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

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Decide whether DFA accepts all Strings

Wikipedia : Because DFAs can be reduced to a canonical form (minimal DFAs), there are also efficient algorithms to determine: whether a DFA accepts any strings. whether a DFA accepts all strings. ...
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1answer
44 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| ...
2
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1answer
56 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 ...
2
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1answer
81 views

(Transformation) Semigroups, the semigroup $\mathbf D_n$ and the wreath product

I have some trouble understanding the following proof, were I can't even figure out how some terms are defined. But first I state some definitions and preliminary lemmas. A transformation semigroup ...
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1answer
88 views

Reducing A$_\text{TM}$ to REGULAR$_\text{TM}$

We can solve A$_\text{TM}$ problem using REGULAR$_\text{TM}$. Assume $R$ is a Turing machine that decides REGULAR$_\text{TM}$. We construct $S$ to decide A$_\text{TM}$ as follows On input ...
2
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1answer
263 views

DFA Rejection State

I'm being asked to construct a DFA for the language over $\{0,1\}$ such that each string of five consecutive symbols contain at least two zeroes. In my construction, it seems to me that it would make ...
2
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1answer
49 views

Proving that $L=\{w\in \Sigma^*: |w|_a= 2^n +273$, $n\in \mathbb{N} \}$ is irregular. [duplicate]

I am trying to prove that $L=\{w\in \Sigma^*: |w|_a= 2^n +273$, $n\in \mathbb{N} \}$ is irregular, whereas: $\Sigma=\{a,b\}$. I tried to use the pumping lemma with no success. I have also tried to ...
2
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1answer
264 views

Turing Machine for comparing, copying, and operating

If one wants to design a Turing Machine for a function such as this: Where $x>0,y>0$ and are both integers represented in unary, so an example movement in this TM on the read-write head would ...
2
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1answer
285 views

DFA Transition Function Inductive Proof

Show for any state $q$, string $x$, and input symbol $a$, $\hat\delta(q, ax) = \hat\delta(\delta(q, a), x)$, where $\hat\delta$ is the transitive closure of $\delta$, which is the transition function ...
2
votes
2answers
668 views

Lambda productions in grammar

I tried removing the $\lambda$ productions from this grammar: $S \rightarrow a A b \mid B B a$ $A \rightarrow b b \mid \lambda$ $B \rightarrow A A \mid b A a $ It seems like you just take away the ...
2
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1answer
551 views

Does closure under the union and concatenation operations imply closure under the star operation?

Given any two languages $A$ and $B$, recall the following regular operations: Union: $A \cup B = \{x \mid x \in A \text{ or } x \in B\}$ Concatenation: $A \circ B = \{xy \mid x \in A \text{ ...
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2answers
55 views

Regular composition of non-regular language

I've got the following problem: Let's take language $L$. Is it posible that $L$ is not regular itself, but it's composition $L\cdot L$ becomes regular? I suspect that's correct, yet I ...
2
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3answers
985 views

Find push down automata and context free grammar

I have the following language: $$ L = \{a^nb^{2n+1} \mid n \ge 0\} $$ I must find the push down automaton and a context free grammar for the language. For the push down I have no idea how to ...
2
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1answer
463 views

Show that this language cannot be accepted by a deterministic push-down automaton [duplicate]

How do you show that there exists no DPDA that accepts $ L = \{0^n1^n \} \cup \{ 0^n1^{2n}\}$ ?
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1answer
67 views

Constructing PDA with either one state or two states

If $L$ is a context-free language and $\epsilon \notin L $, how do you show that there exists a PDA that accepts the language by final state such that it has not more than two states and makes no ...
2
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2answers
384 views

Regular Languages Algorithm?

I need help proving the following question: Let $L$ be any regular language on $\sum{a,b}$. Show that an algorithm exists for determining if L contains any strings of even length. So far, I know ...
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2answers
3k views

Pushdown automata for the language $\{ w001w^\text{R} : w = \{0,1\}^* \}$

I'm trying to make a PDA that accepts the language $\{ w001w^\text{R} : w = \{0,1\}^* \}$ by empty stack. (Here $w^\text{R}$ denotes the reverse of the string $w$.) Our stack symbol s $\#$. I've come ...
2
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2answers
1k views

Existence of NFA of a reverse of a language

Hi, I'm really stuck on how to prove the following: Given that $L$ is a language and $L'$ is a set such that $L' = \{w \mid w' \in L\}$ where $w$ is the reverse of $w'$, e.g., if $w = a a b$ then $w' ...
2
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2answers
52 views

Derivation of a property of a language

I am confused how this relation is derived for a language on alphabet V A,B The relation is $$ (A\cup B)^*=(A^*B^*)^* $$ I am confused how this is derived. Any pointers?
2
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1answer
76 views

Is $f$ is computable by a finite automaton, the dual of $f$ is thus computable also?

et $A$ be a finite alphabet. Let $A^*$ denote the language of all words in $A$, and $\epsilon$ the empty word. Let $\rho : A^* \to A^*$ denote the "reverting" map, that transforms $a_1a_2\ldots a_n$ ...
2
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1answer
110 views

Conditions for: $xy=zw$ and $yx=wz$

Let $x,y,z,w$ be finite strings. Find the necessary and sufficient conditions for the following two equations to hold simultaneously: $$xy=zw$$ and $$yx=wz$$ Automata Theory is new to me and i am ...
2
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1answer
139 views

Two elementary question on automaton and language

1.What is the definition for a semigroup(or monoid) recognizing a set of words(or language)?2.Are recognizable,rational and regular equivalent to each other with respect to a language? PS:The reason ...
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1answer
592 views

Construct PDA that accepts the language $L = \{w_1cw_2 : w_1, w_2 \in \{a, b\}^*, w_1 \neq w_2^R\}$

Problem Construct PDA that accepts the language $L = \{w_1cw_2 : w_1, w_2 \in \{a, b\}^*, w_1 \neq w_2^R\}$ For the language $wcw^R$, it's much easier because the stack is always empty after ...
2
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2answers
158 views

Is it possible to prove that $L$ is a regular language?

Let $L = \{a^{f(m)} | m \geq 1 \}$ where $f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ is monotonically increasing and complies that for all $n \in \mathbb{Z}^+$ there is $m \in \mathbb{Z}^+$ such that ...
2
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1answer
262 views

Generating all words in a language from CFG

I have a non-ambiguous context-free grammar. Is there some standard algorithm to create list of all the words in the language the CFG defines? This can be done with an abvious brute-force search by ...
2
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1answer
29 views

pumping lemma for CFL vs pumping lemma for regular languages

Is the pumping lemma for context free languages a generalization of the pumping lemma for regular languages (for instance if we set $u=\epsilon, v=\epsilon$ we can then relate $wx$ in the pumping ...
2
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1answer
19 views

Prove that regular languages are closed under operation

Operation $random$ is defined on two words with equal length in the following way: $\forall w_1,w_2 \in \Sigma^* s.t. |w_1|=|w_2|=n, w_1=a_1a_2...a_n, w_2=b_1b_2...b_n:$ $Random(w_1,w_2) = ...
2
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1answer
61 views

How can a Moore machine be converted into an equivalent Mealy machine and vice versa?

Moore machine is a finite-state machine whose output values are determined by its current state only. Mealy machine is a finite-state machine whose output values are determined both by its current ...
2
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1answer
267 views

Construct a Turing-Machine for Factorial(unary)

I am designing a turing machine which calculates the factorial of any given input for example, $3! = 3.2.1$, on tape it will look like this $Blank|1|1|1|Blank$ What I have done so far is that, I made ...
2
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1answer
56 views

Designing a Turing machine for $a^n$ $b^{2n}$ $a^n$

I am new to theory of automata, and i have a little knowledge of designing a turing machine, I am stuck in this question which is given to us as an assignment, $\{$$a^n$ $b^{2n}$ $a^n$ : ...
2
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1answer
61 views

$L=${$a^nb^nc^n : n \geq 0 $} CFG Recognizing

Suppose $L=${$a^nb^nc^n : n \geq 0 $} and I. $h(L), h(a)=a, h(b)=bb, h(c)=b$ II. $L^R$ III. $L^*$ IV. $h(L), h(a)=a, h(b)=bb, h(c)=a$ Why just I is a CFG and other is not? anyone can help me to ...
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1answer
132 views

Can you draw the e-NFA from the following definition?

I am trying to understand the solution, because I think I got it completely wrong. I wrote we could take the initial DFA and replace the normal transitions with epsilon transition except for all ...
2
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2answers
45 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 ...
2
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2answers
74 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 ...
2
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1answer
1k 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 ...
2
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1answer
75 views

Prove that RE is closed under reduction

Prove that the class RE is closed under reduction. Definitions: A language $ A \subseteq \Sigma^*$ is called reducible to $ B \subseteq \Gamma^*$ ( denoted by $A \leq B$) if there is a computable ...
2
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1answer
60 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 ...
2
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1answer
76 views

Formal proving of languages accepted by a finite automata.

Suppose $L_1 \cup L_2,L_1 \subseteq E^* $ are languages accepted by finite automata and $L_1\cap L_2 =\emptyset $. We need to prove that $L_2 $ is also accepted by a finite automaton. So I've started ...
2
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1answer
415 views

Verification: DFA/NFA that accepts all strings over $\{0,1\}$ with exactly one block of $00$

I am trying to design a DFA or NFA that accepts all strings over $\Sigma = \{0,1\}$ in which the block $00$ appears only once. Here is what I've tried. Can you verify that this accepts all string ...
2
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1answer
72 views

Is there an (explicit?) bijection from the set of all automatons to the set of all regular expressions that conserves the recongnised language

Let $\Sigma$ be an alphabet, $R$ be the set of regular expressions on $\Sigma$ (that is, trees with leave's values in $\left\{\varepsilon\right\}\cup \Sigma$ and three types of interior nodes, one ...
2
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1answer
133 views

DFA worst case states

Suppose an NFA which accepts language of the form L(N) = {w| w has 1 in n$^t$$^h$ from last symbol.} Then the corresponding DFA would have 2$^n$ states(worst case of subset construction). If we are to ...
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1answer
709 views

How to construct a grammar $G$ such that $L(G) = \{ a^nb^m|n \neq 2m,m,n \ge 0\} $?

Construct a grammar $G$ such that $$L(G) = \{ a^nb^m|n \neq 2m,m,n \ge 0\}$$ My attempt: I first constructed a grammar for the langugage $L(G_1) = \{ a^nb^m|n = 2m,m,n \ge = 0\}$, $G_1 = (\{ S\}, ...
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1answer
65 views

Is regularity is preserved under reversal?

When talking about languages and regular languages. Can I say that reversal preserved regularity since if the language L is regular, we can generate it by right linear grammar. Therefore, the ...
2
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1answer
155 views

Is the set of codes of Deterministic Finite-State Automata a regular language?

Let $\Sigma$ be a given alphabet. Is there a way to code up Deterministic Finite state Automata (DFA) over $\Sigma$ as strings of $\Sigma$ in such a way that the corresponding subset of $\Sigma^*$ is ...
2
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1answer
238 views

Constructing finite state automata corresponding to regular expressions. Are my solutions correct?

I have drawn my answers in paint, are they correct? (4c) For the alphabet {0, 1} construct finite state automata corresponding to each of the following regular expressions: (i) 0 My Answer 4ci (ii) ...
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1answer
36 views

Why is this language not regular?

I am studying Automata using the Coursera course created by Jeff Ullman. On slide 36 of this presentation: http://spark-public.s3.amazonaws.com/automata/slides/3_fa2.pdf it says that the language is ...
2
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1answer
38 views

Is the Champernowne constant an automatic number?

The Champernowne constant in base $b \geq 2$ is obtained by concatenating the $b$-ary expansion of every integer. For example, in base $10$ this is $$ 0.123456789101112131415\dotsc $$ Question: Is the ...
2
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1answer
43 views

Designing a Turing Machine - low level transitions

I couldn't figure out how to proceed with this question. Preparing for the finals, can someone explain how to do this step by step? Design a TM, write low level transitions for $\{a^i b^j :i ≤ j ≤ ...
2
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1answer
18 views

Finding regular expressions for which a given Turing machine halts and accepts, halts and rejects, and diverges.

Consider the Turing machine M = (Q,Σ,Γ,δ,q,F) F = {t} Q = {q,r,s,t,v,x} Σ = {a,b,c} Γ = {B,a,b,c} δ = [q,a,q,a,R] [q,b,q,b,R] [q,c,v,b,R] [q,B,r,B,L] [r,a,s,B,L] [r,b,s,B,L] [r,c,s,B,L] ...
2
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0answers
59 views

How to show $L = \{a^n b^m c^p: n\neq m\} \cup \{a^n b^m c^p: m\neq p\}$ is ambiguous?

I'm recently familiar with this site and prefer to ask a very hard problem :) How can we prove that the following language is inherently ambiguous? $$L = \{a^n b^m c^p: n\neq m\} \cup \{a^n b^m ...