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

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Nondeterministic Finite Automata to Deterministic Finite Automata?

I am unfamiliar with the general process of converting NFA to DFA. I have general understanding of the theory, but I don't have the method established. Please help explain the process required to ...
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43 views

The Mathematics of Finite State Automata

I am a final year undergraduate mathematics student preparing to undertake my BSc-HONS project, provisionally titled for the time being, "Finite State Automata and Regular Languages". Having had a ...
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Question about deterministic finite automaton (DFA) [closed]

For $n \in \mathbb N$, an "$n-$DFA" is an automaton with exactly $n$ accepting states. Let $\Sigma=\{0,1\}$. Prove that the language $\mathcal L=\{0,00,0000\}$ cannot be accepted by any $2-$DFA.
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Show a language is regular with Myhill-Nerode Theorem

I understand how to show a language is not regular using Myhill-Nerode Theorem (proof by contradiction), but how do you show the language is regular? Take language $0^*1^*$ for example. I know this ...
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60 views

What are annihilators?

I was recently listening to Automata lecture, there it was told told that an empty set is an Annihilator for concatenation just like $0$ is for multiplication. What do we mean by this statement?
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What are closure properties of regular languages?

In class we've been talking about DFA's and NFA's and being closed under ____. The homework problems say to "use closure properties of regular languages to show that a regular languages are closed ...
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Set of all factors of regular language regular?

If $L$ is a regular language (i.e. acceptable by a finite automata), is the the set of its factors (infixes) also regular?
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Is the language $\{yxzx^Ry^R \mid x,y,z \text{ belongs to } \{0,1\}^+ \} $ regular?

This is a question from Iran's national grad school entrance exam. In the answers key, the answer was that the following language is regular but I doubt it is true, I proved using pumping lemma that ...
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126 views

Concatenation of Finite Languages and Regular Languages

I know that the following statements regarding Concatenation are false. However, I'm having difficulty explaining why they are false with simple counter-examples. I'm able to find simple ...
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60 views

A Regular Expression for all strings that…

I got a problem I have to solve, the problem says that given an alphabet $\Sigma = \{a, b, c\}$ I have to build a regular expression that describes the string with: An even number of a's. A 4k + 1 ...
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3answers
26 views

Name for a state of an automaton that can't be left

In an automaton, we might have a state that once reached cannot be left. It is for example for Ø in Is there a common/official name for such a state ?
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regular language proof, a language partial to a regular one

So, I've been trying to solve a question I got and I think I'm correct but I'm not positive. Is the language {w| www belongs to L' and L' is regular} regular? I couldnt find any way to prove it isnt ...
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35 views

Turing machine that goes left on first symbol

I have a turing machine with transitions given by the following table I'm inputting the string aaaa. So if I look at the first symbol "a" in state A, it says to replace it with an X, go into state ...
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33 views

Why is $\left\{a\right\}^*\cap L(A)=\emptyset$? where $\delta(q,a)=q$

I am trying to solve this particular problem from Automata Theory by Ullman, Hopcroft, it is as shown below : Let $A$ be a $DFA$ and $a$ be a particular input symbol of $A$, such that for all states ...
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34 views

Stack automata with nonequality

I encontered the following problem that I'm unable to solve. The goal is to construct a stack automata, that accepts language $$ L = \left\{u\mathrm{\underline{c}}v \mid u,v \in ...
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81 views

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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876 views

Finite state machine

I am doing discrete math, and we are studying Finite State Machines. But i am a little confuse on how to do this. Here is a question, Write a regular expression for the language, and define a finite ...
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52 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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65 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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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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107 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 ...
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423 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 ...
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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 ...
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643 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 ...
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388 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 ...
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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 ...
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685 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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60 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 ...
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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 ...
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645 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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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 ...
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538 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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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 ...
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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' ...
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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?
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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 $a_1a_2\ldots a_n$ ...
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116 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 ...
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146 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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Nondeterministic PDA to Deterministic PDA

Are there any resources on how to convert a non-deterministic PDA to a deterministic one, if a deterministic PDA actually exists? Or is there a step by step way on how to do this, kind of like going ...
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774 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 ...
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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 ...
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1answer
280 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 ...
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27 views

Let $Σ=\{a,b,c\}$. Which of the following statements is true?

$1)$ For any $A\subseteq\Sigma^*$, if $A$ is regular, then so is $\{x∣ xx\in A\}$. $2)$ For any $A\subseteq\Sigma^*$, if $A$ is context-free, then so is $\{x∣xx\in A\}$ According to me the ...
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66 views

Let M range over turing machine descriptions .Consider the set REG={M|L(M) is a regular set} which of the statements are true?

The complement of REG is Co-REG REG is recursively enumerable but Co-REG is not REG is not recursively enumerable but Co-REG is Both are recursively enumerable 4.None of them are recursively ...
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Finite automata as dynamical systems

In abstract (deterministic finite) automata theory the set of states of an automaton is an arbitrary set Q, and the transistion function is a specific set δ ⊆ Q × Σ × Q (with alphabet Σ, i.e. another ...
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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 ...
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30 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) = ...
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132 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 ...
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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$ $b^{2n}$ $a^n$ : ...
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$L=${$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 ...