Questions tagged [closure-properties]

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

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334 views

If pref(L) is regular, does that imply L is regular?

I have this exercise for homework: Say we have a language L. we know that the language pref(L) (all the prefixes of ...
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240 views

Closure of Star-Free Languages under Substitution

Let $L$ be a star-free language over finite alphabet $\Sigma$. A substitution will be a map $\sigma : \Sigma \to \mathcal{P}(\Sigma^*)$. It seems obvious that if $\sigma(a)$ is star-free for every $...
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Closure properties of languages

Let $P$ be a regular language and $Q$ be a context-free language such that $Q \subseteq P$(For example, let $P = a^*b^*$ and $Q = \{ a^nb^n | n \ge 0\}$). Then which of the following is always ...
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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 ...
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975 views

Is the union of two non-regular context-free languages always non-regular?

I had this question in my HW: Prove of disprove: If $L_1$ and $L_2$ are non-regular context free languages then $L_1 ∪ L_2$ is not regular. My intuition is that it is wrong. I thought about the ...
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2answers
227 views

XOR of two NP-Complete languages

Given two NP-Complete languages A and B, show that the language: $L = A\bigoplus B =\{a\bigoplus b \mid a \in A, b \in B, |a|=|b|\}$ is not necessarily NP-Complete. Remember $a\bigoplus b$ when $|...
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1answer
445 views

Reverse the input and output of a Mealy machine

Given a Mealy machine $M$, is it possible to construct another Mealy machine $M'$ that generates the reverse outputs from the reverse inputs, and if so, how? That is, for each string $s$, $M(s) = t$ ...
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1answer
567 views

Constructing a decider for a language

I'm confused about the idea of constructing a decider for a language and i need some help with it. For example, if i have an enumerator M1 for a language L and another enumerator M2 for the ...
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518 views

If A is NL-complete then complement of A is also NL-complete?

We know that coNL = NL. But, is this also true? If A is NL-complete then complement of A is also NL-complete? I don't see a reason for that it could be true.
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612 views

Under which operations is the class of non-recursive languages a closure?

I am currently studying turing computability and related problems such as the halting problem with a background in formal languages. I know that the class of recursive (decidable) languages is a ...
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96 views

If $R,F \cap R, F \cap \overline{R}$ are regular, is $F$ regular?

Let $R \in REG$. Is it true that if $F\cap R \in REG$ and $F \cap \overline{R} \in REG$ then $F \in REG$? I took $R = \Sigma^{\ast}$ and because $F\cap R \in REG$, $F$ must be regular. Is this the ...
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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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217 views

PSPACE languages reducible to other PSPACE languages in polynomial space

Intuitively it makes sense that all PSPACE languages are reducible to other PSPACE languages in polynomial space. But how would I go about actually showing this?
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Find a CFG for the set of prefixes of a CFL [duplicate]

How do i generate grammar for Prefix of Langauge L, SupposeG=(V,􏰀,P,S)is a context-free grammar generating a CFL L then pref(L) is defined as pref(L)={x∈􏰀∗ : ∃ y such that xy∈L}. I understand for ...
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Don't understand closure under string reversal

I am trying to learn from http://www.cs.uiuc.edu/class/su08/cs273/lectures/lect_06.pdf #2 and I understand everything except for the 2nd line of delta prime prime function, I having breaking down ...
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3k views

Show that the regular languages are closed against taking “the second half” [duplicate]

Given $L$ is regular, the proof that $\mathrm{HALF}(L)$ is regular is pretty straightforward to me (e.g., #11 in this link): simply making a NFA and meeting in the middle with 2 original DFAs, the ...
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79 views

Is CFL closed against exchanging complementation and reversal?

Let $L$ be a language such that $\overline{L}^R$ (the reversal of the complement of $L$) is context-free. Is then also $\overline{(L^R)} \in \mathrm{CFL}$?
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How can I show a linear languages are closed against concatenating with regular ones?

This was given as a homework problem but I have already submitted the assignment. I'd like to resolve it at this point for my own satisfaction. Given that $L_1$ is a linear language and $L_2$ is a ...
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86 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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1answer
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Proving that Pre(L) is regular using automatas: Should I prove that Pre(L) is the semantic of the new automata?

Let $L$ be a regular language, and $Pre(L)$ be the set of all words that are prefix of some word in $L$. Prove that $Pre(L)$ is regular. My proof: Let $M = (\Sigma, Q, \delta, q_0, F)$ be an automata ...
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387 views

CFL Closure Properties prove or disprove for the following languages

I have following statements which I must prove or disprove : 1) Let $L$ be a CFL and $k \in N$ then $L^k$ is also a CFL. 2) $L_1 \subseteq L_2 \subseteq L_3$ are Languages, if $L_1$ and $L_3$ are ...
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982 views

Show that PTIME and PSPACE is closed under Klenee star

How to show that PSPACE and PTIME are closed under Kleene star ? I can only show that NP is closed, but it is easy because we can use non-determinism to guess partition of word. In these two cases I ...
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146 views

If a language is X-complete, is its complement is X-complete as well?

I'm looking for an information about closure of complexity complete classes. Is it true that any language, if the language is X-complete, then its complement is X-complete? Why? I was thinking ...
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66 views

Decidability language, intersection

I have two langages $ A, B \in \mathrm{coRE}$. How can I prove that $ A \triangle B= ( A - B) \cup (B - A)$ is also in $\mathrm{coRE}\,$?
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67 views

Is P closed under subwords? [closed]

Given a language $L\subseteq \Sigma^*$ in $P$, is the language $subwords(L) = \{v\in\Sigma^* : \text{there exist } u,w\in \Sigma^* \text{ with } uvw\in L\}$ that consists of all subwords of words ...
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Regularity of a language constructed from a know regular language

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|\...
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$L = \{\alpha^i \beta^j \gamma^k \vert i,j,k \in \mathbb{N}_0, (i=1) \Rightarrow (j=k)\}$

I am asking this question here, because I am not allowed to comment on the thread that I am actually interested in, but maybe someone can still help me? I alredy found an anwser to the Problem above (...
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34 views

Decidability of equality of expressions involving exponentiation

Let's have expressions that are finite-sized trees, with elements of $\mathbb N$ as leaf nodes and the operations {$+,\times,-,/$, ^} with their usual semantics as the internal nodes, with the special ...
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131 views

Is star closure of reverse of grammar equivalent to reverse of closure of that grammar

I need to proof if that it's true or not. $ (G^R)^* = (G^*)^R $ If $G$ is a CFG and $ G = \langle V, \Sigma, \delta, S \rangle $ where $ V $ = Set of Variables or Non-Terminal Symbols $ \Sigma $ = ...
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197 views

Unrestricted grammar is closed under intersection

I want to show that unrestricted grammar is closed under intersection and I don't want to use Turing machine or etc. So I think that we have two grammar $G_1$ and $G_2$ that are restricted for example ...
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456 views

Is PSPACE closed under the following forall-there-exists construction?

Suppose I have a language $L$ (over alphabet $\Sigma$), such that $$ w \in L \iff (\forall x \in \Sigma^*) (\exists y \in \Sigma^*) P(x,y,w). $$ and I can give a turing machine that decides $P(x,y,w)$ ...
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189 views

Proving specific prefixes of regular languages are regular

There are particular problems in Kozen that I'm unable to solve, and they seem to be similar to each other. It is showing that sets: $$ \{x \mid \exists y: |y| = 2^{|x|} \text{ and } xy \in A \}$$ $$ ...
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577 views

Are DCFLs closed under concatenation with a regular language?

I have found various opinions saying they are (a link to one is given in D.W.'s comment). However, a proof that DCFLs themselves are not closed under concatenation found here on StackExchange seems to ...
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240 views

Formal language properties and finite state machines [closed]

What are properties of a formal language? Which and how would they be needed to prove that some Non-Deterministic finite state machine can accept a given language?
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Are deterministic context-free languages closed under reversal of languages? [duplicate]

It is well known that context-free languages are closed under the reversal of $L$. My answer to the question "Is the time reversal symmetry of non-deterministic computations important?" notices that ...
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1k views

Turing-recognizable languages closed under star operation

I'm tasked with demonstrating that the class of Turing-recognizable languages is closed under the operation of star, but I'm confused about how this is true. For example, I have a TM to recognize a ...
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130 views

Given a regular language, show another language is regular

I was hoping someone could help me with this question, since I'm having trouble determining what approach to take. Let $L \subseteq \{0,1\}^*$ be a regular language. Show the language $\{w \in \{0,1\}...
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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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Is the union between a regular language and a random language also a regular language?

If not can we draw any conclusion about the newly fromed language, the language that represents the union between a regular language and a non regular language (not context free but a truly random ...
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299 views

The image of a recursive language under a computable function

Let $f:\Sigma^{*}\to\Sigma^{*}$ be a computable function and let $L$ be a recursive language. Is $f(L):=\left \{{f(w)|w\in L} \right\}$ recursive? Here, I see clearly, that $f^{-1}(L)$ is recursive (...
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258 views

Are the regular languages closed against injecting single letters?

Let $L$ an arbitrary regular language and $\qquad L_2 = \{uav : uv \in L\}$. Am I correct to say that this language is not regular by saying: $L$ has an even number of $a$'s. So $u$ and $v$ have ...
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Why $A \cap B = \widehat{\widehat{A}\cup \widehat{B}}$ does not holds for the class of Recursively Enumerable Languages?

"The class of Recursively Enumerable Languages is closed under Union, and Intersection but they are not closed under Complement." I know why they are not closed under Complement & why they are ...
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Prove Context Free languages not closed under difference?

Given two context-free languages $L_1$ and $L_2$, the language given by the difference of the two languages, $L_1 - L_2$, is (in general) not context-free. Is it possible to prove this without using ...
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If $L$ is a $CFL$, then why isn't $L^*$ also $CFL$

I was studying closure properties of CFLs and I came across this. I want to understand why $L^*$ is not a CFL, can anyone explain me in depth with simple examples?
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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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Union of finite and non-regular language [duplicate]

Question: ($B$ and $C$ are languages) $B$ is finite,$C$ isn't regular: Prove/Disprove: $C\cup B$ isn't regular. Thoughts: My intuition says this is true, but I need an idea to prove it. Since I don'...
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222 views

Regular language concatenation with superset

Let $A$ be some alphabet. $A$ itself is a regular language. $E = A^*$ is regular language over $A$. $E$ is a superset of all languages over $A$, regular or otherwise, i.e $E$ contains every possible ...
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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 ...
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470 views

If L1 ⊆ L2 and L2 is regular, then L2 − L1 is regular [closed]

I'm having trouble being 100% sure about this answer. I feel like it's false but I'm having trouble coming up with a complete answer. Can someone explain this step by step? Thanks.
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Finding the mistake(s) within this “proof” of NP being closed for complement

For my classes in theoretical computer science the following proof must be shown to be wrong. However, this is the first time I am attempting myself at this topic, so I would be thankful for some help:...

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