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

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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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1answer
346 views

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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1answer
24 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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2answers
23 views

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
37 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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1answer
38 views

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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1answer
28 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
28 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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1answer
59 views

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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3answers
721 views

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
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 ...
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1answer
27 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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52 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. ...
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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
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1answer
307 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
19 views

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
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 ...
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1answer
51 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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92 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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21 views

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
117 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
26 views

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
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 ...
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1answer
41 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
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) $1111$ ...
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1answer
133 views

Context free grammar to pushdown automata…

<expr> −→ <term> | <expr> + <term> <term> −→ <factor> | <term> × <factor> <factor> −→ (<expr>) | b ...
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2answers
68 views

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
80 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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2answers
128 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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1answer
47 views

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
27 views

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
59 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
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 ...
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103 views

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
25 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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2answers
65 views

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 ...
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2answers
94 views

Combining two DFAs into an NFA to recognize concatenation

Suppose we have two separate DFAs that each recognize their own language. What is the most efficient way to combine these two DFAs into one NFA that recognizes the concatenation of both languages?
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1answer
57 views

How to construct a grammar G such that L(G)={x^ny^mx^my^n/m,n>1}?

construct a grammar $G$ such that $L(G)=\{x^ny^mx^my^n/m,n>1\}$? I don't have much idea how to approach this one. Could some help me to understand how to approach these kinds of problem?
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2answers
331 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 ...
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2answers
94 views

Why is a 3-PDA no more powerful than a 2-PDA?

A k-PDA is a pushdown automata with k stacks. My textbook on Computation Theory has an exercise that asks to prove that 3-PDA is no more powerful than a 2-PDA. Now consider the language that I made ...
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1answer
44 views

Proof of a Known Claim About Languages

I would like to know how to prove that there is no non-trivial language $L$ that satisfies the following condition: $${\large \left(\overline{L}\right)^* = \overline{L^*}}$$ "Non-trivial" is ...
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1answer
20 views

Reversed Language of a Non Regular Language

Is the following saying true or false? In any case why? Thanks!
2
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1answer
58 views

Context free grammar for language

I'm learning how to generate context-free grammar for a language. $L=\{{a}^i {b}^j {c}^k\, |\,i=j\lor j=k$ Here is how I tried ...
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2answers
38 views

What method should you follow to show what a DFA does?

I'm specifically looking for help analyzing the following DFA. What steps would one follow to show what language this particular DFA accepts? To me it seems quite random, and I can't figure out a ...
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2answers
54 views

Reduction between two languages and a common one

My question is as following : Let $A$ and $B$ be some languages, there exist a language $C$ such that $A\le C$ and $B\le C$, where "$\le$" means "reducible to", so $A\le C$ means there is a mapping ...
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35 views

Determining a language with a Turing Machine

How can I build a Turing Machine that determines the following language? $$L_{E - DFA} = \{\langle A \rangle | \text{$A$ is a $DFA$ and $L(A) = \varnothing$}\}$$ Thanks alot
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1answer
77 views

Why isn't this a sufficient proof?

So basically, we have a question that asks us to prove that given a particular Deterministic finite automaton (DFA), there is a symbol for which we can get to a state $q$ from a state $p$ given a ...
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3answers
25 views

If $s \geq 3$, $3$ divides $s$, and $t = s/3$, then $t+1 < s$.

I am using the pumping lemma to prove a language is not regular, and would like to assert what I have stated in the title of the question to complete my proof. That is, if $s \geq 3$, $3$ divides $s$, ...
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3answers
95 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 ...
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54 views

Is the PDA I drew correct?

Could you tell me if the language $L=\{a^{m}b^{n}:m \neq n,m>0,n>0 \} $ is accepted by the following pushdown automaton, where the alphabet of the stack is $\{a,z\}$ and $z$ is the initial ...