# Questions tagged [closure-properties]

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

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### Decidable languages kleene star closure - question on a proof

I read a proof on the closure of decidable languages under kleene star. It begins by saying that the turing machine we want to find would non-determistically split the input string and then use the ...
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### Closure against the operator $A(L)=\{ww^Rw \mid w \in L \wedge |w| \lt 2007\}$

I would like your help with the following question: Let $L$ be a language, and operator $A(L)=\{\,ww^Rw \mid w \in L\ \wedge\ |w| \lt 2007\,\}$ where $x^R$ is the reversed string of $x$. Which of ...
362 views

### The operator $A(L)= \{w \mid ww \in L\}$

Consider the operator $A(L)= \{w \mid ww \in L\}$. Apparently, the class of context free languages is not closed against $A$. Still, after a lot of thinking, I can't find any CFL for which $A(L)$ ...
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### Question on NP $\cap$ coNP

I'm struggling with a past paper question and would appreciate any hints: Suppose $L_1, L_2 \in$ NP $\cap$ coNP and $L_1 \oplus L_2 = \{ x : x$ is in exactly one of $L_1$ or $L_2 \}$. Then ...
306 views

### Reference request: proof that if $L \in DCFL$, then $L \Sigma^* \in DCFL$

So, it's fairly easy to prove that if $L \in DCFL$, then $L \Sigma^* \in DCFL$. Basically, you take the DPDA accepting $L$. You remove all transitions on final states, and then for each $a \in \Sigma$ ...
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### Closure under intersection of context free binary trees

Some sets of ordered binary trees can be represented as a CFG with rules of the form A -> aBC A -> b Where A,B,C are ...
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### Deterministic context-free languages are closed under regular right-product

I am looking for a proof for the following problem: For languages $L$ and $R$, if $L$ is deterministic context-free and $R$ is regular, then $LR$ is a deterministic context-free language. Note:...
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### 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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### Why NP is not closed under Turing reduction

The notion of polynomial time Turing reductions (Cook reductions) is an abstraction of a very intuitive concept: efficiently solving a problem by using another algorithm as a subroutine. For example,...
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### Proving that if L is regular. Then L′ = {ww : w ∈ L} is regular

I believe this statement to be true. And here's my reasoning: Based on regular languages properties, the concatenation of two regular languages is regular. And since L′ = L · L, it follows that L′ ...
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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 ...
322 views

### Is NEXP = co-NEXP?

It is known that $\mathsf{NL}=\mathsf{Co{-}NL}$ and unknown if $\mathsf{NP}=\mathsf{Co{-}NP}$. But what about $$\mathsf{NEXP}=\mathsf{Co{-}NEXP}?$$ Is it known whether these two classes are equal?
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### Closure properties of the class of inherently ambiguous CFLs

is set of inherently ambiguous context free languages close under operations such that union, intersection, kleene star, concatenation, reverse, complementation and etc. how many of theme are answered?...
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### $\mathbf{NC}$ is closed under logspace reductions

I am trying to solve the question 6.12 in Arora-Barak (Computational Complexity: A modern approach). The question asks you to show that the $\mathsf{PATH}$ problem (decide whether a graph $G$ has a ...
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### Proving that recursively enumerable languages are closed against taking prefixes

Define $\mathrm{Prefix} (L) = \{x\mid \exists y .xy \in L \}$. I'd love your help with proving that $\mathsf{RE}$ languages are closed under $\mathrm{Prefix}$. I know that recursively enumerable ...
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### Inverse Homomorphisms and Kleene star

The exercise is to prove or give a counterexample to the following proposition with $L \subseteq \Gamma^*$ regular and $h: \Gamma \to \Sigma^*$ a homomorphism. Is there any regular language $L'$ such ...
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### Closure operator and set of fixpoint

In chapter 2.2 of Giacobazzi, Roberto; Ranzato, Francesco, Uniform closures: Order-theoretically reconstructing logic program semantics and abstract domain refinements, Inf. Comput. 145, No.2, 153-...
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### L closed under logspace reduction

Given two languages $A$ and $B$ I have been asked to show that, if $B \in L$ and we have a logspace reduction $f$ from $A$ to $B$ then $A \in L$. I read the proof that $L$ is closed under logspace ...
903 views