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

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7
votes
1answer
79 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 ...
1
vote
0answers
38 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?
2
votes
1answer
36 views

Prove that regular languages and context-free languages aren't closed under $Perm(L)$

Let the operation $$Perm(L) = \{ w | \exists u \in L \text{ such that } u \text{ is a permutation of } w \}$$ Prove that both regular languages and CFLs aren't closed under $Perm(L)$. I've tried ...
3
votes
3answers
148 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 ...
2
votes
2answers
36 views

Prove/ Disprove: If $L$ is a CFL then $A(L)$ is a CFL too

Consider the operation $A(L)$: $$A(L) = \{ w: w\in L \land w_R \notin L \}$$ where $w_R$ is the reverse of $w$. Prove/ Disprove: if $L$ is a CFL language so does $A(L)$. I am almost certain ...
5
votes
1answer
35 views

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 ...
5
votes
1answer
73 views

Closure properties of linear context-free languages

Under what operations are linear context-free languages closed? Suppose $L_1, L_2$ are two linear context free languages. Are there any guarantees about $L_1 \cup L_2$, $L_1 \cap L_2$, ...
3
votes
2answers
62 views

Proving regular languages are closed under a form of interleaving

I've got the following definitions: $$\mathrm{Interleave}\,(x,y) = \{w_1\dots w_n\mid w_i\in\{x_i,y_i\} \text{ for }i=1, \dots, |x|\}$$ when $x$, $y$ and $w$ are words with $|x|=|y|$ and $w_i$ means ...
2
votes
1answer
36 views

prove that a language is context free given a regular language

R is a regular language over $\Sigma=\{0,1\}$ $Sub(R)=\{0^i1^j \mid \exists w\in R.|w|=i-j \}$ I need to prove that Sub(R) is context free. I know that the quotient of a context free language with a ...
0
votes
1answer
43 views

Proving that non-regular languages are closed under concatenation

How can I prove that non-regular languages are closed under concatenation using only the non-regularity of $L=\{a^nb^n|n\ge1\}$ ?
0
votes
1answer
47 views

Confusing example of a language which may be Context-free or not context-free

Hi so consider the language $L= \{(0^i)(1^j)\mid i=k*j \text{ for some positive }k\}$ Could I not rewrite this as $\{((0^k)^j)(b^j)\mid k>1\}$. Seeing it in this form makes me think of a form $a^n ...
-1
votes
1answer
32 views

Prove a Language is Regular [duplicate]

For a language $L\in\Sigma^*$ we define $$ L^*=\{w\mid \exists k\in \mathbb{N}\cup\{0\}, ∃x_1,...,x_k\in L \ (w=x_1...x_k) \} $$ Let $L$ be a regular language over some alphabet $\Sigma$. Prove that ...
1
vote
1answer
87 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?
2
votes
1answer
83 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 ...
1
vote
0answers
18 views

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 ...
3
votes
1answer
75 views

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 ...
-3
votes
1answer
69 views

Prove A² is regular [duplicate]

Suppose that $A$ is a regular language. How can I show that $A^2 = A \cdot A$ is a regular language? Is there a construction?
1
vote
2answers
63 views

Are regular languages closed against an intersection that keeps words with the same number of ones?

How can we show that the class of regular languages is closed under the following operation? Let $L_1$ and $L_2$ be laguages over $\Sigma=\{0, 1\}$. The operation is: $$\{x \in L_1 \mid \text{ for ...
0
votes
2answers
83 views

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 ...
0
votes
1answer
38 views

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 ...
0
votes
1answer
81 views

If L1 ∪ L2 and L1 are regular, is L2 also regular?

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 ...
2
votes
1answer
250 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. ...
1
vote
1answer
102 views

If $g ∘ f$ is primitive recursive, are $f$ and $g$, too?

Assuming I have functions $f, g : \mathbb{N} \to \mathbb{N}$ and I know that $g \circ f$ is a primitive recursive function. What can I tell about $f$ and $g$? Are they primitive recursive as well? Or ...
0
votes
1answer
61 views

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 ...
4
votes
1answer
71 views

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 ...
0
votes
0answers
55 views

Union, Intersection, Difference, etc. of different types of languages

I am preparing for a competitive exam (GATE) in which questions are asked in Automata about operations among different types of languages. For example, If $L_1$ is recursive & $L_2$ is ...
0
votes
1answer
30 views

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?
1
vote
1answer
24 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 $\mathrm{coRE}\,$?
2
votes
1answer
153 views

are regular languages closed under division

I am trying to solve this question which appeared in previous exam paper Can someone help me what i am failing to understand For languages $A$ and $B$ define $A \div B = \{x \in \Sigma^{\ast} : xy ...
2
votes
1answer
75 views

How to prove that the Myhill-Nerode equivalence classes for L are the same as for its complement?

Given language $L$, I want to show that its Myhill-Nerode equivalence classes are the same as for its complement $\overline{L}$. I am thinking of constructing a DFA $M$ for the Language $L$ so the ...
0
votes
1answer
39 views

prove decidability and recognizability

I want to prove that for any language $L_1$ described by a Turing machine and any regular language $L_2$, $L_1 \cap L_2$ is described by a Turing machine that its recognizability and decidability is ...
1
vote
1answer
47 views

Is the extension of every undecidable theory undecidable?

While it is not the case that the extension of every decidable theory is decidable, is it true that: the extension of every undecidable theory undecidable? In other words, given an undecidable ...
1
vote
1answer
60 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 ...
1
vote
1answer
207 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 ...
1
vote
1answer
173 views

Concatenation of languages in NP

I have a hard time to understand why the concatenation of two languages over an alphabet (concatenation is in NP), doesn't imply that each of the languages for themselves are in NP. I talked with my ...
6
votes
3answers
874 views

How to XOR automata?

Say we have 3 DFAs. We know how to OR, AND, or NOT them. But how does one XOR them? There is not one single mention of this online. $x\; \mathrm{XOR} \;y\; \mathrm{XOR} \;z = ((x|y)(\neg x|y)|z) ...
5
votes
1answer
58 views

Are the complements of $NP$-languages with only $n$ words of length $n$ also in $NP$?

Assuming $\Sigma = \{ 0, 1\}$.. Given a language $L$, such that for each $n\in \mathbb{N}$ we have $n$ words of length $n$ in $L$ and assuming $L\in NP$, can we prove also that $L\in Co-NP$? So it ...
1
vote
1answer
162 views

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 ...
6
votes
1answer
134 views

If L is context-free, must FH(L) be context-free?

Define $FH(L) = \{x \in \Sigma^* : \exists y \in \Sigma^* \text{ with } |x| = |y| \text{ such that } xy \in L\}$. In other words, $FH(L)$ is the set of first halves of even length strings in $L$. ...
0
votes
1answer
280 views

Is Context Free Language closed under perfect shuffle?

Note that this is not shuffle but perfect shuffle, defined as follows: Let $w = a_{1}a_{2} \ldots a_{n}$ and $x = b_{1}b_{2} \ldots b_{n}$ be two strings of the same length. Then the perfect shuffle ...
0
votes
1answer
186 views

Kleene positive closure - help in proofing this claim

I just started a course called 'Automata and Formal Languages'. I'm having difficulty in proofing\disproofing this equality. $ (L_{1} \circ L_{2})^{+} = L_{1}^{+} \circ L_{2}^{+} $ Where: $ ...
3
votes
2answers
528 views

If both the concatenation of two languages and the second “half” are regular, is the first too?

Given that $L_2$ is regular and infinite and $L_1 \cdot L_2$ is regular, then $L_1$ is also regular. I need some help on getting started on proving this is the case. My intuition is that if $L_1 ...
2
votes
3answers
186 views

Prove that the equal-length concatenation of regular languages is context free

If A and B are regular, then prove that $A@B = \{xy \mid x \in A \text{ and } y \in B \text{ and } |x|=|y|\}$ is always context free. So I'm trying to come up with the proof that looks something like ...
4
votes
1answer
348 views

Are DCFLs closed under reversal?

According to this chart, DCFLs are closed under reversal. However, I am not convinced as the intuitive proof (reversing the arrows of the controlling finite state machine and switching the pushes and ...
9
votes
3answers
2k views

Union of regular languages that is not regular

I've come across that question : "Give examples of two regular languages which their union doesn't output a regular language. " This is pretty shocking to me because I believe that regular languages ...
2
votes
2answers
98 views

Prove that REG is closed against removing all but lexicographicaly largest words (per length)

Let $\Sigma_n = \{0, 1, ... , n-1\}$. Suppose $L \subseteq$ $\Sigma^*_n$, and let $\qquad\displaystyle\mathcal{B}(L) = \{ x \in L : x = \textrm{lex}\max L_m, m \in \mathbb{N}_0 \}$, ...
9
votes
1answer
289 views

Constructing all context-free languages from a set of base languages and closure properties?

One way of looking at regular expressions is as a constructive proof of the following fact: it's possible to construct the regular languages by starting with a small set of languages and combining ...
1
vote
1answer
75 views

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 ...
1
vote
0answers
111 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 ...
0
votes
1answer
75 views

For two regular languages, why is the set of words from one that don't have a subsequence in the other also regular?

In general, a string $x$ is a subsequence of $w = w_1\dots w_n$ if there are integers $i_1<\dots< i_k$ such that $x = w_{i_1}\dots w_{i_k}$. The subsequence is proper if $k < n$ and $k > ...