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

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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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171 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

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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85 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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17 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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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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23 views

Give a regular expression for $A = \{1^{k}y|k \geq 1, y \in \{0,1\}^{*}$ and $y$ contains at least $k$ $1$'s $\}$

The regular expression that is given is $1(0 \cup 1)^{*}10^{*}$. I'm having trouble realizing why this regular expression describes the language given. For example, the string (for $k$ = 4) $1111$ ...
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22 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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58 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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52 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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96 views

Lambda productions in grammar

I tried removing the $\lambda$ productions from this grammar: $S \rightarrow a A b \mid B B a$ $A \rightarrow b b \mid \lambda$ $B \rightarrow A A \mid b A a $ It seems like you just take away the ...
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50 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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62 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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30 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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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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74 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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108 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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72 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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88 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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596 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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263 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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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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142 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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242 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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66 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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177 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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15 views

having hard time reading symbols in this Turing Machine

I am reading few books and I am looking at different examples of a Turing Machine, and I am getting frustrated reading symbols especially in this example...What does ...
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26 views

Constructing PDA for a language

I want to prove or disprove that for a given two PDA's (Pushdown Automata) $M_1$ and $M_2$, we can build a PDA $M$ such that $$L(M) = \{w \in L(M_1) \mid w\text{ contains some string in ...
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23 views

Is there a prefix-free regular language whose set of proper prefixes is not regular

Is there a regular language $L$ such that the language $$L^\prime := \{ w : w\text{ is a proper prefix of a word in }L \}$$ has the following properties: $L^\prime$ is not regular, and $L^\prime$ ...
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35 views

How to construct a grammar G such that L(G)={x^ny^mx^my^n/m,n>1}?

construct a grammar $G$ such that $L(G)=\{x^ny^mx^my^n/m,n>1\}$? I don't have much idea how to approach this one. Could some help me to understand how to approach these kinds of problem?
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50 views

Is the PDA I drew correct?

Could you tell me if the language $L=\{a^{m}b^{n}:m \neq n,m>0,n>0 \} $ is accepted by the following pushdown automaton, where the alphabet of the stack is $\{a,z\}$ and $z$ is the initial ...
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40 views

is this language regular or not?

I have problem with this language $$L = \{ a^n b^m : \text{$n+m$ is odd} \}$$ is it regular or not My Solution I used pumping lemma, w = a^2p b^2p+1 (the same for a^2p+1 b^2m ) ...
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65 views

Push down automata for context free grammar

I'm having trouble finding the PDA for this language $L = \{x^{3i} y^j z^k\ |\ i \ge 0 \land k \gt 2j \gt 0\}$ The ...
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41 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 ...
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35 views

Consider this grammar

Consider this grammar: \begin{array}{l} S \to aabBba \mid aAb \mid aab \\ bBb \to bCa \mid aaa\\ aA \to aC \mid bba\\ C \to aab \mid Cb \end{array} This is clearly context-sensitive (CS). It's not ...
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74 views

Is there a DFA with $k+2$ states which its reverse has $2^k$ states

I am trying to figure out if there exists a DFA $M$ with $k+2$ states (for every $k\in \mathbb{N}$ ) so that every automaton which accepts $L(M)^R$ has at least $2^k$ states. I am trying to find an ...
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41 views

Prove the existence of $C\in L_{regular}$ so that: $A \prec C \prec B $

Given $A,B$ regular languages. Prove the existence of $C\in L_{regular}$ so that: $A \prec C \prec B $ Whereas $A\prec B$ stands for: $A\subset B $ and $B\setminus A $ is infinite regular language. I ...
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36 views

If $L\cdot\{\epsilon,0\}$ regular language, is $L$ regular?

I've encountered a question during my studies: If $L\cdot\{\epsilon,0\}$ regular language, is $L$ regular? I thought to disprove it by using $A\subseteq 2\mathbb{N}, L=\{w\in\{0\}^*:|w|\notin A\}$ ...
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70 views

Proving a set is language generated by given grammar

I have grammar $G$ with productions $S\rightarrow aS|aSbS|\epsilon$, and task is to prove that $L(G)=\{w|$every prefix of $w$ has at least $a$'s as $b$'s$\}$ (I'm not sure if translation is correct, I ...
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40 views

Equivalence of two PDAs

I want to show that if $P$ is a PDA, then there exists a PDA $P_2$ with only two stack symbols, such that $L(P) = L(P_2)$. As I want only two stack symbols for $P_2$, it seems intuitive to encode in ...
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If $L \cdot \{\epsilon, a, b\}$ is regular, is $L$?

Given that $L \cdot \{\epsilon, a, b\}$ is regular, is $L$ regular too? (Our alphabet is $\Sigma = \{a,b,c,d\}$ What I thought was yes, and here is why: If it is regular, then we know there ...
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126 views

Greibach normal form conversion

I'm trying to convert this into GNF: $S \rightarrow ASaa | bab$ $A \rightarrow Ba | bAB$ $B \rightarrow abba$ So I'm getting this, but I'm not sure understanding and applying correctly the concept ...
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42 views

NFA from regular expression

I'm trying to make an NFA from the following regular expression. I'm not sure about the edges between nodes $q2$ and $q4$, maybe someone can point out where everything went wrong.
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22 views

Regular expression for a language

I made a regular expression to match this language but I'm not sure it's right. Perhaps someone can show me where it deviates. The language: $L = {a^{n} c b^{m} (cc)^{p} : n \geq 1, m \leq 1, p\geq ...
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52 views

Is there an (explicit?) bijection from the set of all automatons to the set of all regular expressions that conserves the recongnised language

Let $\Sigma$ be an alphabet, $R$ be the set of regular expressions on $\Sigma$ (that is, trees with leave's values in $\left\{\varepsilon\right\}\cup \Sigma$ and three types of interior nodes, one ...
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153 views

Best book for automata theory and compiler design?

I am currently pursuing my M.Tech in Digital Image Processing, I want to take admission in PhD program using subjects either Formal Language and Automata Theory or Compiler Design, Can anyone please ...
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134 views

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

How to construct a context free grammar $G$ such that $L(G) = \{ a^i b^j c^k| j \gt i+k\} $? I don't have much idea how to approach this one. Could some help me to understand how to approach these ...
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36 views

Can every regular language have a linear bounded automaton

As the question states: I am trying to understand automata. Can every regular language have a linear bounded automaton?
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120 views

How would I go about proving for this NFA?

I am struggling on this one question, where it is asking to define an XOR automata which is defined as an NFA and it is defined as the following: N accepts the string x if the number of distinct ...
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237 views

Finite state machine to report when the last 4 inputs were 1011

Suppose you want to construct an FSM containing one input and one output. Consider the example: The machine should assert the input (set to 1) when the last four bits taken in as input match the ...