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

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Question About Pumping Lemma used on a FA

I am learning about pumping lemma and I am trying to solve a problem. I need to use pumping lemma to show that: the Language L(M) defined by the following machine is infinite. Here is the dfa: ...
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2answers
134 views

Show that, given regular expression $R$, we can decide whether $L(R)$ is prefix-free

Suppose language $L$ is called prefix-free if no member is a proper prefix of another. For instance, cat is a proper prefix of category and so $L = \{cat,category,ego,go,rye\}$ is not prefix free. ...
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67 views

Creating a Push Down Automaton from a Grammar

I have the following grammar, but I'm not sure what exactly it is that it does: $\qquad\begin{align} S &\to S \vee T \mid T \\ T &\to T \wedge F \mid F \\ F &\to p \mid \thicksim p ...
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366 views

Regular expression and DFA/NFA questions

If a language L is generated by a regular expression, then L is recognized by a DFA. I think this is true, because regular expressions describe regular languages, those of which are exactly ...
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24 views

Reversed Language of a Non Regular Language

Is the following saying true or false? In any case why? Thanks!
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Finding regular expressions

I'm given the DFA shown below and need to find regular expressions for the following languages: $L_{1,2}^0, L_{2,1}^6, L_{2,5}^4, L_{2,3}^5, L_{1,3}^5$. The language $L_{p,q}^r$ is defined as ...
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31 views

Help Specify possible definitions for this Boolean Function

My math is rusty, but I need some guidance here. Problem I wish to design a stochastic, boolean procedure $f(state)$, that picks a winner, $f(state_{win})\to 1$ or loser, $f(state_{loss})\to 0$. I ...
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141 views

What is the language generated by G and how to draw the finite state automaton that recognizes this language?

G = (V, T, S, P) V = (0, 1, A, B, S) T = {0, 1} S is start S -> 0A S -> 1A A -> 0B B -> 1A B -> 1 For the drawing, I am confused about the last ...
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105 views

DFA with $k$ states, words of length $k$

Is this statement true? If we have a DFA with $k$ states, and if $L(M) = L$ is infinite, then there is a word of length at least $k$ and at most $2k-1$. Isn't this a trivial answer? Take the ...
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435 views

Converting NFA to DFA

Im trying to convert a NFA to DFA. This is the NFA and this is the DFA to which i converted Is this right? Also when converting if i write a state as [q0,q1] is this same as [q1,q0] edit: ...
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85 views

Proving Equivalence of DFA and NFA

Im trying to learn Equivalence of DFA and NFA.The problem is that in the below explanation Q' is given as the power set of Q.But this statement seems to be contradictory to the previous statement ...
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25 views

How is Finite Automation Linked to Lexical Analyser

I understand that Finite Automaton is a Mathematical model of a system with discrete number of input and outputs. Also the system has finite number of states.My question is how is this linked with ...
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82 views

Designing PDAs to Accept Languages

I want to design PDAs to accept the following two languages: $L_1 = \{a^ib^jc^k \mid i=j \text{ or } j=k\}$ $L_2 = $ The set of all strings with twice as many $0$s as $1$s. I am especially ...
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75 views

Prove/Disprove: $vwvw=vvww$ iff $\{v\}^*\{w\}^*=\{vw\}^*$

Let $\Sigma$ be an alphabet and $v,w\in \Sigma^*$. I'm trying to prove that: $$vwvw=vvww\quad\text{iff}\quad\{v\}^*\{w\}^*=\{vw\}^*.$$ I tried to do it by induction, with no success. Any help will ...
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38 views

Grammar derivation

Given these grammar productions: $$\begin{align*} &S\to A1B\\ &A\to 0A\mid\lambda\\ &B\to 0B\mid 1B\mid\lambda \end{align*}$$ And given string $w = 01101$ If I wanted to make a) ...
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37 views

S-grammar for this regular expression

Given this regular expression: $r = a a^* b + b^* c b$ I think this is the simple grammar, but I was getting a little lost with the productions: $S \rightarrow S_1 | S_2$ $S_1 \rightarrow a A b$ ...
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32 views

Regular Language Operation

I need to show that the given regular language is closed under the following operation. For example: AllSuffixes(L) = {v : uv in L for some u in (0+1)* } I do not ...
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78 views

Can someone explain this automaton?

I have a question about constructing an automaton for given language: $$L = \{000, 010, 100, 110\}$$ Solution for this was given below. Can anyone explain why this automaton accepts the language? This ...
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336 views

Is it true that any infinite subset of a non-regular unary language is non-regular?

My question is very similar to this: Is there a subset of a non regular language that is regular My claim is that because the subset is infinite, Myhill Nerode says that the language is not regular. ...
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129 views

Pumping Lemma to show a language is not regular

Let $\Sigma = \{a, b\}$. Use the Pumping Lemma to show that $\mathcal L = \{ a^pab^q: p < q \}$ is not regular. Not sure how to apply PL here, if someone can give some direction.
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206 views

Right-linear grammar from regular expression

I made a right-linear grammar that from this regular expression: The alphabet is: $Σ = \{a, b, c\} $ Regular expression: $r = cc^*(ba)^*bb$ My solution, it seems a little too short like I'm ...
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30 views

Language made by a regular expression

I created a language from this regular expression but I'm not sure about it, especially where I wanted to use the $w$ to express a sequence of terminals. The expression: $r = a a ^{*} (b + bb + bbb) ...
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2answers
479 views

Finite automata for any even number of a's followed by any even number of b's

I'm new to formal languages. I'm stuck with the following question. Any help is appreciated. Find finite automata for $$L = \{a^i b^j \mid i, j\text{ are even, }j\ge0\}$$ Thank you
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723 views

To design a Finite State machine

Design a FSM for a binary number in which the input is valid if no. of 0's divisible by 5 and no. of 1's divisible by 3
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161 views

Is this language regular?

Given $m,n∈Z$, A is a finite alphabet set ,and $L=\{(a^m,a^n)\}^*$ is subset of $A^*\times A^*$ . Is this language regular ? For example, is $L=\{(a^3,a^7)\}^*$ regular ? Here L is not the set ...
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214 views

Prove $L=\{ 1^n| n\hspace{2mm}\text{is a prime number} \}$ is not regular.

Prove $L=\{ 1^n| n\hspace{2mm}\text{is a prime number} \}$ is not regular. It seems to use one Lemma: Pumping Lemma.
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46 views

Given a DFA $\mathcal{M} = (S, \Sigma, q_0, \delta, F)$, is there an algorithm that finds the pumping length of $L(\mathcal{M}$)?

This question has been bugging me for a while, and I'm curious what such an algorithm would look like, if it exists. My guess is that it does exist, but I'm not sure how it would look.
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399 views

Describe a Turing Machine that accepts the language of all non-negative decimal integers that are multiples of 3.

I have exam coming up and I need help with this: Describe a Turing Machine that accepts the language of all non-negative decimal integers that are multiples of 3 Thank you :)
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46 views

Automata: Proof

Here is the problem: Consider a NFA, M = (K, Σ, Δ, s, F) with (p, a, q) ∈ Δ. Prove that (pʹ, aw) ⊢∗ (qʹ, w) for any w ∈ Σ∗, q′ ∈ E(q) and p′ with p ∈ E(p′). Thanks in advance.
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How can I prove a DFA accepts a certain mininum number of states?

We know that if there are two languages, L1 and L2, if L1 and L2 are regular, the intersection of those two is also regular. Suppose we have two machines, M1 and M2, and using them, a new machine M3 ...
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60 views

Büchi Pushdown System Accepting Run

From the following definitions: Definition (Büchi Pushdown System) A Buchi pushdown system (BPDS) is a tuple BP = (Q,S,→,Qf) with (Q,S,→) a PDS (where S is the stack content) and Qf ⊆ Q a set of fi ...
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83 views

L is a context free language over {0, 1}, prove, disprove:

cont... L is a context free language over {0, 1}, prove, disprove: L1 is a CFL over {a, b}, where L1 is the language of all words from L, that 0 is converted to a and 1 is converted to bba. Thanks ...
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92 views

Prove min(L) = all words in L that they don't have any prefix of themselves in L

We define the minimal words language of $L, \min(L)$, to be the language of all words in $L$ that don't have any prefix in $L$. Assume $L$ is regular language. I need to prove by building an ...
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3answers
143 views

showing that a regular language is regular after taking a letter off or after adding letters

I'll be happy to recieve help with this one: Given the regular language $L$ defined over alphabet $\{a,b\}$, show that the following languages are also regular: $\{xy\mid xay\in L\}$ ...
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2answers
482 views

Proof Two Grammars Generate the Same Language

I have two right linear grammars and I need to prove they both generate the same language. What is the right way to do it? L1: $S \rightarrow 0A$ $S \rightarrow 1B$ $A \rightarrow 0A$ $A ...
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487 views

Creating a minimal dfa from a regular expression

Having a bit of difficult with the following question: Create a minimal dfa for the language $L(r)$ where $r = a^*\bigl((ab+b)^*\bigr)$?
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84 views

What is the difference between the automatas for the regular expressions (a + b)* and (a* + b*)?

I know that the automaton for the regular expression (a + b)* will just have one state, where the initial state = the accepting state and there is one edge going into that state labelled a,b. Sorry, ...
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157 views

Proving that for every context-free language there exist a pushdown automata $M$ s.t. $L=L_{e}(M)$

The book I am reading have proof for the statement Every context-free language there exist a pushdown automata $M$ s.t. $L=L_{e}(M)$ For the case $\epsilon\not\in L$. The proof uses greibach ...
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1answer
216 views

Proof related to a finite state machine

I have this confusion related to a finite state machine M such that if the number of states n>=2, then there exits i $ \overset{i}\equiv{}= {\overset{i+1}\equiv{}}$ I mean the $i^{th}$ equivalence ...
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119 views

Finding if two machines can be equivalent

I have this problem: Consider the following machines M1 and M2. M1 has initial state A and the initial state of M2 is unspecified. Can the machines be made equivalent by the correct choice of ...
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1answer
126 views

Help with set notation?

I want to describe the set of all words in the following format: a0w1 where a represents EITHER 0 or 1, and w represents {0,1}* So 00011 is valid as is 1010011, etc. etc. I'm really new to set ...
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79 views

something that looks sort of symmetrical but also not

Given the set $S_0$ of finite binary strings whose digit sum is congruent to 0 mod 2 and the set $S_1$ of finite binary strings whose digit sum is congruent to 1 mod 2, what are the implications of ...
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62 views

Modelchecking on Automata, $\phi$ not SAT and $\phi \models$ False

Given a formula $\phi$ Is $\phi \models FALSE$ equivalent to $\phi$ not SAT? Or does $\phi \models FALSE$ means that $\phi$ is never $TRUE$ and $\phi$ not SAT means, that there existst at least one ...
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229 views

Proving that a grammar generates a language

Since every context free grammar is equivalent to a Push down automaton, to show that a grammar $G$ generates a language $L$, is it sufficient to draw a PDA equivalent to $G$ and then show the PDA ...
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319 views

How do I write this formally?

For every number N in a sequence of numbers it is true that each odd N is followed by 0 or more other numbers (not including 0) then the number N+1. How do I write this formally? This is my attempt ...
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72 views

How many neighbours should a cell have in a cellular automata?

So, I'm currently working with cellular automata but I started to wandered, what's the perfect amount of neighbours each cell should have if I'm working in a bi-dimensional space? Up to now I was ...
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1k views

Extended transition function of a DFA - a proof

I would like to write a proof of the following statement $$ \delta^+(q,PQ) = \delta^+(\delta^+(q,P),Q) $$ $\delta^+$ - Extended transition function I have to do it by induction. However, I'm not ...
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126 views

How to represent a formal proof

I know how I want to do the proof, but I don't know how to represent it. It's an automata proof, so all I need to do is show that it is regular. How could I represent a DFA as a copy? So I have the ...
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878 views

Find the DFA for the language $L = \{a^nb: n \geq 0\} \cup \{b^na : n \geq 1\}$

Problem Find the DFA for the language $$L = \{a^nb: n \geq 0\} \cup \{b^na : n \geq 1\}$$ This is a problem from the book "An Introduction to Formal Languages amd Automata 4th edition", ...
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289 views

Complementary language of a context free grammar

First post on Mathematics ;) I'm stucked with a problem related to automata theory / formal grammars. The problem ask the student to design a Pushdown automaton that accepts the complementary ...