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

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224 views

Properties of a valid DFA

Is a DFA required to have transitions on each input symbol from each state defined? If there isn't a path from state q1 to another state on input a for example, does that invalidate the DFA itself. ...
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1answer
224 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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1answer
835 views

If L is regular, so is $L-\{λ\}$?

A language is regular, by definition, if you can create a DFA for it. Then I need to prove that if $L$ is regular, then so is $L-\{\lambda\}$ for any $\lambda\in L$. Any ideas?
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1answer
388 views

Eilenberg's rational hierarchy of nonrational automata & languages

In the preface to his very influential books Automata, Languages and Machines (Volumes A, B), Samuel Eilenberg tantalizingly promised a Volume C dealing with "a hierarchy (called the rational ...
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1answer
132 views

Difference between $(a|b)^\ast$ and $a^\ast b^\ast$?

What is the difference between $(a|b)^\ast$ and $a^\ast b^\ast$? Can you show more examples of Kleene star and patterns and explain a little bit? I've searched so many sites in Google, but it returns ...
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1answer
281 views

Is it always possible to convert a non-deterministic PDA to a deterministic one?

Is it always possible to convert a non-deterministic PDA to a deterministic one? What is the significance of this observation for the computing power of contex-free grammars?
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1answer
508 views

Turing Machine Vs Linear Bounded Automata

Example of language accepted by Turing Machine but not by Linear Bounded Automata ? Is there any EXPSPACE language?
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2answers
410 views

A magic trick with synchronizing words

See the following magic trick. http://www.speedyadverts.com/SAEntertainment/html/realmagic4.html Spoiler Alert Believe it or not, the lady didn't really read your mind; she is not even a real lady ...
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2answers
193 views

Density of black cells in rule 110

Is there a way to compute the limit of the ratio (number of black cells)/(number of white cells), in the rule 110 or rule 30 automaton? With initial state = 1 black cell. Simulation of first 120000 ...
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1answer
67 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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3answers
173 views

Word problem in a free group

Can the word problem in a free group be solved by a finite state automaton? I know it can be solved by a pushdown automaton.
4
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1answer
132 views

Minimal DFA satisfying a finite view of a language.

Say one has a language $L \subseteq \Sigma^*$, but one doesn't know what strings are actually part of the language. All one has is a finite view of the language: a finite set of strings $A \subseteq ...
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1answer
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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2answers
432 views

Help in constructing a DFA equivalent to this NFA

First post here, woot. I've been a member of Stack Overflow for a while, so hopefully you guys are just as friendly! I'm having issues converting simple NFAs to DFAs... I just don't get it. ...
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2answers
419 views

String matching automata preprocessing

I have an alphabet A = {a,b,c} and a pattern P = "abcaab". The task is to build a finite automaton of the transition function (delta) for {0,6} (the length of the pattern) and each element of the ...
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2answers
68 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
161 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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1answer
769 views

Nondeterministic PDA to Deterministic PDA

Are there any resources on how to convert a non-deterministic PDA to a deterministic one, if a deterministic PDA actually exists? Or is there a step by step way on how to do this, kind of like going ...
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1answer
172 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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2answers
225 views

How can I prove that this language is regular?

Given a language $L$, define the language $K$ as the language $L$ where every second character is replaced with a $\#$. (Note: $\#$ is not part of the alphabet of $L$.) For example, if $L = \{ab, ...
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1answer
975 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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2answers
525 views

NFA to DFA conversion, half the power set

Is there a way to tell when a NFA will use at least half the power set when converted to a DFA. I tried to create a few examples, but i just can't see a pattern that would say whether an NFA will use ...
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3answers
148 views

Having trouble with “languages”

Okay, so I come across a question: What language is represented by this regular expression: $$(((a*b)b) \cup b)$$ An example given prior to this is: L(anguage) = $\{w | w \in \{0,1\}\}$ L(anguage) ...
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1answer
653 views

Unable to construct Context-free Grammar from Pushdown Automaton

I have a problem in constructing a Context-free Grammar for the Language $$L = \{a^mb^n : m≠n,m>0,n>0\} .$$ Though I can able to construct a Pushdown Automata. I can construct a CFG, but it ...
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1answer
115 views

What is this operation between sets {a, b}{c, d} do?

Not sure what this operation does, which is why i'm on here. It's not the cartesian product and no idea what it's called. I need to know to prove: For any language L, (Null set)L = L(null set) = ...
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3answers
274 views

What is the language of this DFA?

How would you write the language for this DFA as L(M) = {...}? I think in English I would say L(M) is defined as {a,b}* ending in b, ba or aa.
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2answers
122 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 ...
2
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1answer
477 views

Construct PDA that accepts the language $L = \{w_1cw_2 : w_1, w_2 \in \{a, b\}^*, w_1 \neq w_2^R\}$

Problem Construct PDA that accepts the language $L = \{w_1cw_2 : w_1, w_2 \in \{a, b\}^*, w_1 \neq w_2^R\}$ For the language $wcw^R$, it's much easier because the stack is always empty after ...
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2answers
2k views

Construct PDA that accepts the language $L = \{ a^nb^{n + m}c^{m}: n \geq 0, m \geq 1 \}$

Problem Construct PDA that accepts the language $L = \{ a^nb^{n + m}c^{m}: n \geq 0, m \geq 1 \}$ My initial idea was, If we read an $a$ push a $x$ onto stack If we read a $b$, there are two ...
4
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1answer
586 views

Question regarding stack operation notation in PDA

I'm currently reading two books: An Introduction to Formal Languages and Automata, 4th Edition by Peter Linz. Introduction to the Theory of Computation, 2nd Edition by Michael Sipser. What ...
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1answer
192 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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2answers
156 views

Is it possible to prove that $L$ is a regular language?

Let $L = \{a^{f(m)} | m \geq 1 \}$ where $f: \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ is monotonically increasing and complies that for all $n \in \mathbb{Z}^+$ there is $m \in \mathbb{Z}^+$ such that ...
3
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3answers
696 views

Is the language of all strings over the alphabet “a,b,c” with the same number of substrings “ab” & “ba” regular?

Is the language of all strings over the alphabet "a,b,c" with the same number of substrings "ab" & "ba" regular? I believe the answer is NO, but it is hard to make a formal demonstration of it, ...
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4answers
10k views

Convert from DFA to NFA

For this language $\{ w | w \text{ contains at least three } 1's \}$, its DFA diagram is defined as follows: While trying to convert it to NFA, but I realized that its NFA would be identical to ...
2
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1answer
402 views

Understanding and using the transfer-matrix-method

Let $G = (V,E,\Phi)$ be a weighted directed graph and $\mathcal{W}' : E \rightarrow \mathbb{C}$ the weighting. Let additionally $m = \# V$, $E_m$ the $m \times m$ identity matrix. Let $v,w \in ...
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3answers
2k views

How to prove two regular expressions are identical in mathematical way?

I'm currently working on "regular expression" exercises in the textbook ("An Introduction to Formal Languages and Automata"), and the problem that I'm facing is, most of the time, my solution is ...
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2answers
689 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", ...
3
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3answers
838 views

How to compute the transition function in non-determinism finite accepter NFA?

I'm currently teaching myself Automaton using Peter Linz book - An Introduction to Formal Languages and Automata 4th edition. While reading chapter 2 about NFA, I was stuck this example (page 51): ...
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1answer
776 views

Proving regular expressions to be equivalent

I'm trying to prove that two regular expressions are equivalent. I mean prove in the rigorous sense of the word (i.e. this is a legit proof). The process is to show that R1 is a subset of R2, and ...
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1answer
241 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 ...
2
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1answer
208 views

Generating all words in a language from CFG

I have a non-ambiguous context-free grammar. Is there some standard algorithm to create list of all the words in the language the CFG defines? This can be done with an abvious brute-force search by ...
4
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1answer
76 views

Enhancing the monoid structure over a finite alphabet to prove Arden's rule

Suppose you have a finite-state, deterministic automaton, that you wish to convert to a regular expression. A common method, perhaps easier to apply by hand that Yamada's algorithm, is to reduce the ...
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0answers
456 views

Regular Expression Simplification

Hi my question about Theory of Computation is: Let $M$ be defined as follows $$M = (K; E; s; D; F )$$ for $$ \begin{align*} K &= \{q_0, q_1, q_2, q_3\}, \\ E &= \{a, b, c\}, \\ s &= ...
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1answer
2k views

How to show that $ALL_{DFA}$ is in P

How can I show that $ALL_{DFA}$ is in P ? $ALL_{DFA} = \{ \langle A \rangle \mid A \text{ is a DFA and } L(A) = \Sigma^* \}$
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2answers
263 views

Is this DFA correct

I'm supposed to construct a DFA which accepts { w | w is a word except 'aa' and 'aaa' } Is this the correct solution? The thick line state is supposed to be the end state. EDIT Sry, somehow ...
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2answers
3k views

Understanding $\epsilon$ transitions in a finite state automaton

I am trying to understand how $\epsilon$ transitions work. From what I've read, when you "go" to a state S that has arrows pointing outwards with $\epsilon$'s in it, you automatically go to those ...
2
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1answer
790 views

Nondeterministic Finite Automata to Deterministic Finite Automata?

I am unfamiliar with the general process of converting NFA to DFA. I have general understanding of the theory, but I don't have the method established. Please help explain the process required to ...
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5answers
5k views

Program for working with DFA/NFA/PDA?

I'm looking for a program or utility to help construct DFA/NFA/PDA etc. and the like. The main reason I'm interested in this, is because I want to be able to move the various states around without ...