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

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Show that a language is regular

Show that language $B$ is regular: $$B = \left\{1^ky\mid y\in \{0,1\}^*\right\} $$ $y$ contains $\ge k$ symbols $1$ So I try in following way - I'll draw DFA: What about my solution? Is it good?
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Did I find the right expression for the regular language for this FSA?

I have the following FSA, and the regular language that I found for it: Is this language correct? It doesn't match the solution in the book, but my teacher says there can be multiple equally ...
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2answers
30 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 $$ X = \{s \in \{0,1\}^* \mid \text{$s$ contains at least two $0$'s and at most two $1$'s}\}\ ? $$
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43 views

Can someone explain to me how this is a proof?

I honestly don't understand how this proves anything.
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43 views

Closed regular languages

Are regular languages closed under the following construction? $f(L) = \{w \mid w \in L$ and for all prefixes $x$ of $w$ it holds that $x \notin L$ $\}$
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26 views

Subset of regular languages

Assume $L_1$ and $L_2$ two regular languages, and $L_1\subseteq L\subseteq L_2$. Does this imply that $L$ is a regular language? Thanks in advance.
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100 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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71 views

New automaton that runs two others step by step?

If I have two automatons for two languages ($M_1, M_2, L_1,L_2$ respectively), what would be the procedure to mix them by defining a new automaton such that the new automaton would accept the words ...
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187 views

Is $\epsilon$ in every alphabet?

Given a $\Sigma$ an alphabet, is $\epsilon$ in it logically? For example, if I have a function $ f : \Sigma \to \Sigma $, can I define it for example $ f(\sigma) = \epsilon$? even if my alphabet is ...
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162 views

Whether $L=\{(a^m,a^n)\}^*$ is regular or not?

I am condidering the automatic structure for Baumslag-Solitar semigroups. And I have a question. For any $m,n \in Z$, whether the set $L=\{(a^m,a^n)\}^*$ is regular or not. Here a set is regular means ...
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1answer
691 views

How to prove CYK algorithm has $O(n^3)$ running time

I have a final coming up in few days, and the professor mentioned the CYK algorithm. I want to be prepared for the final. I'm trying to find out how to prove the algorithm has worst case running time ...
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764 views

Showing that a language is regular - Pushdown Automaton

So what I have to prove is that $L$ is regular given that the stack of PDA for $L$ never grows beyond $n$ entries on any input, and in this case $n=200$.
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132 views

Solve right linear set equations

Trying to solve this set of equations. I'm feeling like I'm making it so complicated. Of course + is union. Am I on the right track? A = 0B + 1D B = 0C + 1A C = 0A + 1B + λ D = OD + 1C + λ A ...
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906 views

Can _any_ NFA be converted to a DFA?

I was wondering if for every NFA there exists an equivalent DFA? I think the answer is yes. How would one prove it? Since I'm just starting up learning about Automata I'm not confused about this and ...
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210 views

Pushdown Automaton

Can someone help me construct a pushdown automaton to recognize the following regular expression representing the language $(a^3+a^5)$* using as few states as possible? How can this be done using a ...
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1answer
86 views

Show that $a^k w b^k$ when $|w|_a$ is divisible by $3$ is not regular

I want to show that $L = \{ a^k w b^k \mid k \geq 0, w \in \{a,b\}^*, |w|_a \text{is divisible by } 3 \}$ is not regular. I tried to use Pumping lemma as follows: Let $p$ be pumping length. $a^pb^p ...
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1answer
50 views

what is the effect of adding another stack to a PDA

does it increase the power of a push down automata? or does it have no effect on the power of the PDA ?
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1answer
191 views

Finite Automata, basic question with semigroups

If we use the notation where when we say: $$M = M(G)$$ We mean to say that $M$ is a automata with states and alphabet elements of $G$. From here, I am posed this question (Abstract Algebra by ...
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181 views

CFG pumping lemma

I can't figure out how to prove this is a non CFG. $\{xy : x, y \in \{a,b\}^*, n_a(x) = n_a(y), n_b(x) = n_b(y) \}$, Where the number of a's in x = number of a's in y and number of b's in x = to the ...
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34 views

How to design a finite state automaton that recognises the languages like $1^n 0^n$

The question goes like this: Design a finite state automaton that accepts binary strings with at least two $0$s and at most two $1$s. I can easily design an NFA which accepts at least two $0$s OR at ...
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1answer
44 views

How n (1+b) is not prime?

Here is the complete proof taken from this link How do I convince myself that n(1+b) is not prime when b>=1? Here is what I did: if n is 3 and b is 3. Then ...
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43 views

Proving that $L := \{a^n\mid\exists k \geq 0\ n = k^3\}$ is not a regular language

I'm having trouble using the pumping lemma to prove this language $L := \{a^n\mid\exists k \geq 0\ n = k^3\}$ is not regular. Assume $L$ is regular. Thus there is a DFA $M$ for it. Choose $m$ as the ...
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77 views

Why are Recursively enumerable languages closed under intersection?

when Recursively enumerable sets are not closed under complement ,then how come they are closed under intersection because the definition of intersection comes as L1 intersection L2= complement(L1' ...
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1answer
45 views

Write grammar for given language

I'm trying to write a grammar for the language below: $$(a+b)^*−\{𝑤𝑤𝑤 ∶ 𝑤\in\{a,b\}^*\}$$ could anyone help me?
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71 views

What the way studying math to automata theory

Good day everyone. I need to know automata theory. Can you advice me the best way to study math? What themes will I need to know to understand automata theory. What a sequence of study? What level ...
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1answer
11 views

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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1answer
90 views

$NP^{PP} vs. PP^{NP}$, which one subsumes the other?

I understand why P with an NP oracle ($P^{NP}$) subsumes $NP$: because it contains co-NP. But how about NP with a P oracle? Can it be any different from NP? (I'm guessing they are the same otherwise ...
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51 views

NFA for $L = \{ w \in \{a,b,c\}^* : \text{at least one symbol appears only once in }w \}$

Write an NFA to recognize the language $$L = \{ w \in \{a,b,c\}^* : \text{at least one symbol appears only once in }w \}.$$ I'm not quite sure how to do this question. I don't know how to keep ...
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57 views

Is this DFA correct?

I am reviewing. I need to write a DFA that accepts a string w such that bab is not a substring. Is there any error. Also, any guideline or tips? Since it's a DFA I don't make more than 2 ...
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113 views

Euclid's Proof that there are an infinite number of primes

I'm trying to clarify my understanding of decidability of a language. The following question is totally made up by me so I hope it makes sense. Let $L = \{A: A \text{ is an algorithm that can ...
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25 views

How to prove the following related with regular languages

How can we prove the following. If $$\sum$$ is any alphabet and L is any language $$L \subset \sum*$$ Then L*L* = L* ?
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44 views

Particular Problem for Context Free Grammars

Consider the context-free-grammar $G$ defined by productions: $$ S \rightarrow aS\,|\,Sb\,|\,a\,| b $$ Prove by induction on the string length that no string in $L(G)$ has $ba$ as a substring. I ...
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31 views

Showing that all regular languages are closed under reversal

I'm trying to show that $L^{reverse} = \{w^{reverse}:w \in L\}$ is a regular language. The first argument I can come up with is simply: if we have an NFA for $L$, then an NFA for $L^{reverse}$ can be ...
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669 views

Languages and Grammar (Finding a language)

I have a trivial question (that I have actually solved, hopefully) although I am a bit curious if my result is alright. We have $N= \{S , C ,D\}$, $T=\{c, d\}$ and $P = \{S \to Dc, D \to Dd, D \to ...
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1answer
84 views

About described DFA

I need to find DFA (or NFA, $\epsilon$-NFA, it's not improtant (I know how to convert between them)) that accept all strings of $0$'s and $1$'s such that every block of five consecutive symbols ...
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55 views

If $v^6w^8 = w^{12}v^4$ then $(vw)^2 = v^2w^2$?

Given the words $v,w \in \sum^*$, is this correct? If $v^6w^8 = w^{12}v^4$ then $(vw)^2 = v^2w^2$ If $vw^2 = wv^2$ then $v=w$ For one, I tried $v=\epsilon, w=\epsilon$ and it worked, and ...
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75 views

Why is this the $L(M)$ of this DFA?

Why is this the $L(M)$ of this DFA? Can someone please explain it? I am new to this course. When I tried alone answering the question of "What is special about the words that get accepted by this ...
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1answer
45 views

Post-concatenation of the languages represented by the null set

I have a small question regarding concatenation of regular languages: Is it true that the concatenation $L\varnothing$, where $L$ is any regular language, result in $\varnothing$? Namely, does ...
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1answer
137 views

Constructing a parallel composition from a given transition system and automaton

I am looking at an exercise, where it asks me to construct a parallel composition from a given transition system and an automaton. The transition system looks like this: and the automaton (with ...
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1answer
46 views

Stochastic Automaton accepting every word with same probability

I am looking for a stochastic automaton, which induces the same probability $c \in [0,1]$ for all words in $\Sigma^*$, where $\Sigma$ is some finite alphabet. A stochastic automaton over an alphabet ...
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1answer
115 views

Give a regular grammar for L

Give a regular grammar for L= {a^n b^n : n<=100} I would do something like this : S---> A | empty string A---> aB| empty String B---> Ab but How do we keep count of the number in the grammar? ...
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1answer
121 views

How to construct a context free grammar $G$ such that $L(G) = \{ a^i b^j a^k| j \gt i+k\} $?

How to construct a context free grammar $G$ such that $L(G) = \{ a^i b^j a^k| j \gt i+k\} $? My attempt: $G_1 = (\{ S,A,B\}, \{a,b\},P,S)$ where $P$ consists of: $$ S\to AbBC $$ $$A \to ...
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85 views

Proving this language is regular?

Is $$L =\big\{x^ny^m : |n-m| = 2\big\}$$ a regular language? I can't seem to figure this question out, and i've tried drawing a dfa but I still can't seem to find it. If there is a possible dfa, ...
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100 views

regular language?

I need help proving whether this language is regular or not. $$L = \big\{ w \mid w \in \{a,b\}^*, n_a(w)\text{ is even}, n_b(w)\text{ is even}\big\}$$ That is, the number of $a$'s is even and the ...
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777 views

How to prove by pumping lemma these languages are not regular?

$L_1 = (a^k * b^r \mid k \neq r^2)$ $L_2 = (a ^{\sum_i ^n t} \mid n > 0 )$
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429 views

Show that every finite automaton is a one-state pushdown automaton

I'm reading a book that states: Every finite automaton is a one-state push-down automaton How can I go about proving it?
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113 views

Pushdown automata - definition and definition of $\vdash$

I am reading about pushdown automata and I don't understand the definition of $\vdash$. My book writes that $$(q,aw,Z\alpha)\vdash(p,w,\beta\alpha)$$ if $$(p,\beta)\in\delta(q,a,Z)$$ Can someone ...
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162 views

Which automata recognise the algebraic numbers?

I am reading historical stuff on the algebraic and transcendental numbers. Descartes, in his Geometry, excluded all curves not expressible as algebraic equations. Later, Leibniz called such curves ...
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330 views

Showing that two regular expressions represent complementary regular languages over {0,1}

How do up you show that two that the regular expressions, such as $(01+1)^*$ and $(0+1)^*\left(0 + 00(0+1)^*\right)$ represent complementary regular languages over $\{0,1\}$? I'm trying to do some ...
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70 views

Closure property of Alternating language

Problem Given a language $L$ is context-free, must $\operatorname{alt}(L)$ is also context free? where $$\operatorname{alt}(L) = a_1a_2a_3 \ldots, \quad L = a_1b_1a_2b_2a_3b_3 \ldots$$ I couldn't ...