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

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14
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1answer
418 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 ...
9
votes
2answers
456 views

Why is it undecidable whether two finite-state transducers are equivalent?

According to the Wikipedia page on finite-state transducers, it is undecidable whether two finite-state transducers are equivalent. I find this result striking, since it is decidable whether two ...
8
votes
3answers
11k views

Difference between NFA and DFA

In very simple terms please, all resources I'm finding are talking about tuples and stuff and I just need a simple explanation that I can remember easily because I keep getting them mixed up.
8
votes
1answer
351 views

Proof that an automaton stops

I discovered an automaton that produced interesting, random-seeming patterns. The rule was as follows. Given a grid of points, some occupied and some not, and a current point and a previous point; ...
7
votes
3answers
5k views

Why does this FSM accept binary numbers divisible by three?

This final state machine accepts binary numbers that are divisible by three. In theory the states should equal to the value $n$ mod $3$, but how does this work for binary numbers? What I don't get ...
7
votes
1answer
131 views

Challenge on Some Language and PDA

Suppose We have Some language as follows: $L_1=\{w^* | w=x \text{ and } x \in \Sigma^*\}$ $L_2=\{ww^R ww^R | w \in ( \Sigma + \Sigma)^*\}$ $L_3=\{w | w=xy, x,y \in \Sigma^*, y \text{ is a ...
7
votes
1answer
214 views

Does there exist a universal pushdown automaton?

Let $\Sigma$ be a fixed alphabet and let $PDA(\Sigma)$ be the set of all Push-Down-Automata (PDA's) having input alphabet $\Sigma$. Is there an alphabet $S$ and a function $f:PDA(\Sigma) \to S^∗$ such ...
7
votes
2answers
429 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 ...
7
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2answers
197 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 ...
7
votes
1answer
270 views

What is the class of languages accepted by DFAs whose transition monoids are transitive permutation groups?

In the Wiki page A permutation automaton, or pure-group automaton, is a deterministic finite automaton such that each input symbol permutes the set of states. ..... A formal language is p-regular ...
6
votes
1answer
106 views

Existence of NFA for this language

I'm given a task to find (and prove) such language $L$ in the alphabet $\Sigma = \{a,b\}$ with all words less than $1000$ in length, for which any DFA/NFA will have more than $10^{10}$ of states. For ...
6
votes
1answer
310 views

Automata theory on infinite words: any video lectures?

I am fun of automata theory. Can you suggest good video lectures on the subject? (there is a good one here, but it is accessible from RWTH University only)
5
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3answers
1k views

Finite Automaton that accepts only the words baa,ab, abb and no other strings longer or shorter

I am trying to understand the answer here for FA that accepts only the words baa, ab, abb and ...
5
votes
1answer
79 views

Is there any relationship between the bounding box and the period of an oscillator in the Conway's Game of Life?

Is there any relationship between the bounding box and the period of an oscillator in the Conway's Game of Life? In particular I am interested in this case: what is the maximum period for an ...
5
votes
1answer
523 views

Connecting finite automata and regular languages in teaching/applications

I am considering giving a presentation to middle schoolers, aged about ten to fourteen, about finite automata and regular languages. Average American students have no problem with uses of the ...
5
votes
1answer
28 views

$L=${$a^nb^nc^n : n \geq 0 $} CFG Recognizing

Suppose $L=${$a^nb^nc^n : n \geq 0 $} and I. $h(L), h(a)=a, h(b)=bb, h(c)=b$ II. $L^R$ III. $L^*$ IV. $h(L), h(a)=a, h(b)=bb, h(c)=a$ Why just I is a CFG and other is not? anyone can help me to ...
5
votes
2answers
71 views

Language of a grammar vs regular expression vs nfa

I read some note about Automaton Course. i see this note, that following all is the same. but i think the L(g) is not equal to NFA and regular expression. anyone could help me with defining the ...
4
votes
3answers
3k 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 ...
4
votes
1answer
2k views

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

How can I show that $ALL_{DFA}$ is in P ? $ALL_{DFA} = \{ \langle A \rangle \mid A \text{ is a DFA and } L(A) = \Sigma^* \}$
4
votes
3answers
990 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): ...
4
votes
2answers
428 views

Push down automata problem

Informally describe the Nondeterministic PDA that generates: $$\{x\#y\ \mid x,y\in\{a,b\}^{*}\text{and}\space x\ne y\}$$ My initial plan was to use nondeterminism to go through each character before ...
4
votes
3answers
256 views

Deciding equivalence of regular languages

Given two regular expressions $R$ and $S$ on an alphabet $\Sigma$ it is possible to decide their equivalence as follows: build two finite automata $M_R$ and $M_S$ such that $L(R) = L(M_R)$ and $L(S) ...
4
votes
1answer
124 views

Is $(p,\epsilon,p)$ a path of an automaton?

$A$ is an alphabet. An automaton over $A$ can be defined as a set $A_0 = (Q, E, I, T),$ where $Q$ is the set of states, $E \subseteq Q \times A \times Q$ is the set of edges or transition, $I, T ...
4
votes
1answer
674 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 ...
4
votes
2answers
104 views

Have action/predicate systems (or similar) been considered in the literature?

Question. Has the following concept, or anything similar, been considered in the literature? Definition. An action/predicate system consists of sets $A$ (the actions) and $X$ (the predicates) such ...
4
votes
1answer
777 views

Give a push down Automata for this language: the length of is odd and it's middle symbol is 0

Give a push down automaton for this language: {w| the length of w is odd and it's middle symbol is 0} Here is the CFG I wrote for this language: ...
4
votes
1answer
80 views

Turing Machine Problem

We know, A Turing machine is a hypothetical device that manipulates symbols on a strip of tape according to a table of rules I Draw a TM for input $x=(0+1)^*$ i want to implement ...
4
votes
1answer
379 views

The shortest word in context free language

Let $G=(\Sigma,N,R,S)$ be a context-free grammar. For every production rule A --> w, we say that its length is $r$ if $|w|=r$. In addition $n = |N|$, and $k =$ the maximal length of a production rule ...
4
votes
1answer
671 views

Routing Automaton

Is there a formal proof for the following question? For a DFA $M= (Q,\Sigma,\delta,s,A)$, we extend the function $\delta : Q \times \Sigma^* \to Q$, such that every $w \in \Sigma^* $, ...
4
votes
2answers
119 views

Number of states required to recognize $\{ ss : s \in \{ 0 , 1 \}^*, |s| = i \}$ and its complement

$$\Sigma = \{0,1\}\;\\ S_{i} = \left\{ss: s\in {\Sigma}^{*} \text{and $s$ has length $i$}\right\}$$ Prove that for any $i$, any DFA recognizing $S_{i}$ must have $2^{i}$ or more states. Design a ...
4
votes
1answer
67 views

Gliders, static structures in various (dynamic) systems

Structures, i.e. symmetries over time, appear in various systems: gliders in cellular automata, like Game of Life or Rule 110, unmatched string's parts in rewrite systems – unchanged in multiple ...
4
votes
1answer
596 views

Drawing a PDA for a language

I am initiating myself into TOC and using sort of random resources from the web. I was looking at this problem from a Berkeley problem set: Construct a PDA to accept $$ L = {a^ib^j|i \neq j , 2i ...
4
votes
1answer
134 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 ...
4
votes
0answers
50 views

predicate logic with assumption NP $\neq$ CO-NP?

Anyone could describe why: Set of All Tautology in propositional logic with assumption NP $\neq$ CO-NP is CO-NP Complete. Thanks. I ask it here before: Is the language of tautologies NP-complete? ...
4
votes
1answer
61 views

Proving that a language is not context-free

Given the language $$L = \{ a^p \mid p\, \text{IS NOT prime} \}$$ is $L$ Context free? If not, prove that it's not. May I have some suggestions on how to use the pumping lemma to prove this, ...
4
votes
2answers
80 views

Showing that 2 languages are context free

I have these 2 languages: $$L_1 = \left\{a^ib^jc^k: k\ge i+j\right\}\\ L_2 = \left\{w_1cw_2 : w_1,w_2\in\{a,b\}^\ast\land |w_1|_a = |w_2|_a\right\}$$ How can I determine that they are context free ...
4
votes
1answer
78 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 ...
3
votes
3answers
748 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, ...
3
votes
2answers
66 views

Does $L=\{(\langle M \rangle,k)| M$ is a TM and $\exists w\in \sum^*$ s.t when $M$ runs on $w$, $M$ visits some state at least $k$ times$\} \in R$?

I'd like your help with understanding , how come the following language is decidable (in $R$): $L=\{(\langle M \rangle,k)| M$ is a TM and $\exists w\in \sum^*$ such that when $M$ runs on $w$, $M$ ...
3
votes
3answers
128 views

Showing $L=\{uw \mid \exists v:uv\in L_{1},vw\in L_{2}\}$ is regular

Let $L_{1,}L_{2}$ be regular languages and define $L:=\{uw \mid \exists v\in\Sigma^{*}:uv\in L_{1},vw\in L_{2}\}$. I wish to prove that $L$ is regular using only closure properties (such as ...
3
votes
2answers
406 views

Question about regular languages and finite automata

We say a language $L$ is regular if it is accepted by some finite automaton $M$. I would like someone to clarify this definition. Given a finite automaton $(Q, \Sigma, \delta, q_0, F)$, we define the ...
3
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2answers
94 views

If $L\in REG$ then $M$ has a finite number of distinct rows

Let $L \subseteq \Sigma^{\star}$ and let $M^{\Sigma^{\star} \times \Sigma^{\star}}(\{0,1\})$ an infinite matrix such that for each $x,y\in \Sigma^\star$: $$ m_{x,y}=\begin{cases} 1 & x y\in L\\ 0 ...
3
votes
4answers
6k views

Intersection of two deterministic finite automata?

I'm trying to solve a problem where I have to create a DFA for the intersection of two languages. These are: $$\{s \in \{{\tt a}, {\tt b},{\tt c}\}^\ast : \mbox{every ${\tt a}$ in $s$ is ...
3
votes
3answers
77 views

Is $L = \{(x,y,z) | x+y=z\}$ a regular language?

Suppose $x,y,z$ are coded as decimal or their binary representations in an appropriate DFA. Is $L$ regular? My intuition tells me that the answer is no, because there are infinitely many combinations ...
3
votes
1answer
174 views

Context Free Language? Proving through grammar?

I need help solving this question: Is $L = \{ w \in \{a,b,c\}^* \mid n_a(w) = n_b(w) = 2n_c(w)\}$ a context-free language? That is the number of $a$'s equal the number of $b$'s equal twice the ...
3
votes
1answer
328 views

Would the following NFA accept all strings?

The question asks the following: "Let N be a nondeterministic finite automaton with s states. Suppose than N accepts all strings of length s or less. Does it follow that N accepts all strings? (If so, ...
3
votes
2answers
509 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. ...
3
votes
1answer
103 views

reverse automata mininum states

There is a formal proof for the following sentence? For every $k$ there is a DFA (deterministic finite automaton) $M$ with $k+2$ states such that every automaton that accepts the language $L(M)^R$ ...
3
votes
2answers
169 views

Automata: 1=2, 2= 26, 3=1054, 4=5768, 5 =139314069504, 6 = ???

I am in my own Automaton (finite-state deterministic automata) research, so i have four sets of automata. 2 states automata, 3 states, 4 states and 5 states. Input alphabet $\{0,1\}$ so... the ...
3
votes
2answers
135 views

Is this proof using the pumping lemma correct?

I have this proof and it goes like this: We have a language $L = \{\text{w element of } \{0,1\}^* \mid w = (00)^n1^m \text{ for } n > m \}$. Then, the following proof is given: There is a $p$ ...