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

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2
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1answer
93 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
votes
2answers
41 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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1answer
318 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
votes
1answer
61 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
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 ...
2
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1answer
50 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
55 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
287 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
118 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 ...
2
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1answer
504 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\}, ...
2
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1answer
62 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
142 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
votes
1answer
214 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) ...
2
votes
1answer
31 views

Pushdown Automata and Challenge in Grammar

I read one old-midterm exam on Automata. consider: the language that accepted by above pushdown automata is not generated by which of the following grammar? 1) S->aBaa|a$\epsilon$ ...
2
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1answer
40 views

A CFG Grammar for One Language

Suppose : $w_1,w_2 \in \{a,b\}^∗$ and $ L=\{w_1w_2 \mid w_1,w_2 \in \{a,b\}^* \land n_a(w_1)=n_b(w_2)\}$ $n_a$ is number of $a$'s and $n_b$ is number of $b$'s. This is a Entrance Exam question. I ...
2
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0answers
65 views

Determine the language that corresponds to the following automata.

I want to determine that language that corresponds to the following automata Note: $q_{6}$ have arrow to $a$ to himself. I started with the minimal words: $aaabb$ $aaba$ $aaaba$ $bababa$ the ...
2
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0answers
52 views

This proof in my textbook involving the pumping lemma appears incorrect - is it?

It states Let $B$ be the language $\{0^n1^n2^n | n \geq 0\}$. We use the pumping lemma to prove that $B$ is not regular. The proof is by contradiction. Assume to the contrary that $B$ is regular. ...
2
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1answer
58 views

Context free grammar for language

I'm learning how to generate context-free grammar for a language. $L=\{{a}^i {b}^j {c}^k\, |\,i=j\lor j=k$ Here is how I tried ...
2
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0answers
41 views

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 ...
2
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0answers
62 views

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 ...
2
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1answer
60 views

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 ...
2
votes
1answer
114 views

Converting CFG to CNF

I have the following problem of converting CFG to CNF: $$ \begin{aligned} S \Rightarrow\,& bA \mid aB\\ A \Rightarrow\,& bAA \mid as \mid a\\ B \Rightarrow\,& BB\mid bs\mid b ...
2
votes
1answer
23 views

Properties of “fail-safe” languages

I'm wondering if anyone has any experience with the concept of a "fail-safe" language. And, if so, where could I find more information on the subject. To explain what a "fail-safe" language is: Let ...
2
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0answers
26 views

Is there a 2D 3-colorstate mobile automaton that grows like $ln^{0,5}(t)$?

Define an integer function $f(t)$ for an integer $t>25$ such that $|f(f(t)) - ln(t)| < \sqrt {ln(t)}+2$. Define $L(X(t))$ as the number of nonwhite states at iteration $t$ of mobile automaton ...
2
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0answers
48 views

Expressiveness of finite memory programs

Assume we have a simple programming language with while, if, := (assignment), ...
2
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0answers
49 views

Quality of Reduction of finite automata

I am looking for an example, which corresponds to what I've learned in my Applied Automata Theory Class: Given a NFA $\mathcal{A}$, a $\approx _\mathcal{A}$ quotient automaton can be bigger then a ...
2
votes
2answers
120 views

An NFA with $\Sigma = \{1\}$ with $x^2$ accepting runs on strings $1^x$ for all $x \geq 0$ - how to construct?

One of my homework assignments requires us to construct an NFA over the alphabet $\{1\}$ which has exactly $x^2 + 3$ accepting runs over the input string 1^x for all $x \in \mathbb{N}$. Now, the +3 ...
2
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1answer
419 views

Understanding and using the transfer-matrix-method

Let $G = (V,E,\Phi)$ be a weighted directed graph and $\mathcal{W}' : E \rightarrow \mathbb{C}$ the weighting. Let additionally $m = \# V$, $E_m$ the $m \times m$ identity matrix. Let $v,w \in ...
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2answers
256 views

Converting from NFA to a regular expression.

This is a NFA, I have been working to covert it to a regular expression. After I'am done, I arrive at an expression as follows $$ \left(((a\cup b)a^*b) (ba^*b)^*a\right)^* \left(((a\cup b)a^*b) ...
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4answers
151 views

Prove this language is not regular [closed]

How do I prove that this language = {1^k | k is a perfect square} is not regular by showing that no DFA can accept the language?
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2answers
84 views

If $L_2L_1$ is accepted by a DFA, is $L_1$ too?

Given that $L_2, L_2L_1$ are accepted by a DFA, is $L_1$ accepted by a DFA too? What is the general approach to such question? What if instead of $\cdot$ we are given that $L_2 \cup L_1$ is ...
1
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2answers
107 views

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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4answers
138 views

Non-Deterministic Turing Machine Algorithm

I'm having trouble with this question: Write a simple program/algorithm for a nondeterministic Turing machine that accepts the language: $$ L = \left\{\left. xw w^R y \right| x,y,w \in \{a,b\}^+, ...
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2answers
64 views

Why this lemma is true?

Let $\Sigma$ be an alphabet of size $|\Sigma|=k$. Let $w\in\Sigma^*$ be a word over $\Sigma$. If $|w| > 2^k$, then $w$ contains an infix $y$ with $|y|\ge 2$, such that every letter occurring in y ...
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2answers
265 views

Is this DFA correct

I'm supposed to construct a DFA which accepts { w | w is a word except 'aa' and 'aaa' } Is this the correct solution? The thick line state is supposed to be the end state. EDIT Sry, somehow ...
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2answers
43 views

Convert NFA with only accepting states to regular expression?

Suppose you wanted to find a Regular Expression that defines the language accepted by the folowing Finite State Automaton. Where S1 and S2 are both accepting states. How would I go on doing this?
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2answers
309 views

Constructing a finite automata from a subset of its language

I am attempting to solve the following problem: Let $M=(Q,\Sigma,\delta,q_0,F)$ be a deterministic finite automata which accepts $L(M)$, and let $E$ be the subset of $L(M)$ consisting of all words of ...
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2answers
25 views

Regular expression translation.

Given a set {1,2,...9} how can I construct a regular expression starts with a 3 has no 8's and has even number of 6's? Here's what I tried: $$$$ Define a new set no8 = {1,2,3,4,5,6,7,9} ...
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2answers
65 views

The halting problem for tapes that are or are not completely blank

Is it possible to construct a Turing machine that halts only if the tape is not completely blank? Also, is it possible to construct one to halt if the tape is completely blank? Intuitively, I think ...
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3answers
95 views

NFA for $(ab|a)^{*}$ using only 2 states

In Introduction to the Theory of Computation by Michael Sipser, there's an example which shows how to convert the regular expression $ (ab|a)^{*}$ into an NFA. The "standard" method results in 8 ...
1
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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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2answers
25 views

Construct a deterministic finite automation

The question asks to: construct a DFA which accepts exactly $\frac{n(n-1)(n-2)}{6} + \frac{n(n-1)}{2}+1$ many members of $\{0, 1\}^n$ for every n. I have no idea where to start to constructing the ...
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2answers
1k views

Concatenation of 2 finite Automata

I have some problems understanding the algorithm of concatenation of two NFAs. For example: How to concatenate A1 and A2? A1: ...
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2answers
310 views

If $L$ is regular, prove that $\sqrt{L}=\left\{ w : ww\in L\right\}$ is regular

Let $L$ be a regular language. Prove that $\sqrt{L}:=\left\{ w : ww\in L\right\}$ is also a regular language. I suppose I need to modify state machine for $L$ to accept $\sqrt{L}$, but I've been ...
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2answers
166 views

Prove that any finite set of words is regular. How many states is sufficient for a single word $w_1…w_m$?

DFA Prove that any finite set of words is regular. How many states is sufficient for a single word $w_1...w_m$? For part 2, wouldn't it require M states if the word length is M?
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1answer
98 views

Is the language $\{i|f(i)=1\}$ recursive, function $f$ is described further inside. [duplicate]

Possible Duplicate: Show $f$ is primitive recursive, where $f(n) = 1$ if the decimal expansion of $\pi$ contains $n$ consecutive $5$'s $$L = \{i\mid f(i)=1\}$$ $f(i)$ equals $1$ if ...
1
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1answer
324 views

A regular expression for the words that don't contain the sequence $ab$ over $\{a,b,c\}$

The following is an exercise in a book I am reading: Let $\Sigma=\{a,b,c\}$, define $L$ to be the language of all words over $\Sigma$ that do not contain $ab$ as a sub-word. Find a regular ...
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2answers
3k views

Understanding $\epsilon$ transitions in a finite state automaton

I am trying to understand how $\epsilon$ transitions work. From what I've read, when you "go" to a state S that has arrows pointing outwards with $\epsilon$'s in it, you automatically go to those ...
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2answers
26 views

FSM to be regular with atleast two 0's and at most two 1's

Is it possible to construct a FSM to prove that the set $X$ is regular, where $$ X = \{s \in \{0,1\}^* \mid \text{$s$ contains at least two $0$'s and at most two $1$'s}\}\ ? $$
1
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1answer
40 views

Can someone explain to me how this is a proof?

I honestly don't understand how this proves anything.