# Questions tagged [closure-properties]

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

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### What happens with trios, full trio, (full) semi-AFL, (full) AFL if we require closure under intersection?

Wikipedia says: A trio is a family of languages closed under e-free homomorphism, inverse homomorphism, and intersection with regular language. A full trio, also called a cone, is a trio ...
1answer
2k 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 ...
1answer
39 views

### How can the union of two 'context-free but not regular' languages be regular?

I cannot understand how the union of two languages which are context-free but not regular, can result in a regular language: If $L_1$ and $L_2$ are 'context-free but not regular' languages, defined ...
1answer
597 views

### Complexity classes that are closed under subtraction

Are NP or P closed under subtraction? Im having a hard time deciding whether they are or aren't. Question was edited Original question: Im having some hard time figuring out what languages are closed ...
0answers
42 views

### Intersection of decision problems?

Say we have two problems $\Pi_1\in NP$ and $\Pi_2\in coNP$. Where does $\Pi_1\cap\Pi_2$ live?
1answer
29 views

### Is the difference of two context-free languages still context-free?

i am asking myself the following question: Assuming: A and B are context-free languages, then A - B (difference) must also be context-free language, right? but I do not know how to prove it.
2answers
81 views

### Can the regular image of a context-free language be undecidable?

I just need to know the truth or falsity of a simple statement. Let $L_1$ be a context-free language over an alphabet which contains some number of characters $\Sigma$, as well as a single, special ...
1answer
21 views

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674 views

### Proof that the language $ww^R$ is not regular without using the pumping lemma

I am breaking my head over this. Let the alphabet $A$ be given by $A = \{a,b,c\}$ and let $$L = \{ww^R \mid w \in A^* \}.$$ Prove that the language $L$ is not regular without using the pumping ...
2answers
52 views

### if $L_1$ and $L_2$ are languages over the same alphabet and $L_1 \cap L_2$ is context free, at least one of them must be context free

I am having a hard time understanding if this would be true or false, can someone point me in the right direction?
1answer
54 views

### unambiguous context-free languages and complementation

I was considering the following two natural questions about the relationship between unambiguity and complementation for the class of context-free languages: Is the complement of an unambiguous ...
2answers
3k views

### Show that NP is closed under concatenation

Show that NP is closed under concatenation. This is a homework problem and I would appreciate some guidance. I began by saying the following: Let $A$ and $B$ exist in NP. Let $V_1$ and $V_2$ be ...
4answers
10k views

### Is the class NP closed under complement?

Is the class $\sf NP$ closed under complement or is it unknown? I have looked online, but I couldn't find anything.
2answers
11k views

### Why isn't the class of Turing-Recognizable languages closed under Complement?

I'm studying Turing Machines and I've already showed how Turing-Decidable is closed for the operations of Union, Intersection, Concatenation, Complement and Kleene Star. Next I did some demonstrations ...
2answers
14k views

### Union and intersection of a regular and a non-regular language

Lets say we have $L_1$, which is a regular language and $L_2$ which is not. Are $L_1 \cap L_2$, $L_1 \cup L_2$ , $L_1$ \ $L_2$ and $L_1 \cdot L_2$ are always non-regular languages? We know that two ...
1answer
119 views

### Closure of regular languages under deleting a 1 from each even run of 1s

Let $R$ be a regular set over the alphabet $\{0, 1\}$. Give a machine construction to prove that the set obtained by deleting one 1 from each even length block of 1’s is also regular, and using ...
2answers
100 views

### Finding if the given language is regular or not

I have the language $$L = \{a^mb^nc^o| \, m + n + o > 5\}$$ where $m,n,o$ are non-negative integers. I have to find whether the language is regular or not. My attempt: I feel it should be non ...
1answer
166 views

### Finite state automaton for the reverse of the language, and multiple starting states

I am studying about Finite State Automaton, and I found that the when reversing a language (i.e., transforming $L$ to $L^R$), I have to add new start state. Why is that? Also, can a Finite State ...
1answer
110 views

### Show that RP is closed under concatenation

I'm trying to prove the following problem: Show that $RP$ is closed under concatenation Now, let's say that the two languages are $L_{1}$ and $L_{2}$ (both in $RP$). Then I accept a word iff the ...
1answer
368 views

### Why is the intersection of these two Languages Recursively Enumerable, not Recursive?

I am only several days exposed to computational theory, so my understanding is quite slim: in a question, it says that for a regular language L1 and a recursively enumerable but not recursive language ...
2answers
93 views

### Proving $L'=\{uv : u \in L \; \land v \in L^R \; \land |u|=|v|\}$ is a CFL with closure properties [duplicate]

Given a language $L$ over $\Sigma=\{a,b\}$ let us define $L'=\{uv : u \in L \; \land v \in L^R \; \land |u|=|v|\}$ Prove: if $L$ is regular, then $L'$ is a context free language. I know how to ...
1answer
190 views

### Is the union of NP Complete language and a finite language (in P) NP Complete?

Let there be a language $A$ which is NP complete and language $B$ which is a finite language, is the union of $A \cup B$ NP complete language?
1answer
35 views

### 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 ...
0answers
32 views

### Dependency of operations of languages

I've struggled in the closure properties of the general class of languages because I couldn't use any automata concept and grammars. In specific, I'm interested in dependency of operations. (The ...
1answer
603 views

### 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′ ...
3answers
11k views

### Context-free Languages closed under Reversal

In class this week we've been learning about the CFLs and their closure properties. I've seen proofs for union, intersection and compliment but for reversal my lecturer just said its closed. I wanted ...
2answers
2k 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 $L=\{a^nb^n|n\ge1\}$ ?
4answers
778 views

### Why can't we say that NP is closed under complement given that we can say it is closed under intersection

I found online solutions which prove the closure of NP under intersection in the following way: given machines $M_1,M_2$ for accepts nondeterministically languages $L_1,L_2$, we construct the ...
1answer
543 views

### Class P is closed under concatenation

Proving that Class P is closed under concatenation. The answer is given below: But I do not know why stage 2 is repeated at most O(n), could anyone explain this for me please?
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220 views

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### Show that the collection of Turing-recognizable languages is closed under homomorphism [duplicate]

I have seen this question here, Closure of Turing-recognizable languages under homomorphism But actually this question answers the question of "What is the relation between homomorphism and ...
2answers
142 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 \$|...
1answer
541 views

### Closure of Turing-recognizable languages under homomorphism

I've proven that the Turing-recognizable languages are closed under concatenation and I need to show that they are closed under homomorphism. But what's really the difference? Doesn't closure under ...