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

learn more… | top users | synonyms

0
votes
1answer
104 views

Stronger Pumping Lemma for Context Free Languages

Hi Math Stack Exchange, Taking a class in automata theory, and having real trouble proving the following strong automata theorem for context free languages (from Sipser, Problem 2.37): If L is a ...
1
vote
1answer
20 views

Are computable sets closed under XOR?

How do I prove if computable are/are't closed under XOR?
1
vote
1answer
39 views

Proving that reguarity is closed under prefixes?

Show that regularity is closed under prefixes. That is, if $L$ is regular, then so is $$L_1 = \{x \mid \exists y: xy\in L\}$$ I am having a hard time trying to work this through. Can you please ...
1
vote
1answer
26 views

How to identify context free language?

Consider the following context-free grammars$:$ $G_1: S → aS|B, B → b|bB$ $G_2: S → aA|bB, A → aA|B|ε, B → bB|ε$ Which one of the following pairs of languages is generated by $G_1$ and $G_2$, ...
1
vote
1answer
46 views

Proving that the given language is non regular

We had a question today as follows: Let $L$ be a nonregular language and $X$ a finite set of strings from the same alphabet as $L$. (a) Prove that $L ∪ X$ is nonregular. (b) Prove that $L - X$ is ...
0
votes
1answer
41 views

Identify language of given PDA?

Consider the transition diagram of a PDA given below with input alphabet $Σ =\{a,b\}$ and stack alphabet $Γ = \{X,Z\}. Z$ is the initial stack symbol. Let $L$ denote the language accepted by the PDA. ...
0
votes
1answer
52 views

Reduction to/from REC and RE language?

Let $X$ be a recursive language and $Y$ be a recursively enumerable but not recursive language. Let $W$ and $Z$ be two languages such that $\overline{Y}$ reduces to $W$, and $Z$ reduces to $\overline{...
1
vote
1answer
26 views

confusion of an example for Powerset construction

I have an example of Powerset construction from the lecture. Powerset construction is applied on automata A1. The result is automata A2. You could see I do Powerset construction myself below the ...
1
vote
1answer
27 views

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 } ...
0
votes
0answers
33 views

Queue automaton algorithm for accepting primes

What is an example of a queue automaton algorithm that accepts prime numbers, encoded as strings of prime length? For example, if the input is either of ...
0
votes
1answer
19 views

power + operator for binary

What is the specific definition for power $+$ operator in automata theory? For example, when $x$ is a binary what does it mean that $x = 0^+$. Does it mean that x is a string with at least one $0$?
0
votes
2answers
28 views

Reverse of binary number

Let us say that x is a set of binary numbers $$x = \{0, 1, 1001\}$$ Am I correct that $x^R$ is equal $$x^R = \{0,1,1001\}$$ or is it: $$x^R = \{1,0,0110\}$$ What I mean by that is: do we create a ...
0
votes
0answers
14 views

FIRST of a grammar with righthand recursions

$L$ is defined by the grammar $$ S \rightarrow SAB \mid \epsilon \\ A\rightarrow Ba \mid \epsilon \\ B \rightarrow Ab \mid \epsilon $$ What is a $FIRST$ for elements of this grammar? I have two ...
1
vote
1answer
24 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 $L^R$ (with,...
1
vote
1answer
31 views

Finite Automata for regular expression

I am trying to construct finite automata for this regular expression: Every block consisting of 5 characters need to contain at least two zeros. The regular expression would look sth like this: (00(1|...
3
votes
1answer
89 views

How many DFA's exist with two states over the input alphabet $\{0,1\}$?

How many DFA's exist with two states over the input alphabet $\{0,1\}$? My attempt : Input set is given. So, we have 3 parts of DFA which we can change: Start state Transition Function Final ...
2
votes
1answer
52 views

Is the complement of a given language context-free?

I have a problem with finding out if the complement of language L is context free. $L = \{ ww : w \in \{a,b\}^{*} \wedge \text{ }w \text{ number of }a\text{'s in }w \equiv \text{number of }b\text{'s ...
3
votes
1answer
105 views

The Mathematics of Finite State Automata

I am a final year undergraduate mathematics student preparing to undertake my BSc-HONS project, provisionally titled for the time being, "Finite State Automata and Regular Languages". Having had a ...
0
votes
1answer
45 views

why is the below language a regular set? [closed]

Given a set S={x∣ there is an x-block of 5's in the decimal expansion of π} (Note: x-block is a maximal block of x successive 5's) In the question it is mentioned that there is x-block of ...
0
votes
1answer
15 views

Consider the given regular grammar what are the Myhill-Nerode equivalence classes for the language generated by the grammar?

S → bS | aA | ϵ A → aS | bA A) {w ∊ (a + b)* | #a(w) is even) and {w ∊ (a + b)* | #a(w) is odd} B) {w ∊ (a + b)* | #a(w) is even) and {w ∊ (a + b)* | #b(w) is odd} C){w ∊ (a + b)* | #a(w) = #b(w)...
0
votes
0answers
42 views

Regular and context free language

Let $A$ be a set represented by a regular expression $0^*1^*$ and let $B=\{0^n1^n\mid n\geq 0\}$. It is known that $A$ is a regular language, while $B$ is a context-free language. I understand that $...
-1
votes
2answers
49 views

Automata Theory - Designing a Non-deterministic Finite automata [closed]

I have been combing through youtube looking for a simple explanation on how to "crack" the "NFA" myth. Please help. Since drawing the schematic can be cumbersome, if you can show me how I can derive ...
0
votes
1answer
30 views

Prove that $\mathcal L=\{a^ib^jc^k|i+k=j\}$ is context-free

Prove that $\mathcal L=\{a^ib^jc^k|i+k=j\}$ is context-free I am trying to prove it without to build a pushdown automaton First I tried to look which words are in $\mathcal L$, $\{\varepsilon,ab,...
0
votes
0answers
29 views

How many states needed for this FSM, and how do I define them?

I want to define states for FSM that gives an output $1$ if and only if $X$ is dividable by $5$ with a residue of 3. where $X$ is the binary number that the machine got until now (Not Including ...
0
votes
1answer
31 views

how to build a PDA for the language of the intesection of PDA and DFA

lets say we got PDA $M=<Q, \Sigma,\Gamma,\delta,q_0,+,F>$ ('+' marks the end of the stack) with $L_1 = L_f(M)$. and we got $A=<P,\Sigma,\delta_1.p_0.F_1>$ with $L_2 = L(A)$. how can I ...
1
vote
1answer
36 views

Converting an NFA to a DFA

I am trying to convert this NFA to DFA: So I built the power automata, and this is what I got: This should be the answer: I don't understand where am I wrong since $$\delta_{\text{ND}}(...
1
vote
1answer
31 views

Using Pumping Lemma to prove that $L=\{a^mb^{3m}:m\in\mathbb{N}\}$ is not recognizable over $A=\{a,b\}$

[Pumping Lemma]: Let $\mathcal{A}=(Q,A,\cdot,i,T)$ be a (complete and deterministic) automaton and let $L=L(\mathcal{A})$ be the language recognized by $\mathcal{A}$. If $L$ is infinite and $\mathcal{...
0
votes
1answer
68 views

What is the nature of given language?

$$L=\{a^n b^n :n\geq0, n\neq100 \}$$ I just wanted to know that through pda. How will we make sure that $n\neq100$ or say I put a restriction that $n\geq100$. How to design a PDA using these ...
1
vote
1answer
57 views

Question about deterministic finite automaton and accepting states

For $n \in \mathbb N$, an "$n-$DFA" is an automaton with exactly $n$ accepting states. Let $\Sigma=\{0,1\}$. Prove that the set of the languages that can be accepted by "$1-$DFA" is a subset of the ...
1
vote
1answer
76 views

Question about deterministic finite automaton (DFA) [closed]

For $n \in \mathbb N$, an "$n-$DFA" is an automaton with exactly $n$ accepting states. Let $\Sigma=\{0,1\}$. Prove that the language $\mathcal L=\{0,00,0000\}$ cannot be accepted by any $2-$DFA.
0
votes
1answer
50 views

Is $L=\{o^{i}1^{j}o^{j}1^{k}o^{k}1^{i} | i,j,k>0\}$ a context free language?

I need some help in finding and proving (by the pumping lemma or by building a grammar) if $L=\{o^{i}1^{j}o^{j}1^{k}o^{k}1^{i} | i,j,k>0\}$ is a context free language. thanks!
1
vote
1answer
39 views

Is $L=\{o^{i}1^{i}o^{j}1^{i} | i,j>0\}$ a context free language?

I need to find and to prove (by the pumping lemma or by building a grammar) if $L=\{o^{i}1^{i}o^{j}1^{i} | i,j>0\}$ is a context free language. I would like to get some help. thanks!
1
vote
1answer
50 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 (q,...
1
vote
1answer
61 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 ...
0
votes
2answers
34 views

which of these languages are regular sets?

$$ L_1 = \{a^p b^q\ |\ p+q \ge 10^6\} \\ L_2 = \{a^m b^n\ |\ m-n \ge 10^6\} $$ According to me both of these languages require comparison between number of $a$'s and $b$'s so both of them should ...
0
votes
1answer
34 views

Pumping lemma for two words that “completely different”

Let $"x"$ and $"y"$ be a words, we will say that two words are "completely different" if for all $1\leq i\leq |x|$ the $i$ letter in $x$ diffrent from the $i$ letter in $y$. Prove that the ...
0
votes
0answers
17 views

Is the given language decidable or not?

L={|L(M) ={1} } Converting this in terms of program terminology I gee that given any input program we have to see whether it accepts "1" and nothing else . So for input 1 , if it accepts it then we ...
1
vote
0answers
23 views

Converting CFG to PDA for $S\to aSd|aBd\\B\to bBc|\varepsilon$

I need to build a pushdowm automata for the context-free-grammar $$S\to aSd|aBd\\B\to bBc|\varepsilon$$ My attempt: I'm not sure if my attempt is correct or not.
0
votes
1answer
39 views

DFA, best practice

Consider the following language {A,B,C} and the following regex (A|B)+C. I'm a little in doubt about which of my two examples is more correct. Or are they both equally correct? e.g., Is it allowed ...
0
votes
1answer
18 views

Determining whether a given language is regular, and finding a regular expression

I know there are a lot of questions similar to this one, Proving a language is regular is just one example. However, I have not managed to find an answer that really answers my question. I'm currently ...
0
votes
4answers
89 views

Determining if a binary string represents a prime integer

Let $\Sigma = \{0,1\}$ and $w$ be the string $0011101$ over $\Sigma$. If we work out what $w$ is, $w$ is the binary representation of $57$, which is not prime. It is remarked in Introduction to ...
0
votes
1answer
102 views

Is there a subtle difference between NOEXTEND(A) vs NOPREFIX(A)?

My question originates from Sipser's book. Let A be a language with the DFA $(Q, \Sigma, \delta, q_{0}, F)$ and define: NOPREFIX(A) = {w $\in$ A| no proper prefix of w is a member of A} NOEXTEND(A) ...
1
vote
1answer
34 views

Prove that a given CFG grammer $G$ is equivalent to language $L$

I need help to prove that the given CFG grammar $G$ is equivalent to language $L$: as $S\to 0S1 \mid SS \mid \varepsilon$ and $L=\{w\in\{0,1\}^* \mid \#_0(w)=\#_1(w)\text{ and for any prefix } u \...
1
vote
1answer
61 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 E+E|E*...
0
votes
0answers
23 views

How to find a push-down automata that describes $f_{\sigma}(L)$? (given that $L$ is context free language)

Let it be $L$ a context free language. Definition: $f_{\sigma}(L)$={$w:σw∈$$L$}. Need to find a push-down automata $M'$ so that $f_{\sigma}(L)$=$L(M')$. Ok, so here is my idea for ...
3
votes
0answers
18 views

Finding a language the accepts a given push-down automaton [duplicate]

Ok, so given the following automaton: I need to find the language that accepts it (no need for formal prove, a short intuitive explanation is good enough). I think the answer here is {$a^{11+6k}, k≥...
0
votes
0answers
37 views

Language resulting in the same NFA as the DFA

I have tried to construct the NFA and DFA from the same language term, and they keep coming out the same, I was wondering if this is correct for: {w | w has an even length and an odd number of a's} ...
1
vote
1answer
22 views

Construction of DFA using an odd bit of language

I am working through a lecture and it constructs a DFA using the language: $$\{w\mid w\textsf{ is any string not in }(ab^+)^\ast\}$$ What does the $(ab^+)$ mean?
1
vote
1answer
41 views

Prove that if you can derive w from α in n steps, it's possible with n left-derivations as well

My university's automata theory book claims that the following claim can be proved by induction but it doesn't bother showing the proof. I've tried to prove it myself but I got stuck at the ...
0
votes
3answers
32 views

Language of Regular Expression

I'm trying to teach myself Regular expressions for Automata, I'm struggling to work out what the output of $L((1+01)^*)$ would be Would it be the star closure of $\{1,01\}$ or star closure of $\{1,...