Automata Theory, including abstract machines, grammars, parsing, grammatical inference, transducers, and finite-state techniques

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Are these two languages equivalent?

$L = \{w \in \{a,b\}^{*}$ such that $w = R(w)\}$ R(w) = the reverse of w $L = \{u \in \{a,b\}^{*}\}$ If not can you draw a pda or define a grammar for each? I don't see how they could have ...
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Euclid's Proof that there are an infinite number of primes

I'm trying to clarify my understanding of decidability of a language. The following question is totally made up by me so I hope it makes sense. Let $L = \{A: A \text{ is an algorithm that can ...
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80 views

how to convert Finite Automata into Push Down Automata

I am trying to convert Finite Automata into Push Down Automata and I am not sure if I am doing this right. There are not many good tutorials on this topic that I can find, but this is what I have. I ...
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58 views

Cartesian product of a DFA and PDA

I am just wondering how to do it. I am not quite sure if this is the right way. 1) Write down all the combination of states. 2) Take into consideration the "input" side of the DFA and write the ...
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72 views

Convert NFA with only accepting states to regular expression?

Suppose you wanted to find a Regular Expression that defines the language accepted by the folowing Finite State Automaton. Where S1 and S2 are both accepting states. How would I go on doing this?
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Is it possible to make a PDA for $\{ ww : w \in \{ 0,1 \}^* \}$?

Consider the language $L = \{ ww : w \in \{ 1,0 \}^* \}$. I know it's easy to make a PDA for $\{ w w^\text{R} : w \in \{ 0,1 \}^* \}$ where $w^{\text{R}}$ is the reverse of $w$, but I can't think of ...
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Designing a pushdown automata

I think I realized something, but I am not sure. So what I think I've realized is that if you have to design a L such that it accepts any string with the same amount of 1 and 0. You need 2 states: ...
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162 views

What is the difference between regex operations in math and regex in UNIX / Linux?

What is the difference between regular expression operations (union, concatenation, kleene star) and regular expression (implemented in UNIX and can be used together with the grep command)? Are there ...
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82 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). ...
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41 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 ...
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Is this grammar ambiguous?

$S > SS|10|0$ $S > SS > SSS > 0SS > 00S > 000$ $S > SS > 0S > 0SS > 00S > 000$ $2$ leftmost derivations
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Relationship between pushdown automata and CFG

I thought that pushdown automata and CFG were separate things in that one was the graphical expression of the other, but I saw a pushdown automata that uses CFG terms such as epsilon, S, S->1S0, or ...
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1answer
50 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-) ...
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1answer
59 views

Using the Pumping Lemma to prove a language is not regular.

I want to know if my proof is wrong and whether what I am doing works. $$\sigma = \{0, 1\}$$ $$A = \{0^n1^m \mid n < m\}$$ Claim: A is not regular. Proof: Assume A is regular. Let p be the ...
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25 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* ?
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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 ...
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Proof that an automaton stops

I discovered an automaton that produced interesting, random-seeming patterns. The rule was as follows. Given a grid of points, some occupied and some not, and a current point and a previous point; ...
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30 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 ...
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Proof that suffix of $ab$ needs at least 3 states in NFA

I need to prove that for an NFA that accepts all languages $L(M)=\{w \in \{a,b\}^* \mid wab \}$ with a suffix of $ab$ needs at least 3 states. The smallest automata would look like this: $\to(s) \to ...
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1answer
43 views

Proving that a language having a particular CFG grammar is equivalent to a particular L

I think we need to prove that L(G) is a subset of L and then we need to prove that L is a subset of L(G). For the first part, I think we need to say for any w in L(G) we have an even number of as ...
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Particular Problem for Context Free Grammars

Consider the context-free-grammar $G$ defined by productions: $$ S \rightarrow aS\,|\,Sb\,|\,a\,| b $$ Prove by induction on the string length that no string in $L(G)$ has $ba$ as a substring. I ...
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36 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| ...
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1answer
31 views

Create a general NFA for $M_n$

I need to create a general NFA $M_n$ where $n \in \mathbb{N_0}$ with the following language defined: $$L(M_n) = \left\{ w \in \{0,1\}^* \big | x1y \textit{ for } x \in \{0,1\}^* \textit{ and } y \in ...
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find a regular expression and FA that each define L1 ∩ L2

from the following pairs I am trying to find a regular expression and FA that each define ...
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Finite Automaton that accepts only the words baa,ab, abb and no other strings longer or shorter

I am trying to understand the answer here for FA that accepts only the words baa, ab, abb and ...
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1answer
26 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 ...
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1answer
37 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: ...
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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. ...
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3answers
81 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
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1answer
685 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 ...
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1answer
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Pumping lemma clarifications

http://www.cs.oberlin.edu/~asharp/cs383/handouts/pumping.pdf I came across this and realized that what some people told me was completely wrong. We get to choose xyz, but the demon choose uvw and uvw ...
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1answer
80 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 ...
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1answer
74 views

How do we choose a good string for the pumping lemma?

we have a Language $$\mathscr L: \{a^mb^n: m \ne n\}$$ we need to choose a good string $w$. apparently $w = a^{n+1}m^{n}$ is not a good string. can someone explain why? I also found this and I ...
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114 views

Show that, given regular expression $R$, we can decide whether $L(R)$ is prefix-free

Suppose language $L$ is called prefix-free if no member is a proper prefix of another. For instance, cat is a proper prefix of category and so $L = \{cat,category,ego,go,rye\}$ is not prefix free. ...
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Can the NFA constructs be minimized?

When converting a regular expression to a NFA you need to use certain constructs. My question is can these constructs be minimized? We have one with 4 states, I want to use the one with 2 states ...
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2answers
395 views

How do we prove that a particular DFA is minimal?

I guess we need to prove that there is no redundant state, so can we use state elimination and prove that the regular language is minimal? We could prove that the regular language is minimal by ...
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1answer
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transition graph that accepts only Λ and language a*

I am trying to have a transition graph that accepts only Λ and also one that accepts language a* ...is this ok ??? transition graph that accepts only Λ transition graph that accepts language a*
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1answer
48 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 ...
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1answer
57 views

Creating a Push Down Automaton from a Grammar

I have the following grammar, but I'm not sure what exactly it is that it does: $\qquad\begin{align} S &\to S \vee T \mid T \\ T &\to T \wedge F \mid F \\ F &\to p \mid \thicksim p ...
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1answer
264 views

Context free grammar to pushdown automata…

<expr> −→ <term> | <expr> + <term> <term> −→ <factor> | <term> × <factor> <factor> −→ (<expr>) | b ...
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Turing machine true/false questions

There is a non-regular language that is recognized by a Turing Machine. I believe the answer to this is true, because Turing machines can "count" computations and ...
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3answers
89 views

CFG for language l

please solve this question.thanks Consider the language L expressed by (a+b)*a defined over Σ = {a, b}. Draw FA and construct the CFG corresponding to the language L.
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145 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 ...
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Proof by pumping lemma

Let's say that we have to prove that $L = \{ww^Rv |w,v\in \Sigma^*\}$ is irregular. I would take a string such that $w = baba^m$ and $w^R=a^mbab$ and $v = a$ and then I would pump divide $w$ into ...
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1answer
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Showing that all regular languages are closed under reversal

I'm trying to show that $L^{reverse} = \{w^{reverse}:w \in L\}$ is a regular language. The first argument I can come up with is simply: if we have an NFA for $L$, then an NFA for $L^{reverse}$ can be ...
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1answer
64 views

Proving that a language is not context-free

Given the language $$L = \{ a^p \mid p\, \text{IS NOT prime} \}$$ is $L$ Context free? If not, prove that it's not. May I have some suggestions on how to use the pumping lemma to prove this, ...
2
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1answer
48 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 ...
4
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2answers
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Have action/predicate systems (or similar) been considered in the literature?

Question. Has the following concept, or anything similar, been considered in the literature? Definition. An action/predicate system consists of sets $A$ (the actions) and $X$ (the predicates) such ...
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1answer
27 views

If the language is context free?

i believe intuitively the following language is CF. But there is a book (without more description) that states the language is not CF. If I'm in a wrong way? $L=\{W_1cW_2 | W_1,W_2 \in (a+b)^* W_1 ...
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The halting problem for tapes that are or are not completely blank

Is it possible to construct a Turing machine that halts only if the tape is not completely blank? Also, is it possible to construct one to halt if the tape is completely blank? Intuitively, I think ...