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

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proving that if a FFA accepts L=> L is a regular language

Ok, so after wasted time for nothing on this question that I asked yesterday: proving that a regular language can be accepted by a fast finite automaton Now comes the more interesting prove: ...
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36 views

If $L_1$ and $L_2$ are regular then $L_1 \cup\;L_2\; = L$ is regular. Is the converse true?

The following is an answer I found to the question. For instance, $\sum^*$ is a regular language; but it can be decomposed into two languages $L_1= \{0^i1^i,\;i\ge0\}$ and $L_2 = ...
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60 views

Why is $L=\{xww^r \mid \text{ $x$, $w$ belongs to }(a+b)^+) \}$ not regular?

If I have a language $L$ such that $L= \{xww^r \mid \text{ $x$ and $w$ belong to }(a+b)^+) \}$. Then can't we write regular expression for this to be $(a+b)^+(ab+ba)$? What's wrong in this regular ...
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62 views

recurrence relation of a language

I am looking at the following: Consider a language $X$ which consists of all bitstrings with no more than 2 consecutive zeros (represented by the above automaton). Next consider a sequence $s_n$ ...
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45 views

Alphabets of Turing Machine

I'm not completely sure about equivalence of two definitions of Turing machine. The first one states that Turing machine has a finite alphabet $\Sigma$, set of states and some rules. Turing machine ...
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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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51 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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47 views

Write grammar for given language

I'm trying to write a grammar for the language below: $$(a+b)^*βˆ’\{𝑀𝑀𝑀 ∢ 𝑀\in\{a,b\}^*\}$$ could anyone help me?
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100 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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12 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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93 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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70 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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74 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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114 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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30 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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51 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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37 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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1k 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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158 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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56 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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76 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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51 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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175 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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47 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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124 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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124 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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93 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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107 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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818 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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489 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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116 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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164 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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366 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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71 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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271 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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A proof question involving a regular set and a context free language

Claim: Let $L \subseteq \Sigma^*\{\#\}\Sigma^*$ be a context-free language, where $\# \notin \Sigma$. Suppose that for each $x \in \Sigma^*$, $\{y|x\#y \in L\}$ is finite. Then $\{y|\text{ for some } ...
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22 views

Parser for reversed language

Language $L$ is specyfied by grammar : $(\{S,A,B\},\{c,d\},S,\{S \rightarrow SA, A \rightarrow Bc | \epsilon, B \rightarrow d\})$. My task is to construct LR(1) parsing table for language $L^R$ ...
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28 views

Synchronizing sequence

From Sipser's book: Let $M=(Q,\Sigma,\delta,q_0,A)$ be a DFA and let $h$ be a state of $M$ called its "home". A synchronizing sequence for $M$ and $h$ is a string $s\in \Sigma^*$ where $\delta ...
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30 views

Language with middle third removed

Originating from Sipser's book: Let $A$ be any language, define $A_{{1\over3}-{1\over3}}$ be the subset of strings of $A$ whose middle third is removed. The solution I came across makes the ...
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39 views

Prove that two arithmetic grammars generate the same language

In my university's automata theory book it is claimed that the following two arithmetic grammars generate the same arithmetic language but no proof is shown. $G_1=(\{E\}, \{a,+,*,(,)\}, \{E\to ...
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11 views

Proving that the language $\{w\in \{a,b\}^* \big|\#_a(w)< \#_b (w)\}$ is non regular using the pumping lemma

I need to prove that the language $\mathscr L=\{w\in \{a,b\}^* \big|\#_a(w)< \#_b (w)\}$ is non regular using the pumping lemma My try: $\{a,b\}^*=\{\epsilon,a,b,aa,ab,ba,bb,aaa,aab,\dots\}$ ...
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44 views

Design a DFA to check whether the Given Number is Even

I have the following question I have designed the following A Binary String is even if it is ending with 0 and odd if its ending with 1.I have applied this.Im i right ? UPDATE:
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Checking if $L_1 \cup L_2$ is regular language

Let $L_1=\{a^n b^r|n \geq 1, r\geq1,n=r\}$ $L_2=\{a^n b^r|n \geq 1, r\geq1,n\neq r\}$ be a non regular languages $L_1 \cup L_2$ is regular? I think that $L_1 \cup L_2$ is regular because we ...
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33 views

How we derive language from grammar by bottom up or any other approach?

Consider a CFG with the following productions. S β†’ AA | B A β†’ 0A | A0 | 1 B β†’ 0B00 | 1 $S$ is the start symbol, $A$ and $B$ are non-terminals and $0$ and $1$ are ...
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111 views

Automata | Prove that if $L$ is regular than $half(L)$ is regular too

I've see couple of approaches to this kind of questions yet I have no clue how to approach this one. Let L be regular language, and let half(L) be: $half(L) = \{u | uv \in L\ s.t. |u|=|v|\}$ Prove ...
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43 views

what are the practical uses of “game of life” or “langton's Ant”

A few questions: Besides looking really cool, what are the practical uses of "game of life" or "langton's Ant"? I understand how agent-based modeling itself is a potentially useful methodoly, not ...
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40 views

Prove that $\{w \mid \text{ w has even length and the first half of w has more 0s than the second half of w} \}$ is not regular?

I have had some difficulties understanding proofs that a language is not regular using the Pumping Lemma, and now I need to prove that the following language $$A = \{w \mid \text{ w has even length ...
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31 views

Prove the infinite union is not regular

Prove $\bigcup _{i=1}^\infty A_i$ is not regular. We know $A_i$ is regular, but how can prove the infinite union is not regular. I think a counter example would work, but I can't think of any. ...
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22 views

Prove that the Language $L= \{ 0^n1^m \;|\; n,m \ge 0 \}$ is regular

I've looked and didn't find an answer. I know that languages like $\{ 0^n1^n \;|\; n \ge 0 \}$ and $\{ 0^n1^m \;|\; m \gt n \ge 0 \}$ are irregular so I don't understand how this language can be a ...
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22 views

Finding a language accepted by an automaton?

I'm currently doing work on discrete mathematics in my free time and am having some difficulties with understanding some questions pertaining to Automata and Finite-state machines. To be specific, I'm ...