views

DSPACE(f(n)) closed under complement

I think you can create the complementary language that is an element of DSPACE($f(n)$), where $f(n) \geq \log(n)$ by adding a step to the algorithm that reverses the answer. By that the function $f(n)$...
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4answers
13k views

Union of regular languages that is not regular

I've come across that question : "Give examples of two regular languages which their union doesn't output a regular language. " This is pretty shocking to me because I believe that regular languages ...
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3answers
173 views

First half of context-free palindromes

If $L\subseteq\Sigma^*$ is a regular language, then $\text{mir}(L) = \{ww^R \mid w\in L\}$ is context-free. This is a nice exercise. Question: does the reverse hold? Thus, if $\text{mir}(L)$ is ...
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2answers
166 views

Proving that L is not regular using closure properties

I need to show that the following language is not regular. $$L = \{\ ab^jc^j\ |\ j \geq 0\ \}\ \cup\ \{\ a^ib^jc^k\ |\ i, j, k \geq 0 \ and\ i \neq 1\ \}$$ There is also a hint that it cannot be ...
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3answers
14k views

What is complement of Context-free languages?

I need to know what class of CFL is closed under i.e. what set is complement of CFL. I know CFL is not closed under complement, and I know that P is closed under complement. Since CFL $\subsetneq$ P I ...
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2answers
47 views

Calculate the number of distinct permutations of length n in the closure of a language

I am studying a distance CS course, but there is no tutor available, so I would appreciate your help... Consider the language $S = \{a, aa, ab\}$ How many distinct words of length $n$ will appear in $...
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1answer
48 views

How to use homomophism in closure proofs?

I am having a hard time understanding homomorphism. All I seem to understand is that it is a substitution. When I look at examples of proving closure of a particular operation over a regular language, ...
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1answer
41 views

operate infinite times over a regular language

Let $T:Σ^*\to Σ^*$ be an operation such that $T(L)$ is regular for all regular languages $L \in Σ^*$. Is it possible to prove $T^∞(L)$ is regular? $T^∞(L)=\bigcup_{i=1}^{\infty}{T^{i}\left(L\right)}$...
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1answer
69 views

Why proving that two languages used to merge into a regular language are not necessarily regular isn't possible with closure properties?

Let $L$ be a regular language over alphabet $\Sigma$. $L$ is the result of merging $2$ languages letter by letter that is for $a_1a_2...a_n\in L_1, b_1b_2...b_n\in L_2, L=a_1b_1a_2b_2...a_nb_n$. $\...
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3answers
217 views

If $L$ is a regular language, then $s(L)$ is also regular

...where $s$ is a substitution that replaces each symbol of each string in $L$ with a regular expression. For example, if $L=a^*b$ and $s(a) =ab, s(b) = b^*$, we have $s(L) = (ab)^*b^*$. My ...
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3answers
4k 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)...
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2answers
79 views

How to prove certain parts of one regular language restricted by another regular language is also regular?

I’ve encountered the following difficult question that I don’t know how to solve. $L_1$ and $L_2$ are regular languages over the same $\Sigma$. $$\begin{align}L^\wedge=&\{σ_1σ_2...σ_n\mid n\ge1, \...
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2answers
863 views

Proof that the language $ww^R$ is not regular without using the pumping lemma

I am breaking my head over this. Let the alphabet $A$ be given by $A = \{a,b,c\}$ and let $$L = \{ww^R \mid w \in A^* \}.$$ Prove that the language $L$ is not regular without using the pumping ...
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2answers
52 views

if $L_1$ and $L_2$ are languages over the same alphabet and $L_1 \cap L_2$ is context free, at least one of them must be context free

I am having a hard time understanding if this would be true or false, can someone point me in the right direction?
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1answer
65 views

unambiguous context-free languages and complementation

I was considering the following two natural questions about the relationship between unambiguity and complementation for the class of context-free languages: Is the complement of an unambiguous ...
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2answers
3k views

Show that NP is closed under concatenation

Show that NP is closed under concatenation. This is a homework problem and I would appreciate some guidance. I began by saying the following: Let $A$ and $B$ exist in NP. Let $V_1$ and $V_2$ be ...
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4answers
10k views

Is the class NP closed under complement?

Is the class $\sf NP$ closed under complement or is it unknown? I have looked online, but I couldn't find anything.
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2answers
12k views

Why isn't the class of Turing-Recognizable languages closed under Complement?

I'm studying Turing Machines and I've already showed how Turing-Decidable is closed for the operations of Union, Intersection, Concatenation, Complement and Kleene Star. Next I did some demonstrations ...
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2answers
17k views

Union and intersection of a regular and a non-regular language

Lets say we have $L_1$, which is a regular language and $L_2$ which is not. Are $L_1 \cap L_2$, $L_1 \cup L_2$ , $L_1$ \ $L_2$ and $L_1 \cdot L_2$ are always non-regular languages? We know that two ...
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1answer
146 views

Closure of regular languages under deleting a 1 from each even run of 1s

Let $R$ be a regular set over the alphabet $\{0, 1\}$. Give a machine construction to prove that the set obtained by deleting one 1 from each even length block of 1’s is also regular, and using ...
2
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2answers
106 views

Finding if the given language is regular or not

I have the language $$L = \{a^mb^nc^o| \, m + n + o > 5\}$$ where $m,n,o$ are non-negative integers. I have to find whether the language is regular or not. My attempt: I feel it should be non ...
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1answer
245 views

Finite state automaton for the reverse of the language, and multiple starting states

I am studying about Finite State Automaton, and I found that the when reversing a language (i.e., transforming $L$ to $L^R$), I have to add new start state. Why is that? Also, can a Finite State ...
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1answer
127 views

Show that RP is closed under concatenation

I'm trying to prove the following problem: Show that $RP$ is closed under concatenation Now, let's say that the two languages are $L_{1}$ and $L_{2}$ (both in $RP$). Then I accept a word iff the ...
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1answer
679 views

Why is the intersection of these two Languages Recursively Enumerable, not Recursive?

I am only several days exposed to computational theory, so my understanding is quite slim: in a question, it says that for a regular language L1 and a recursively enumerable but not recursive language ...
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2answers
120 views

Proving $L'=\{uv : u \in L \; \land v \in L^R \; \land |u|=|v|\}$ is a CFL with closure properties [duplicate]

Given a language $L$ over $\Sigma=\{a,b\}$ let us define $L'=\{uv : u \in L \; \land v \in L^R \; \land |u|=|v|\}$ Prove: if $L$ is regular, then $L'$ is a context free language. I know how to ...

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