# Questions tagged [closure-properties]

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

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### Is the set of languages satisfying the pumping lemma closed under concatenation?

Let $L$ be the set of all languages that satisfy the pumping lemma, including non-regular languages that satisfy it. Is the set $L$ closed under concatenation? I couldn’t prove it or find a ...
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### Is NL closed under complemenrt?

I am trying to understand if NL is closed under complement or not. By NL i mean the non-deterministic-logspace complexity. I suppose that the answer is linked to the fact that we don't even know if L =...
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### Prove that $\{xyz \mid zyx \in A \}$ is regular if $A$ is regular

Does the following work and is there anything possibly simpler? Let $X = (Q, \Sigma, \delta, s, F)$ be a DFA for $A$. Intuitively, we want to "remember" (or guess) two states $p$ and $q$ ...
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### Prove that the class of regular languages is closed under three operation

We define an operation three on strings as three(c1c2c3c4c5c6...) = c3c6... then the above-described definition is extended to languages. Prove that the class of regular languages is closed under this ...
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### Prove the class of regular languages is closed or not closed under the operations below

Suppose $A$ and $B$ are both languages over $\Sigma=\{0,1\}$. We use $n_0(x)$ and $n_1(x)$ to represent the number of $0$s and $1$s in the string $x$ respectively. Consider the following two ...
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### Why are Recursive Enumerable Languages closed under union?

Union of two REL is closed under union. I don't understand how is it closed. I followed this link. The have stated: Here the trick is to simulate both M1 and M2 “simultaneously”. In other words, we ...
1 vote
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### Is there distinction between $(C/poly)\cap(coC/poly)$ and $(C\cap coC)/poly$?

Let $C$ be an uniform complexity class for example $NL$ or $NP$. Is there distinction between $(C/poly)\cap(coC/poly)$ and $(C\cap coC)/poly$?
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### 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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### Prove by contradiction that the language with unequal number of a's and b's is not regular

Consider the language $$L = \{w \mid w \text{ has an unequal number of a’s and b’s}\}$$ where Σ = {a, b}. Prove that L is not regular. Hint: Try proof by contradiction. Would this be the right Answer: ...
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### Classes of Functions Closed Under Polynomial Composition - Papadimitriou Exercise 7.4.4

I am studying Computation complexity using Papadimitrious's book: "Computational Complexity". I am trying to do Problem 7.4.4: "Let $C$ be a class of functions from nonnegative ...
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### Definition of Closed Under Left Polynomial Composition

I am studying Computation complexity using Papadimitrious's book: "Computational Complexity". While doing Problem 7.4.4, I came across the definition of what it means for a class of ...
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### Given L is a regular language, prove that Perm(L) is Context-Free

Given a regular language $L$ defined over $\Sigma = \{0, 1\}$. We define a new language $$Perm(L) = \{w \mid \exists x \in L, w \in perm(x)\},$$ where $perm(x)$ is the set of all permutations of the ...
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### Language equivalency for modified CFG closed over intersection

Suppose "CFG+" was created, where it is identical to standard context-free grammars in every way, but rather than rules being limited to unions, was also closed over intersections, both ...
1 vote
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### Show that the Language is irregular

I was solving some problem from past test, there was this question: Use the closure property of regular language to show the language $L$ is not regular $$L =\{ a^3 b^n c^{n-3} \mid n>3\}$$ I ...
1 vote
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### Regular languages closed under prefix operation

Suppose that $D$ is a regular language over an alphabet $A$. How can I prove that the following language is also regular? $$\mathrm{LANGUAGE}_2(D) := \{ d \mid d,t \in A^* \text{ and } dt \in D \}$$ ...
1 vote
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### Using closure properties, prove that $L=\{a^kb^ra^m|k,r,m\ge0 \text{ and } m=k+r\}$ is not regular

I'm trying to prove that $L=\{a^kb^ra^m|k,r,m\ge0 \text{ and } m=k+r\}$ is not regular and, although it's trivial to prove it via pumping lemma, I'm having troubles trying to find a way to prove it ...
1 vote
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### Which closure properties are always valid between regular, context-free and non context-free languages?

I am making a scheme that respresents some closure properties (union, intersection, complement and concatenation) for regular languages, context-free languages, decidable languages and RE languages. ...
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### Prove that the class of regular languages is closed under the Kleene + operation. That is, show that if L is regular, then so is $L^{+}$

This is my attempt at a proof: Let $E$ be a $REGEX$ accepting $L$. We claim the $REGEX$ $E^{'} = E^{+}$ accepts L. i.e. $L(E^{+}) = (L(E))^{+}$ $L^{+}$ is regular since there is a $REGEX$ $E^{+}$ ...
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### How do i prove this language is regular? [duplicate]

I have this language {0+1+0+} and i need to prove it is regular,i had the idea to use the closure properties but i can find any regular languages to unify perhaps.Any ideas?
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### Relationship between Kleene Star of a subset of regular language and the regular language

If $L(R_1) \subseteq L(R_2) \subseteq L(R_3)$ then $L(R_1)^* \subseteq L(R_2)^* \subseteq L(R_3)^*$. Does this also imply that $L(R_1)^* \subseteq L(R_3)$?
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I'm working through so textbook questions on regular languages, and came across a problem that amounts to showing the following language is regular, given that $A$ is a regular language: $$\{x|\... 0 votes 1 answer 155 views ### PDA kleene star construction I know how to prove that CFL are closed under kleene star operation using CFG. I can't find online or in class notes a proof using PDA. I would appreciate description of the basic idea (not formal). 0 votes 0 answers 81 views ### How to show that language L is NOT context-free? True or false: To show that a language L is not context-free, one can alternatively show that the union between L and a known context-free language is not context-free. I know that you can prove ... 1 vote 1 answer 48 views ### closure of Context free grammer to homomorphism using PDA I was looking online, on sipser book, and on lecture notes and I can't find a proof to closure of context free languages to homomorphism that using PDA instead of CFG. I'm not looking for a full and ... 0 votes 1 answer 86 views ### Complement of 0^n1^n | n \in \mathbb{N} I know why A is irregular by Closure properties of irregular language. I also know the complement of  \{ 0^n 1^n | n \in \mathbb{N}\} is A = \{ 0^i 1^j| i \neq j\} \cup (0 \cup1)^*(1)(0 \cup1)^*0(0 ... 2 votes 1 answer 25 views ### Decidability of a language and inclusion between two other languages I have this assignement that asks to say if the following statement is true or false, and possibly justifying the answer: "Let L₁, L₂ be decidable languages. For every language L s.t. L₁ ⊆ L ⊆ L₂, L ... 0 votes 1 answer 46 views ### How to define an automata for zig zag concatenation? [duplicate] I have two DFAs one for language A and one for language B. I'm asked to make an FDA that is the zig-zag concatenation of letters of A and letters of B. This is described by the following: {w: w = a_1 ... 1 vote 1 answer 185 views ### Zigzag concatenation of two languages Given two regular languages A,B on the same alphabet \Sigma, I want to show that the following language is regular:$$ \{a_1b_1 \ldots a_kb_k \in \Sigma^* \mid a_1,\ldots,a_k,b_1,\ldots,b_k \in \...
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Suddenly a thought came to my mind and I thought of resolving it as follows. We know that: String is a finite sequence of symbols from an alphabet $\Sigma$ i.e. we cannot have an infinite sequence ...