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

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Pushdown automaton design

I have to design a PDA that recognizes the language: $$L=\{w \mid \#(a,w) - 3\#(b,w) = 2\} $$ where $\#(a,w)$ means the number of letters $a$ in $w$ My idea is to count $a$'s and $b$'s. I have to ...
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Pumping lemma to prove that a language is not context free

We've got $L = 0^{x^{2}}$. So we let $w = 0^{p^{2}}$, and we know that we can split w into $w = u\cdot v\cdot w\cdot x\cdot y$ , according to the pumping lemma for CFGs. I'd like to know how to ...
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Push Down Automata that recognizes language

I'm struggling on how to use the stack for this push down automata problem. The problem is to design a PDA that recognizes the language: $$A = \{a^ib^{2i}|\,i>0\}$$ So, we will be pushing a's onto ...
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Designing a deterministic finite automata

How would I go about designing a deterministic finite automata to recognize the language L = {λ, ab, abab, ababab, . . . } consisting of strings that start with ‘a’, end with ‘b’, and alternate in ...
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59 views

equivalence class for language in Theory of automata

we say x,y is equivalent to language L, if for any $z \in L$ we have: $xz \in L \Longleftrightarrow yz \in L$. for $ L= (ab \cup aab)^* $, what is the equivalence class for L? my professor ...
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Input and output of a Turing machine

For some machine models of computation there is no question what their input and output is: it's just the contents of some specific "cells", e.g. on a "tape" isomorphic to $\mathbb{N}$. Consider for ...
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Use the power-set construction to find a deterministic automata

Given a nondeterministc automata N, how do you use the power-set construction to find a deterministic automata that recognizes L(N)? Here is my work so far: We can start in state 1, 2. If we get ...
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Soft question, Understanding NFAs and DFAs; Requirements for either.

I have a few quick questions about NFAs and DFAs. Is any automaton with epsilon transitions always a NFA? Is any automaton with two paths for the same symbol from a state always a NFA? Ex. Say state ...
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34 views

Is there a fast way to know whether a language is regular or not?

Or at least have an idea? Because I can't see whether a language is regular before I can disprove it by pumping lemma and it takes me like a hour to try to disprove.
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using pumping lemma to prove that a set is not regular

A={s11s|s $\epsilon$ {0}^*} so the strings 00011000 and 000001100000 are accepted of A but not 00100 or 001100000. Demon chooses k. ...
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231 views

How to convert this NFA to DFA?

What are the steps for converting this NFA to a DFA??
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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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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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170 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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29 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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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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139 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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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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375 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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Reversed Language of a Non Regular Language

Is the following saying true or false? In any case why? Thanks!
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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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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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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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107 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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Converting NFA to DFA

Im trying to convert a NFA to DFA. This is the NFA and this is the DFA to which i converted Is this right? Also when converting if i write a state as [q0,q1] is this same as [q1,q0] edit: ...
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Proving Equivalence of DFA and NFA

Im trying to learn Equivalence of DFA and NFA.The problem is that in the below explanation Q' is given as the power set of Q.But this statement seems to be contradictory to the previous statement ...
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25 views

How is Finite Automation Linked to Lexical Analyser

I understand that Finite Automaton is a Mathematical model of a system with discrete number of input and outputs. Also the system has finite number of states.My question is how is this linked with ...
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84 views

Designing PDAs to Accept Languages

I want to design PDAs to accept the following two languages: $L_1 = \{a^ib^jc^k \mid i=j \text{ or } j=k\}$ $L_2 = $ The set of all strings with twice as many $0$s as $1$s. I am especially ...
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Prove/Disprove: $vwvw=vvww$ iff $\{v\}^*\{w\}^*=\{vw\}^*$

Let $\Sigma$ be an alphabet and $v,w\in \Sigma^*$. I'm trying to prove that: $$vwvw=vvww\quad\text{iff}\quad\{v\}^*\{w\}^*=\{vw\}^*.$$ I tried to do it by induction, with no success. Any help will ...
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38 views

Grammar derivation

Given these grammar productions: $$\begin{align*} &S\to A1B\\ &A\to 0A\mid\lambda\\ &B\to 0B\mid 1B\mid\lambda \end{align*}$$ And given string $w = 01101$ If I wanted to make a) ...
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38 views

S-grammar for this regular expression

Given this regular expression: $r = a a^* b + b^* c b$ I think this is the simple grammar, but I was getting a little lost with the productions: $S \rightarrow S_1 | S_2$ $S_1 \rightarrow a A b$ ...
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Regular Language Operation

I need to show that the given regular language is closed under the following operation. For example: AllSuffixes(L) = {v : uv in L for some u in (0+1)* } I do not ...
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81 views

Can someone explain this automaton?

I have a question about constructing an automaton for given language: $$L = \{000, 010, 100, 110\}$$ Solution for this was given below. Can anyone explain why this automaton accepts the language? This ...
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349 views

Is it true that any infinite subset of a non-regular unary language is non-regular?

My question is very similar to this: Is there a subset of a non regular language that is regular My claim is that because the subset is infinite, Myhill Nerode says that the language is not regular. ...
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131 views

Pumping Lemma to show a language is not regular

Let $\Sigma = \{a, b\}$. Use the Pumping Lemma to show that $\mathcal L = \{ a^pab^q: p < q \}$ is not regular. Not sure how to apply PL here, if someone can give some direction.
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Right-linear grammar from regular expression

I made a right-linear grammar that from this regular expression: The alphabet is: $Σ = \{a, b, c\} $ Regular expression: $r = cc^*(ba)^*bb$ My solution, it seems a little too short like I'm ...
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30 views

Language made by a regular expression

I created a language from this regular expression but I'm not sure about it, especially where I wanted to use the $w$ to express a sequence of terminals. The expression: $r = a a ^{*} (b + bb + bbb) ...
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496 views

Finite automata for any even number of a's followed by any even number of b's

I'm new to formal languages. I'm stuck with the following question. Any help is appreciated. Find finite automata for $$L = \{a^i b^j \mid i, j\text{ are even, }j\ge0\}$$ Thank you
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To design a Finite State machine

Design a FSM for a binary number in which the input is valid if no. of 0's divisible by 5 and no. of 1's divisible by 3
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161 views

Is this language regular?

Given $m,n∈Z$, A is a finite alphabet set ,and $L=\{(a^m,a^n)\}^*$ is subset of $A^*\times A^*$ . Is this language regular ? For example, is $L=\{(a^3,a^7)\}^*$ regular ? Here L is not the set ...
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Prove $L=\{ 1^n| n\hspace{2mm}\text{is a prime number} \}$ is not regular.

Prove $L=\{ 1^n| n\hspace{2mm}\text{is a prime number} \}$ is not regular. It seems to use one Lemma: Pumping Lemma.
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Given a DFA $\mathcal{M} = (S, \Sigma, q_0, \delta, F)$, is there an algorithm that finds the pumping length of $L(\mathcal{M}$)?

This question has been bugging me for a while, and I'm curious what such an algorithm would look like, if it exists. My guess is that it does exist, but I'm not sure how it would look.
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Describe a Turing Machine that accepts the language of all non-negative decimal integers that are multiples of 3.

I have exam coming up and I need help with this: Describe a Turing Machine that accepts the language of all non-negative decimal integers that are multiples of 3 Thank you :)
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Automata: Proof

Here is the problem: Consider a NFA, M = (K, Σ, Δ, s, F) with (p, a, q) ∈ Δ. Prove that (pʹ, aw) ⊢∗ (qʹ, w) for any w ∈ Σ∗, q′ ∈ E(q) and p′ with p ∈ E(p′). Thanks in advance.
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How can I prove a DFA accepts a certain mininum number of states?

We know that if there are two languages, L1 and L2, if L1 and L2 are regular, the intersection of those two is also regular. Suppose we have two machines, M1 and M2, and using them, a new machine M3 ...
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Büchi Pushdown System Accepting Run

From the following definitions: Definition (Büchi Pushdown System) A Buchi pushdown system (BPDS) is a tuple BP = (Q,S,→,Qf) with (Q,S,→) a PDS (where S is the stack content) and Qf ⊆ Q a set of fi ...
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L is a context free language over {0, 1}, prove, disprove:

cont... L is a context free language over {0, 1}, prove, disprove: L1 is a CFL over {a, b}, where L1 is the language of all words from L, that 0 is converted to a and 1 is converted to bba. Thanks ...
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Prove min(L) = all words in L that they don't have any prefix of themselves in L

We define the minimal words language of $L, \min(L)$, to be the language of all words in $L$ that don't have any prefix in $L$. Assume $L$ is regular language. I need to prove by building an ...
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showing that a regular language is regular after taking a letter off or after adding letters

I'll be happy to recieve help with this one: Given the regular language $L$ defined over alphabet $\{a,b\}$, show that the following languages are also regular: $\{xy\mid xay\in L\}$ ...
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500 views

Proof Two Grammars Generate the Same Language

I have two right linear grammars and I need to prove they both generate the same language. What is the right way to do it? L1: $S \rightarrow 0A$ $S \rightarrow 1B$ $A \rightarrow 0A$ $A ...