# Questions tagged [closure-properties]

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

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### Regular languages closed under quotients with arbitrary languages

When proving that that the quotient of a regular language $R$ and an arbitrary language $B$, I understand you take a DFA $M$ accepting $R$, and then construct a DFA that is the same, but its final ...
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### Given a non-deterministic Mealy machine $M$, if $L$ is regular, is $M(L)$ regular?

Consider a nondeterministic Mealy machine, $M$, defined as follows: $M = (Q, \Sigma, \Delta, \delta, \tau, q_0)$ where $Q$ is a finite set of states $\Sigma$ is an input alphabet $\Delta$ is an ...
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### P is closed under power of integer

I'm new in this area of complexity and I'm trying to get into it by understanding basic proofs. I want to prove that if $L\in P$, so $L^k\in P$, where $k$ is non-negative integer. How to prove it in ...
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### Why does the concatenation of the empty set with any language give the empty set? [duplicate]

Why does the concatenation of $\emptyset$ with any language give $\emptyset$. I would like to know the intuitive explanation for it.
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### Closure properties of undecidable languages

I know that the decidable are close under: complementation, union, intersection and concatenation? What about the undecidable languages? I think they are close under complementation, but not under ...
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### If $L_1L_2$ is regular language then $L_2L_1$ is regular to?

We have two languages: $L_1,L_2$. We know that $L_1L_2$ is regular language, so my question is if $L_2L_1$ is regular to? I try to find a way to prove it... I can't assume of course that $L_1,L_2$ ...
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### Prove that $S_2$ is closed under union and complement

I'm having trouble proving that $S_2$ is closed under union and complement, even though in this Wikipedia article it says that: It is immediate from the definition that $S_2$ is closed under union ...
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### mapping reduction for every recursive language [duplicate]

how do I prove that for every 2 languages $A,B\in R$ where $A,B \notin \{ \emptyset , \Sigma^* \}$ I can do a reduction $A \leq_m B$? [EDIT] My try: $A$ is decidable therefore it has a turing ...
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### Are deterministic and nondeterministic Cellular Automata equivalent?

It seems that in CA context nondeterministic (ND) means probabilistic, not ND as in NFSMs. At least I haven't seen a paper or book which discusses NCAs, without talking about probabilistic CAs. I ...
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### A and B are Turing recognizable, is A - B Turing recognizable?

If A and B are Turing recognizable, is A - B Turing recognizable? I think that A - B would be Turing recognizable because they're both in the space of Turing recognizability. For example, if A is ...
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### Is the closure of P under e-free homomorphisms equal to NP?

The context free languages can be obtained as the closure of the Dyck language under the cone operations. The Dyck language $D_2$ is a deterministic context free language, and the cone operations ...
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### Intersection of a language with a regular language imply context free

Lets say you have a language $L$ and you want to determine if it is context free. Context free languages intersected with regular languages are context free. Is that enough to prove that $L$ is ...
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### Language described by inverting accepting states of NFA

Connecting to When states that are not accepting states become accepting states in NFA, what happens?, what is the formal language described by inverting accepting states of NFA? By inverting, I mean ...
This is a problem in a theory of computation book that's stumping me: Suppose that we know that $L_1 ∪ L_2$ and $L_1$ are regular. Can we conclude that $L_2$ is regular? Explain. At first, I ...
### Is there $L$ such that $L$ and $\bar L$ are context free, but $L$ is not deterministic context free?
The usual candidates for context free languages whose complement is also context free, but they are not regular are the Deterministic Context Free Languages ($DCFL$). For example, \$L=\{a^nb^n\mid n\...