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

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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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205 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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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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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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187 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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177 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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24 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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41 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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56 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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87 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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54 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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109 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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42 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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537 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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70 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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54 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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71 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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40 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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123 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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44 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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111 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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119 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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763 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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406 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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103 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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161 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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323 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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69 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 ...
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1answer
234 views

Question regarding the initial stack symbol in Push Down Automaton

Let $L = \{a^nb^n : n \geq 0\} \cup \{a\}$, where $\Gamma = x, \$, \Sigma = {a, b}$, we have the NPDA of $L$ in three states: In the above state diagram, I can break the transtion $\lambda, \lambda ...
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19 views

Minimum Size of DFA

I'm confused about the following DFA problem: Let L denote the set of all strings in $\{a, b\}^∗$ that contain abb or aab as a substring. Show that any DFA that decides L must have at least five ...
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1answer
16 views

Pumping Lemma proof and the union/intersection of regular and non-regular languages

I am still learning the pumping lemma. I have a problem for which I used it. I used it on the first part (a) but I am unsure if it is correct. Parts b-d, I am not sure how to do it. I created a dfa ...
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45 views

NFA (nondeterministic finite automaton) made out of the Bible?

Let B be the language over alphabet {a, ... z} consisting of those words occuring in the Bible. Thus, B = {in,the,beginning,god,created,...}. Describe an NFA whose language is B. Describe what ...
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1answer
26 views

what do we mean by saying that a particular computing machine is more powerful than another computing machine?

I am having confusion in understanding the concept of the word "powerful" in automata theory. Even the confusion arises that if I have a DFA with single final state ,so then is it more or less ...
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45 views

Is it possible to write an Unambiguous Grammar for Two Hard Language ?!?

I came across a very hard interview exam. It was asked wrote an unambiguous grammar for two following language, Who can hint it to solve it? 1) $L = \{a^n b^{2n} c: n\geq 0\} \cup \{a^{2n} b^n d: ...
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29 views

How to construct the DFA for the following question?

Given $L=\{ab^5 w b^4 \}$ where $w=\{a,b\}^*$. I am not able to construct it, I tried it but getting problems with the transitions for input symbol a.
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1answer
24 views

Two DFA to one NFA.

Let assume that I have given DFA $D$ which recognize language $L$. Now I would like create the DFA/NFA which recognize the language $L'$. $$ L' = \{ w \in L : |w| = 2k, k \ge 0 \}$$ In words, $L'$ ...
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42 views

Convert NFA to DFA

I have to convert the following NFA's into the equivalent DFA's. I have done the following: Could you tell me if it is correct??
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33 views

Show that the language is regular - Closure

For languages $A$ and $B$, let the perfect shuffle of $A$ and $B$ be the language $$L=\{w \ \mid \ w=a_1 b_1 \dots a_k b_k, \text{ where } a_1 \cdots a_k \in A \text{ and } b_1 \cdots b_k \in B, ...
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22 views

Language described by inverting accepting states of NFA

What is the formal language described by inverting accepting states of NFA? By inverting, I mean that rejecting states become accepting states and accepting states become rejecting states. Is there a ...
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37 views

NFA construction problem

I need to construct an automaton that recognizes the following language of strings over the alphabet $\{a,b\}$: The set of all strings over alphabet $\{a,b\}$ with the subsequence $abba$. (A ...
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1answer
96 views

Context free grammar to NFA

I've been given an exercise to solve which goes as follows: generate an NFA from the given CFG, S -> AB | c A -> aAb | c B -> bBa | c Now correct me if I'm wrong, but if this language has an NFA it ...
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59 views

Proving languages are undecidable

Let L5 be the language { {G,D} | G is a CFG, D is a DFA, and L(D) ⊆ L(G) } Show that L5 is undecidable. Is L5 R.E.? Is it co-R.E.? I am not quite sure where exactly to start with this. Could ...
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64 views

Can 2 items be added/taken away from a stack in push down automata at once?

Here is a language and 2 ways (I hope) of representing it with a PDA. Can I use the notation (b,a $\to$ ee) or anything of the like, to take away 2 items from the top of a list at once? Such as I ...
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52 views

Automata Language regularity proof by construction.

I've been trying to prove or disprove a question that popped during our last session in Uni, we've been using automaton constructing to prove regularity for a while now and I really do have the handle ...