0
votes
2answers
16 views

Regular Expression Similarity Check

I was solving Formal Language and Automata Theory for a competitive exam, whence I came upon this following question: The regular expression 0*(10*)* denotes same set as: 0(0+10)* (0+1)10(0+1) ...
0
votes
0answers
20 views

A language recognizes by a DFA

So the solution in problem 1.61 in this file http://bsd7.cs.sunysb.edu/~stark/CSE540/Handouts/hw1_notes.pdf uses an argument of the form: "$M$ is a DFA that recognizes the language $C$. Because the ...
1
vote
1answer
7 views

Find a state machine or a regex that accept the following language description

The alphabet of the language L is {a, b} and there has to be an even number of both a's and b's but no other restrictions apply. I've been at this for over an hour, drawing state machines that lead ...
1
vote
1answer
38 views

PDA and Some language Grammar inference

L1={$w^* $| w=x and $ x \in \Sigma^*$} L2={$ww^R ww^R $| $ w \in ( \Sigma + \Sigma)^*$} L3={$w | w=xy, x,y \in \Sigma^*$, y is a substring of x} a) there is a PDA(push down automata) that accept ...
1
vote
1answer
48 views

Regular Language and Unknown Operation

I read the $L$ is a Regular language. Define $L'$ as following: $$L'=\{a_2a_1a_4a_3\ldots a_{2n}a_{2n-1}\mid a_1a_2\ldots a_{2n} \in L\}.$$ why $L'$ is Regular? any hint or idea would be highly ...
2
votes
1answer
40 views

A CFG Grammar for One Language

Suppose : $w_1,w_2 \in \{a,b\}^∗$ and $ L=\{w_1w_2 \mid w_1,w_2 \in \{a,b\}^* \land n_a(w_1)=n_b(w_2)\}$ $n_a$ is number of $a$'s and $n_b$ is number of $b$'s. This is a Entrance Exam question. I ...
2
votes
1answer
60 views

Challenge on Property of Complement in Language

We have: and M be a finite automata. Suppose d(M) be a deterministic automata that equivalence to M. if $M_1$ and $M_2$ be two finite automata, $M_1 + M_2$ is the finite automata such that the ...
1
vote
0answers
32 views

Language of Specific Grammar

I ran into this exercise in Sipser's Note on Computation Theory. Consider the following grammar $G$: $$\begin{align} S &\to aSD \;|\; bB \\ D &\to dS \;|\; a \\ B &\to bB \;|\; ...
-1
votes
1answer
21 views

disproving union of infinitely many regular languages

I want to disprove the following statement: "if $L$ is the union of infinitely many regular languages, then $L$ is guaranteed to be a regular language." I don't know where to start. Any hint will be ...
1
vote
2answers
32 views

NFA for $L = \{ w \in \{a,b,c\}^* : \text{at least one symbol appears only once in }w \}$

Write an NFA to recognize the language $$L = \{ w \in \{a,b,c\}^* : \text{at least one symbol appears only once in }w \}.$$ I'm not quite sure how to do this question. I don't know how to keep ...
1
vote
2answers
99 views

pumping lemma: ww^R not regular

I'm trying to prove that $L = \{ww^R : w \in \{a,b\}^*\}$ ($w^R$ is the reverse of $w$) is not regular using the pumping lemma. Let $p$ be the pumping length and $s = a^pbba^p$. $x = \epsilon$, $y = ...
2
votes
2answers
41 views

Regular Expressions Help [duplicate]

I need a little help with Regular Expressions. The allowed operations are obviously + (union) , * (Kleene star) and concatenation. I have to write Regex for the following 2 examples. I have tried a ...
1
vote
2answers
70 views

Formal Languages - Regular Expression

I've been battling the following two questions for more than a day today. Write a regular expression (comprised of {a, b}) that contain at least two b and do not contain abb. Write a regular ...
1
vote
2answers
26 views

FSM to be regular with atleast two 0's and at most two 1's

Is it possible to construct a FSM to prove that the set $X$ is regular, where $$ X = \{s \in \{0,1\}^* \mid \text{$s$ contains at least two $0$'s and at most two $1$'s}\}\ ? $$
1
vote
2answers
34 views

Closed regular languages

Are regular languages closed under the following construction? $f(L) = \{w \mid w \in L$ and for all prefixes $x$ of $w$ it holds that $x \notin L$ $\}$
1
vote
0answers
44 views

Minimal DFA for a given regular expression

How can I construct a minimal DFA from the following definition? $L=\{w \in \{a,b,c\}^* $: if the second-to-last letter from w is an $a$, then the number of $c$'s $\le 1\}$ I've already made a ...
0
votes
0answers
31 views

Context-free languages

Is the following language context-free? $\{ w \in \{a,b,c\}^* : (\#_a(w) - \#_b(w)) \cdot \#_c(w) = \#_b(w)$ and all c's are encountered before any a$\}$. $\#_a(w)$ = amount of a's in w Thanks in ...
1
vote
1answer
46 views

Is there a prefix-free regular language whose set of proper prefixes is not regular

Is there a regular language $L$ such that the language $$L^\prime := \{ w : w\text{ is a proper prefix of a word in }L \}$$ has the following properties: $L^\prime$ is not regular, and $L^\prime$ ...
1
vote
1answer
22 views

Subset of regular languages

Assume $L_1$ and $L_2$ two regular languages, and $L_1\subseteq L\subseteq L_2$. Does this imply that $L$ is a regular language? Thanks in advance.
1
vote
1answer
62 views

Prove or disprove whether the following is a regular language

I'm given the regular language L, and w being an element of L. If we remove the w from the language L, will the resulting language be still regular? Well I thought to be true. Since initially is a ...
0
votes
1answer
65 views

Prove or disprove that the language $L_1 = \{a^nb^m \mid n < m \}$ is regular

I have possible strategy for a proof that it is not regular. I am wondering if it is valid. Step 1: Prove that the language $L_2 = \{a^nb^n\}$ is not regular (for example with the Pumping Lemma). ...
0
votes
1answer
31 views

Prove L is not a regular language (A Finite State Automaton cannot accept it)

$$\mathscr L = \{x \in \{0,1\}^* \mid \text{there is a } y \in \{0,1\}^* \text{ such that } x = yy\}$$ How can I prove that this is not a Regular language? I tried using proof by contradiction but ...
1
vote
1answer
45 views

Checking some Regular Expression problems

I'm given the alphabet $$ \Sigma = {\{a,b}\} $$ I tried to write a regular expressions for presenting the following sets: All strings in $$\Sigma ^ *$$ with: a-) number of 2s divisible by 4 b-) ...
1
vote
1answer
21 views

How to prove the following related with regular languages

How can we prove the following. If $$\sum$$ is any alphabet and L is any language $$L \subset \sum*$$ Then L*L* = L* ?
1
vote
0answers
58 views

Prove the following by using mathematical induction

If we define the alphabet such that $$ \Sigma = {\{a,b}\} $$ and let $w$ be a string over it. I'd like to prove $$ ( \operatorname{comp}(w))^R = \operatorname{comp}(w^R) $$ where $$ w^R$$ and ...
0
votes
1answer
25 views

Question about Notation in a Regular Language

I am a little confused about the following notation: $L' = \{xy|x\in L \ , y\in L^R\}$. I think this expression is not equivalent with palindrome but I am not entirely sure. For example, I think the ...
2
votes
1answer
29 views

Equivalence of two regular expression

I have a quick question to ask. So I am trying to come up with a regular epxression which represent a language over {a,b} that contains at least one 'b' in it. I came up with this: $$(a| b)^*b(a| ...
0
votes
1answer
22 views

How can I show ithat a language is regular?

I have a very quick question about regular languages, I think $\{a^{2n}| n\geq 1\}$ is regular. I do know that pumping lemma can be used to show something that is not regular. I am wondering what ...
0
votes
1answer
28 views

Question About Pumping Lemma used on a FA

I am learning about pumping lemma and I am trying to solve a problem. I need to use pumping lemma to show that: the Language L(M) defined by the following machine is infinite. Here is the dfa: ...
2
votes
0answers
54 views

This proof in my textbook involving the pumping lemma appears incorrect - is it?

It states Let $B$ be the language $\{0^n1^n2^n | n \geq 0\}$. We use the pumping lemma to prove that $B$ is not regular. The proof is by contradiction. Assume to the contrary that $B$ is regular. ...
3
votes
3answers
73 views

Is $L = \{(x,y,z) | x+y=z\}$ a regular language?

Suppose $x,y,z$ are coded as decimal or their binary representations in an appropriate DFA. Is $L$ regular? My intuition tells me that the answer is no, because there are infinitely many combinations ...
2
votes
1answer
324 views

Proof of Equivalence of NFA and DFA, quick question about the setup

I am looking at the proof of equivalence of non determinstic finite automata(NFA) and deterministc finite automata(DFA). I am have a small quesion about the construction: Let ...
1
vote
1answer
57 views

Pumping Lemma for regular languages proof template

http://www.cs.uiuc.edu/class/fa06/cs273/Lectures/pumping-lemma/pumping-lemma.html So, I went to that site and it says: $w = xyz$ $|xy| \leq p$ $|y| \geq 1$ for all $i$, $xy^iz$ is in ...
0
votes
1answer
46 views

Struggling with Proof Writing. Simple question for demostration.

I am practicing writing proofs over regular expressions. Here is the question: Show that $(r\cup \varepsilon)^*= r^*$, where $r$ is a string. Intuitively, the left hand side is the concatenation of ...
1
vote
1answer
26 views

Give a regular expression for $A = \{1^{k}y|k \geq 1, y \in \{0,1\}^{*}$ and $y$ contains at least $k$ $1$'s $\}$

The regular expression that is given is $1(0 \cup 1)^{*}10^{*}$. I'm having trouble realizing why this regular expression describes the language given. For example, the string (for $k$ = 4) $1111$ ...
0
votes
2answers
132 views

Prove the language $\{a^k b^l : k \neq l \}$ is not regular

Prove that the following language is not regular: $$L=\{a^k b^l : k,l \ge0, k\ne l\}$$ The problem is that I should use "distinguished states" not the pumping lemma, which is usually used for such ...
2
votes
1answer
43 views

A correct proof for this pumping lemma example?

Given the language $L = \{0^{2^n} | n \geq 1\}$ So, the language contains all strings that have $2^n$ $0$s. First of all I take $z = a^{2^p}$ where $p$ is the constant guaranteed by the pumping ...
0
votes
2answers
333 views

Regular expression and DFA/NFA questions

If a language L is generated by a regular expression, then L is recognized by a DFA. I think this is true, because regular expressions describe regular languages, those of which are exactly ...
1
vote
3answers
97 views

NFA for $(ab|a)^{*}$ using only 2 states

In Introduction to the Theory of Computation by Michael Sipser, there's an example which shows how to convert the regular expression $ (ab|a)^{*}$ into an NFA. The "standard" method results in 8 ...
1
vote
1answer
51 views

is this language regular or not?

I have problem with this language $$L = \{ a^n b^m : \text{$n+m$ is odd} \}$$ is it regular or not My Solution I used pumping lemma, w = a^2p b^2p+1 (the same for a^2p+1 b^2m ) ...
0
votes
1answer
73 views

Finding regular expressions

I'm given the DFA shown below and need to find regular expressions for the following languages: $L_{1,2}^0, L_{2,1}^6, L_{2,5}^4, L_{2,3}^5, L_{1,3}^5$. The language $L_{p,q}^r$ is defined as ...
1
vote
1answer
170 views

DFA for Boolean Formula

Let $ f\left( b_{1}, \dots , b_{n} \right)$ be a boolean function. Define $S_{f} = \{\left( b_{1}, \dots , b_{n} \right): f\left( b_{1}, \dots , b_{n} \right)=1; b_{i} \in \{0,1\}, 1\leq i \leq n \}$ ...
0
votes
0answers
25 views

Is the next expression true: $(L_1 \cap L_2 )L_3\subseteq L_1L_3\cap L_2 L_3$?

Let $L_1,L_2,L_3$ be languages, Is the next expression true: $(L_1 \cap L_2 )L_3\subseteq L_1L_3\cap L_2 L_3$? After a half an hour of trying to disprove it, I've decided my intuition might be wrong. ...
3
votes
1answer
35 views

Irregular $a^nb^n$

We studied in class that regular languages closed under intersection. My question is : if we take the irregular language $L =$ {$a^nb^n : n\geq 0$} and the regular finite language $L' = \{a^3 ...
2
votes
1answer
50 views

Language concatenation

We learned in class that the regular languages are closed under concatenation (e.g $L_1L_2 =\{ w_1w_2 : w_1 \in L_1,w_2 \in L_2\}$ is a regular language if $L_1$ and $L_2$ are also regular ...
0
votes
1answer
18 views

proving regular language

let $L$ be a language over the alphabet $\{a,b\}$ that maintains that for each $w \in L$ ,the difference in absolute between the number of apearences of the letter $a$ and the number of apearences ...
4
votes
1answer
670 views

Routing Automaton

Is there a formal proof for the following question? For a DFA $M= (Q,\Sigma,\delta,s,A)$, we extend the function $\delta : Q \times \Sigma^* \to Q$, such that every $w \in \Sigma^* $, ...
1
vote
1answer
67 views

Infinite regular languages

There is a formal proof for the following sentence? For every 2 languages $A,B$, we write A@B if A subset of B and B\A infinite. Prove that if $A,B$ regular languages and A@b, than exists regular ...
1
vote
1answer
79 views

Is there a DFA with $k+2$ states which its reverse has $2^k$ states

I am trying to figure out if there exists a DFA $M$ with $k+2$ states (for every $k\in \mathbb{N}$ ) so that every automaton which accepts $L(M)^R$ has at least $2^k$ states. I am trying to find an ...
1
vote
1answer
41 views

Prove the existence of $C\in L_{regular}$ so that: $A \prec C \prec B $

Given $A,B$ regular languages. Prove the existence of $C\in L_{regular}$ so that: $A \prec C \prec B $ Whereas $A\prec B$ stands for: $A\subset B $ and $B\setminus A $ is infinite regular language. I ...