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

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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 ...
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Theory of Automata concepts

I just started taking Theory of Automata and I'm having a hard time understanding some of the concepts. It's been only a week and the following questions are my homework. I'm not asking you to do my ...
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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 ...
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37 views

Languages that are not comparable in $R$

I want to know if there are $2$ languages $A,B\in{R}$ such that there's no reduction between them. Namely, $2$ languages $A$ and $B$ $\in$ $R$ such that $A\not\le B$ and $B\not\le A$ Thanks a lot!
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Determining whether a language belongs to R or RE (Turing Machines)

Does the language L belong to R, RE or neither? In each case, why? Thanks a lot!
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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 ) ...
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Extended PDA vs TM

We studied in class that PDA is less powerful than TM. My question is: Extended PDA : for every $\alpha,\beta \in \Gamma \cup \{\epsilon\}$, $\sigma \in \Sigma \cup \{\epsilon\}$, $q,r \in Q$, $w ...
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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 ...
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Showing that 2 languages are context free

I have these 2 languages: $$L_1 = \left\{a^ib^jc^k: k\ge i+j\right\}\\ L_2 = \left\{w_1cw_2 : w_1,w_2\in\{a,b\}^\ast\land |w_1|_a = |w_2|_a\right\}$$ How can I determine that they are context free ...
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78 views

Push down automata for context free grammar

I'm having trouble finding the PDA for this language $L = \{x^{3i} y^j z^k\ |\ i \ge 0 \land k \gt 2j \gt 0\}$ The ...
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Language requiring a DFA with a certain number of states to implement

For any function $f\colon\{0,1\}^n\to\{0,1\}$, define a language $S_f = \{(b_1,b_2,\ldots ,b_n)\in\{0,1\}^n : f(b_1,b_2,\ldots ,b_n) = 1\}$. So all words in the langugage has same length $n$. I have ...
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57 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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124 views

Prove that a PDA with accept states accepts all context-free languages

Or in other words that $\forall L: L \in DCFL => L \in CFL$. First of all, does this statement even require a proof? My idea was to let L be an arbitrary language, such that $L \in DCFL$, this ...
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46 views

build finite automaton for language minimize states

I want to build a finite automaton that accepts $a^nb^n, n \gt 0, m \ge 0$. I can't do it unless the FA has two final states, i.e.: $delta(q0, a) = q1 delta(q1, a) = q1 delta(q1, b) = q2 delta(q2, ...
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170 views

DFA for Boolean Formula

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

The shortest word in context free language

Let $G=(\Sigma,N,R,S)$ be a context-free grammar. For every production rule A --> w, we say that its length is $r$ if $|w|=r$. In addition $n = |N|$, and $k =$ the maximal length of a production rule ...
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31 views

Help Specify possible definitions for this Boolean Function

My math is rusty, but I need some guidance here. Problem I wish to design a stochastic, boolean procedure $f(state)$, that picks a winner, $f(state_{win})\to 1$ or loser, $f(state_{loss})\to 0$. I ...
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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. ...
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Oracle Turing machine - $E_{\text{TM}}$ and $PCP$.

$$E_{\text{TM}}=\{\langle M\rangle|M\text{ is a TM and $L(M)=\emptyset$}\}.$$ $E_{\text{TM}}$ is undecidable $$PCP=\{\langle P\rangle|P\text{ is an instance of the Post Correspondence Problem with a ...
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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 $L' = \{a^3 ...
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Reducing A$_\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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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 ...
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Proving $L=\{0^n \mid \text{n is a perfect square}\}$ is not a Regular Language without the Pumping Lemma

Is this a valid way of going about proving the proposition? Let $L = \{0^n \mid \text{n is a perfect square}\}$. The regular languages are closed under concatenation. So if $x \in L, y \in L$, then ...
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94 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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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 ...
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57 views

Grammar outside the Chomsky Hierarchy

This grammar describes a language that may fall outside the Chomsky Hierarchy (CH): \begin{array}{l} S \to abAbba \\ A \to abA \mid bbaB \\ B \to aab \\ \lambda \to Aab \mid aB \\ \end{array} Going ...
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41 views

Consider this grammar

Consider this grammar: \begin{array}{l} S \to aabBba \mid aAb \mid aab \\ bBb \to bCa \mid aaa\\ aA \to aC \mid bba\\ C \to aab \mid Cb \end{array} This is clearly context-sensitive (CS). It's not ...
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Context-sensitive grammar for this language

In order to write a context-sensitive grammar for: $L = \{ a^{n} b^{n} c^{n} d^{n} : n \ge 1 \}$ One possible set of productions is: $S \rightarrow aBCd | abcd $ $aB \rightarrow aaBb | ab | ...
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Assign Integer to Each Turing Machine

I have the following problem: suppose that we have an infinite set of symbols, $A = \{a_1, a_2, ...\}$ from which all Turing Machine input alphabets are chosen. Show how we could assign an integer to ...
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pumping lemma for regular languages and DFA with k states

The Formal statement of the pumping lemma for regular languages: Let $L$ be a regular language. Then there exists an integer $p \ge 1$ depending only on $L$ such that every string $w$ in $L$ of ...
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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^* $, ...
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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 ...
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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 ...
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CERNY CONJECTURE ,reset sequence

given a reset/synchronizing sequence $w\in \Sigma^*$ ( w is a word in the input alphabet of the DFA which sends any state of the DFA to one and the same state), prove for an automata with k states ...
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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 ...
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Turing decidable/undecidable

let $X = \{\langle M \rangle\ |\ M\text{ is a finite state machine and }L(M) = \emptyset\}$ where $\langle M \rangle$ is an encoding of the machine $M$. can you prove whether $X$ is Turing ...
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halting problem

Prove that it is undecidable for the halting problem of a deterministic Turing machine which accepts nonrecursive language or in-other-words: let's say we have a deterministic Turing machine which ...
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What is the language generated by G and how to draw the finite state automaton that recognizes this language?

G = (V, T, S, P) V = (0, 1, A, B, S) T = {0, 1} S is start S -> 0A S -> 1A A -> 0B B -> 1A B -> 1 For the drawing, I am confused about the last ...
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1answer
47 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 ...
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Blanks in the Tape of a Turing Machine

I used to have a lot of trouble with Turing Machines, primarily because I thought that in-between input symbols on the tape, one could have an arbitrary number of blanks, so every input would need to ...
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Build regular expression from language

I have the following language: L = {w $\in$ {a,b}* | aa is not part of w}. I have to construct a regular grammar from this language and I thought about first finding the regular expression from the ...
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One counter automata

A one-counter automaton $M = (Q, S, \Gamma, t, s, A)$ is a pushdown automaton where the stack alphabet $\Gamma$ contains just two symbols $\#$ and $g$. The symbol $\#$ is initially written on the ...
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Turing Machines

Suppose that $\Sigma$ is a finite set and that $L_1$, $L_2$ and $L_3$ are Turing acceptable subsets of $\Sigma^*$ that satisfy the following properties: $L_1 \cup L_2 \cup L_3 = \Sigma^*$; $L_1 \cap ...
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Determine whether $L=\{w:|w|_a=2^n+273\text{ for }n\in \mathbb{N}\}$ is regular.

Given the alphabet $\Sigma=\{a, b\}$ and for the next Language $L=\{w:|w|_a=2^n+273\text{ for }n\in \mathbb{N}\}$ determine whether the language is regular. Firstly, I think this language is regular. ...
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129 views

unrestricted grammar and Turing machines

Let $G$ be an unrestricted grammar. Then the problem of determining whether or not $L(G) = ∅$ is undecidable. Let $M_1$ and $M_2$ be two arbitrary Turing machines. Show that the problem $L(M_1) ⊆ ...
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100 views

reverse automata mininum states

There is a formal proof for the following sentence? For every $k$ there is a DFA (deterministic finite automaton) $M$ with $k+2$ states such that every automaton that accepts the language $L(M)^R$ ...
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95 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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84 views

DFA with $k$ states, words of length $k$

Is this statement true? If we have a DFA with $k$ states, and if $L(M) = L$ is infinite, then there is a word of length at least $k$ and at most $2k-1$. Isn't this a trivial answer? Take the ...
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Is this a context-free language?

Let $L = L_1^* \circ L_2^*$ where $L_1 = \{1^n 0^m 1^n : n,m \in \Bbb N\}$ and $L_2 = \{0^m 1^{2m} : m \in N\}$. Is the language $L$ a context free language? I think I can write automata for $L_1$ ...
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$DFA/NFA$ for $L(OPPOSITE)=\{uv:vu\in L\}$

I'm trying to prove that: $L(OPPOSITE)=\{uv:vu\in L\} \in L_{FA}$ given that: $L \in L_{FA}$ . I'm trying to construct a finite automata that accepts $L(OPPOSITE)$ in order to prove it but I got ...