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

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For $\sum = \{ 0,1 \}$, $A$ has strings which contain a $1$ in their middle third, and a $B$ which contain two $1$'s in their middle third.

Language $A$ can also be represented as, $$A = \{ uvw \mid u,w \in \sum^*\text{ and, }v \in \sum^* 1 \sum^*\text{ and, }|u| = |w| \ge |v| \}$$ Language $B$ can also be represented as, $$B = \{ uvw ...
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Help with proving a language is regular (Sipser problem 1.49a)

I am working through Sipser's Introduction to the theory of computation on my own, so I don't have access to a teacher. Hopefully you can help me! Giving hints is highly appreciated. This question is ...
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Divisibility problem using DFA

Original problem: Create a DFA for every positive integer $k$, so that when DFA takes a binary string (reading from most significant bit), decides whether the number is divisible by $k$. A DFA for a ...
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What is sense of $In(\phi)$ in Rabin's theory of SO-decidability?

In M.O. Rabin's article Decidability of second-order theories and automata on infinite trees in section dedicated to automata on infinite trees there is definition of $In(\phi)$ function: For a ...
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Pumping Lemma for Context Free Languages: Is this language CFL?

I am learning for the first time the Pumping Lemma for CFL, and I thought I understood how it works until I came across this example: "Show that $L = \{a^m b^m c^n \mid m \leq n\}$ is not a CFL." My ...
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25 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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How to prove that a language `L` is not a regular language?

Given the following question: Prove that the following language is not a regular language: A language L in alphabet $\Sigma = \{a, b\}$ where every word ...
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State diagram of DFA

I'm trying to understand how to create state diagram of DFA. I found following example. On the first diagram I dont understand why we need fourth state when third state is final and there is no ...
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how to create transition system/automata for modulo 4

I don't know how to think when to build a transition system/automata to calculate modulo 4 a of binary numbers. I know that the last two binary digits gives the rest but I need to go through hole ...
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593 views

Create DFA that accept language where number of 0's is even and after every 1 goes 0

Alphabet ${} = \{0,1\}$. Language $L = \{ w \in \{0,1\}^* \mid \text{ number of $0$'s in $w$ is even and after every $1$ goes $0$} \}$. I'm trying to create DFA that accepts language $L$. ...
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Size of automata or regular expressions avoiding cross patterns

Let $\Sigma$ be an alphabet of finite size $k$, and $n$ some integer. I am interested in the language of words of size $n$ that do not contain $abab$ as a subword, for any pair $(a,b) \in \Sigma$ (I ...
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Is it possible to build an Pushdown Automata for an Ambiguous Context-Free Grammar?

Say I have the following grammar: $$S \to \epsilon \mid [S] \mid (S) \mid SS$$ This grammar is ambiguous as both the following parse trees yield the empty string $$S \to \epsilon$$ $$S \to SS \to ...
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Nondeterministic finite automaton understanding problem

It is probably a silly question but I have problem understanding it. Let's say I have to design a nondeterministic finite automaton that accepts the language consisting of words containing a string of ...
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48 views

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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36 views

Show a Turing Machine accepts all DFAs

You have a Turing Machine $T = \{ \{ A \} \mid A$ is a DFA and $L(A) = \Sigma^* \}$ i.e. the DFA that accepts all languages. Show it's decidable. Can you use the complement of this, the DFA that ...
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49 views

Deterministic finite automaton algorithm

I am having a little trouble understanding this question. For a DFA M = (Q, Σ, δ, q0, F), we say that a state q ∈ Q is reachable if there exists some string w ∈ Σ∗ such that q = δ∗(q0, w). Give an ...
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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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CFG and PDA for w1#w2

Looking for a Context Free Grammar and Push Down Automata to describe a language made of two words, separated by a #, where the first words is not equal to the second word. For this example, we can ...
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Is this correct way to prove that { $uv$ $\mid$ $u$ and $v$ are strings over $(0,1)$ & $|u| = |v|$ } language is not regular?

I proved this using Pumping Lemma in the following way, Taking $p$ as pumping length then let us take a string $S$ where, $|u| = |v| = p$ Now according to pumping lemma $S$ can be split into three ...
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How to prove that $ \{ 0^n 1^{5n} : n \ge 10000 \} $ is not a regular language?

I proved that $$ \{0^n 1^{5n} : n \ge 0\} $$ is not a regular language using Pumping Lemma by following way. Solve by contradiction that $ L = \{ 0^n 1^{5n} : n >= 0 \}$ is regular language. ...
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Converting the NFA produced from the language $a^nb^n : n\geq 0$ to a DFA to show its regular? Leading to question about pumping lemma.

I am reading about the pumping lemma, and having a hard time understanding it. I noticed that it is used to prove a language is not regular by contradiction. So you must first prove that a language in ...
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37 views

Construct finite Automata

Im trying to construct finite automata in the form of diagrams accepting certain languages. One is in all parts the alphabet is {a, b}. Construct FA {w| w has neither aa nor bb as a subword} I ...
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Building PDA for a Language

I won't deny that this is not my homework question, but I've been thinking for a couple of hours and still have no understand $L = \{w | w ∈ \{0, 1\}^*\}$, w is a list of unary integers separated by ...
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abstract machine, what language does M accecpt?

What language does M accept? 1: {a}3 ∪ {b}3 ∪ {λ} 2: {a}3 ∪ {b}3 3: {a, b}3 ∪ {λ} 4: {a, b}3 ∪ {λ} I'm not completely sure just yet which one would work. I would appreciate it if ...
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55 views

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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60 views

Constructing a NFA from a right linear grammar, is this correct?

Given the right linear grammar G S -> abA | bbB | a A -> bB | aA | b B -> baB | aaaA | [Epsilon/Terminates] Is the NFA in the image below the proper ...
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71 views

Difference between a*b and a*+b? Does the “+” denote Kleene plus or “or”?

Me and a friend are study for a quiz and are trying to determine the difference between the two NFA's produce by the regular exressions a*b and ...
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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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Deterministic finite automata [closed]

For this question about Deterministic finite automata: Is this answer: bbbb, bbba, bbab, bbaa, b, a correct?
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154 views

Challenge on Some Language and PDA

Suppose We have Some language as follows: $L_1=\{w^* | w=x \text{ and } x \in \Sigma^*\}$ $L_2=\{ww^R ww^R | w \in ( \Sigma + \Sigma)^*\}$ $L_3=\{w | w=xy, x,y \in \Sigma^*, y \text{ is a ...
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Checking correctness of finite state automata designed

How to check correctness of finite state automata we have designed for a regular expression with the help of any computer program or prolog?
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Finite State Automata for (0+11+01)*(0)*(01)*

This is my homework problem and I am struggling hard for it. Can anyone please help me out? I need to find out finite state automata for (0+11+01) * (0) * (01)*. Also, if anyone can please tell me the ...
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Proving non-regularity of a language

How can I prove $L = (01^n2^n | n\geq 0)$ is not regular? Would it be sufficient to say that $01^p2^p$ is in $L$ and by pumping lemma, $01^p2^p$ can be written as $xyz$ such that $|y|>0, ...
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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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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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how to draw this DFA?

{w belongs to all string patterns as a^i b^j a^k | i+j=even and j+k =odd} draw a DFA and find its regular language. please note here, i have put comma in between the format of aba string just for ...
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contradiction of pumping lemma on even palidrome language's completion

I have the language of all the words that are not even length palindromes, or more formally: $L= \{ w: \forall u\in\Sigma^*, w\ne uu^r \}$. And I need to prove that the Pumping Lemma for regular ...
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61 views

How can a DFA corresponding to an NFA have a transition that the original NFA does not?

First sorry for the poor pictures, but I think they are ok enough to get the point across. I would like to see the steps involved to convert this NFA to a DFA using the method explained in this ...
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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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Regular Expression Similarity Check

I was solving Formal Language and Automata Theory for a competitive exam, whence I came upon this following question: The regular expression 0*(10*)* denotes same set as: 0(0+10)* (0+1)10(0+1) ...
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Is a DFA a subset of NFA? Why?

After reading this previous question Difference between NFA and DFA, it's clearer to me their relationship/differences. Can one say that a DFA is a subset of a NFA where both recognize a/the same ...
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Number of states of a finite automaton recognizing all words beginning with some fixed string $x$

For a string $x \in \{a,b\}^\ast$ with $|x| = n$, how many states are required for an FA accepting the language of all strings in $\{a,b\}$ that begin with $x$? For each of these states, describe the ...
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Find a state machine or a regex that accept the following language description

The alphabet of the language L is {a, b} and there has to be an even number of both a's and b's but no other restrictions apply. I've been at this for over an hour, drawing state machines that lead ...
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FA that accepts only those words that have an even number of substrings cdc?

I take this question as it could accept cdccdc,cdccdccdccdc,dddcdccdc,etc. I feel that I coveed all the even loops of cdccdc but I am stuck trying to figure out how to add in c* and d* in the mix of ...
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Turing Machine Problem

We know, A Turing machine is a hypothetical device that manipulates symbols on a strip of tape according to a table of rules I Draw a TM for input $x=(0+1)^*$ i want to implement ...
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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 ...
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Language of a grammar vs regular expression vs nfa

I read some note about Automaton Course. i see this note, that following all is the same. but i think the L(g) is not equal to NFA and regular expression. anyone could help me with defining the ...
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context free grammar that describes even numbers

I am learning about context-free grammars and as a toy example I wanted to design one that describes binary digits ending with 0. My attempt : S -> 1S | 0S | e0 - where e is the empty string. Is this ...