Questions tagged [closure-properties]

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

Filter by
Sorted by
Tagged with
4
votes
1answer
327 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?
2
votes
3answers
429 views

Regularity of “middles” of words from regular language

I need some help with the following problem. Let $L \subseteq \Sigma^*$ be a regular language. I have to prove that the language $P = \{\alpha \mid \beta\alpha\gamma \in L, \beta,\gamma \in \Sigma^*\}$...
0
votes
1answer
130 views

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:...
6
votes
1answer
350 views

Why do we study closure properties of formal languages?

In automata theory we study formal languages like Regular, CF, CS and etc. and each of them have their own closure properties under union, intersection, star and etc. . I like to know, why it is ...
4
votes
1answer
354 views

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?...
3
votes
2answers
3k views

Why are palindrome and not-palindrome both context-free?

Both palindrome and its complement are context-free. This is very interesting. Both are non-deterministic context-free, which in general are not closed under complement. What is it about these two ...
1
vote
1answer
145 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 ...
0
votes
1answer
58 views

How to prove that the decidable languages are closed against iteration only by enumerators?

We have the $L\in R$, how can we prove that $L^*\in R$ only by enumerators? I try to use induction, but as I understand I wrong... I'd like to get any help!
2
votes
2answers
2k views

Closure under reversal of regular languages: Proof using Automata

I have been studying the closure properties of regular languages, referencing the book Introduction to Automata Theory, Languages, and Computation by John E. Hopcroft and Jeffery D. Ullman. Under the ...
0
votes
1answer
820 views

If a language is context free, then its complement is decidable

I am having a bit of trouble figuring this out. If L is context-free then we know it is decidable. The class of decidable languages is closed under complement thus, $L$ $\cap$ $L^{c}$, therefore $L^{c}...
1
vote
1answer
2k views

Proving $A$ avoiding $B$ regular if $A$ and $B$ are regular

Suppose we define an operation such that $$A \text{ avoiding } B = \{w \in A \mid w\text{ has no substring in }B\}\,.$$ How can I prove that, if $A$ and $B$ are regular then $A\text{ avoiding }B$ ...
7
votes
1answer
182 views

Smallest class of automata model whose corresponding language class contains CFL and is closed against (dis)allowing nondeterminism in the model

From a comment, an interesting question popped up. The class of CFLs (the languages recognized by PDAs) are obviously not closed under nondeterminism - what I mean by this is that deterministic PDAs ...
4
votes
1answer
589 views

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 ...
-1
votes
1answer
204 views

Is the complement of the given language necessarily in NP?

$A$ is a given language so that $A \in NP$. Assume that $P = NP$. Is $A'$ necessarily in NP? What I did: $A \in NP , P=NP$ $P=coP$ (Can be proven by running a TM $M$ as a decider for P, ...
3
votes
3answers
1k views

Why is the set of all regular expressions classified as context-free, instead of regular?

As I understand regular languages can be closed under concatenation, so can I concatenate the set of all regular expressions to classify them as regular?
0
votes
1answer
97 views

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 ...
1
vote
1answer
594 views

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 ...
4
votes
1answer
500 views

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 ...
-1
votes
1answer
51 views

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 ...
3
votes
1answer
4k views

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.
1
vote
1answer
2k views

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 ...
8
votes
4answers
366 views

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 $L_1,L_2$ ...
1
vote
1answer
188 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: $$ \{x \mid \exists y: |y| = 2^{|x|} \text{ and } xy \in A \}$$ $$ ...
1
vote
1answer
94 views

If two languages together cover all words and one is regular, is the other one as well?

If $L_1$$\subseteq$ $\Sigma^*$, $L_2$$\subseteq$ $\Sigma^*$ , $L_1$ is regular and $L_1$$\cup$ $L_2$ = $\Sigma^*$ then is $L_2$ necessarily regular? I think that the answer is yes, but I'm not sure ...
0
votes
1answer
114 views

How do I prove that a language is deletion closed?

For example, how could I prove that the following language is deletion closed: {$a^k$$b^j$ : $j$, $k$ $\geqslant$ 0} The reason seems obvious to me, I just can't see a way to prove it.
1
vote
1answer
6k views

Why is the complement of a language that is not regular also not regular?

In my lecture notes I we were given two languages and were shown that each of the two languages were not regular. The second was the complement of the first language. To show the second was not ...
7
votes
1answer
174 views

Context-free languages not closed under making them “extension-free”

For a language $L$, define: $$ NE(L) = \{x \in L : x \text{ is not the proper prefix of any string in } L\} $$ I'm trying to show context-free languages are not closed under this operation. I've been ...
0
votes
1answer
355 views

Closure properties of finite state transducers

Given $T_1, T_2\colon \Sigma^* \to \Gamma^*$ ($\Gamma$ is output alphabet), let $\Delta(T_1, T_2)$ consist of all input strings $w \in Σ^*$ where $T_1(w) \neq T_2(w)$. Prove that FSTs ...
2
votes
1answer
380 views

$\mathbf{NC_2}$ is closed under log-space reduction

I actually have to prove the following : $\mathbf{NL} \subseteq \mathbf{NC_2}$ I have the following approach : I will prove that $\mathbf{PATH} = \{〈D, s, t〉 | \text{D is a directed graph with a ...
2
votes
2answers
2k views

Formal construction of PDA intersecting a DFA

Given the PDA $P = (Q_P,\Sigma,\Gamma_P,\delta_P,F_P)$ and the DFA $D = (Q_D, \Sigma, \delta_D,q_D,F_D)$ What is the 6-tuple definition of the PDA such that: $L(P') = L(P) \cap L(D)$ I don't ...
0
votes
1answer
215 views

Infinite Union of non-regular languages

Is infinite union of non-regular languages $L_i$ that form a chain such that $L_i\subseteq L_{i+1}$ always non-regular? Or is there a possibility that it be ever regular? Is there an easy way to ...
0
votes
1answer
147 views

How to prove the linear context free languages are closed under gsm mapping?

I'm stuck on the following question: How to prove the linear context free languages are closed under gsm mapping?
-2
votes
2answers
395 views

Why is the intersection of CFL and RL not always RL?

Suppose M is a CFL and N is aa RL. Then wouldn't the language generated by the intersection of M and N contain strings, some of which are accepted by both DFA and PDA? So if they are accepted by a DFA ...
-3
votes
1answer
291 views

Complement of a Language which is set of Turing Machine descriptions

If $L$ is the set of strings $\langle M\rangle$ such that $M$ accepts all strings of even length and does not accept any strings of odd length. What will be $\overline L$ ? a) set of strings $\...
3
votes
1answer
923 views

How to prove closure property of regular languages using regular expressions?

I know that we can prove closure of two regular languages under operations like union, intersection, concatenation etc. by constructing NFAs for them but how to do the same thing using regular ...
0
votes
1answer
78 views

Regular languages and sets proof

I just have general questions about sets and determining if they are regular languages. i) If A is regular, and A is a subset of B, then B must be regular. ii) If B is regular, and A is a subset of ...
1
vote
1answer
628 views

Does applying a homomorphism to the intersection of two CSLs yield RE languages?

For each language $L \in L(RE)$ there are a homomorphism $h$ and two context-free languages $L_1$ and $L_2$ such that $L = h(L_1 \cap L_2)$. I understand that this is because context-free languages ...
5
votes
3answers
433 views

Is an irregular language concatenated with a language with which it has no common symbols irregular?

Here's an example of what I'm talking about. Suppose I have a languages $$ L_{1} = \{a^ib^i \mid i>0\},\\ L_{2} = \{c^i \mid i>0\} $$ and $$ L_{1}L_{2} = \{a^ib^ic^i \mid i>0\} $$ Is it ...
1
vote
1answer
11k views

Prove transitivity of big-O notation

I'm doing a practice question (not graded HW) to understand mathematical proofs and their application to Big O proofs. So far, however, the very first problem in my text is stumping me wholly. ...
2
votes
2answers
511 views

show that language $L'$ is regular (given $L$ regular)

I am working on the following question: $L$ is regular. Show that $L'=\{x|\exists y,z,\ xyz\in L\wedge |x|=|y|=|z|\} $ is also regular. Firstly I show my idea. When you accept it I will try to ...
0
votes
1answer
6k views

prove language is Context-free and not regular [duplicate]

I have to prove that $\left \{ a, b \right \}^{\ast} - \left \{ a^ib^i | i\geq 0 \right \}$ is a context-free language and it's not regular. So far I've got that this language is not regular because ...
3
votes
1answer
1k views

Symmetric Difference of Turing Recognizable and Finite Languages

Let A be a Turing Recognizable Language and B a finite Language. I want to prove that their symmetric difference is Turing Recognizable. My reasoning: B is finite, therefore the finite number of ...
1
vote
0answers
572 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 ...
4
votes
1answer
989 views

Proof that CFL aren't closed under intersection using synchronous parallel (N)PDA composition

It is well known that the class of CFLs is not closed under intersection as follows e.g. from the following example: $$L_1 \cap L_2 = \left\{ a^mb^mc^n \mid m,n \ge 1 \right\} \cap \left\{ a^mb^nc^n \...
2
votes
1answer
170 views

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 ...
0
votes
0answers
71 views

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 ...
5
votes
3answers
385 views

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 ...
1
vote
1answer
4k views

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 ...
7
votes
1answer
383 views

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 ...
4
votes
3answers
3k views

Show that regular languages are closed under Mix operations

Let $L_1, L_2$, two regular languages and the operations: $$Mix_1(L_1, L_2) =\{ a_1b_1a_2b_2\ldots a_nb_n | n\ge 0 \land a_1,a_2,\ldots ,a_n,b_1,b_2,\ldots ,b_n\in\Sigma\\ \land a_1a_2\ldots a_n\in ...

1 2 3
4
5
7