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

learn more… | top users | synonyms

19
votes
2answers
485 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 ...
16
votes
3answers
22k 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.
9
votes
2answers
578 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 ...
9
votes
1answer
362 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; ...
8
votes
1answer
284 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 ...
7
votes
3answers
7k 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
231 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
446 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
votes
2answers
199 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 ...
6
votes
1answer
112 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
4answers
73 views

Can this set of rules perform all Boolean operations?

I never worked in this field before, I just thought about this set of rules and never saw something similar before. I apologise if I don't use the right mathematical vocabulary for my question. ...
6
votes
1answer
322 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
votes
3answers
2k 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
2answers
3k views

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

How can I show that $ALL_{DFA}$ is in P ? $ALL_{DFA} = \{ \langle A \rangle \mid A \text{ is a DFA and } L(A) = \Sigma^* \}$
5
votes
2answers
653 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
83 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
140 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
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
2answers
3k views

Finite automaton that recognizes the empty language $\emptyset$

Since the language $L = \emptyset$ is regular, there must be a finite automaton that recognizes it. However, I'm not exactly sure how one would be constructed. I feel like the answer is trivial. ...
4
votes
4answers
8k 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 ...
4
votes
3answers
1k 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
3answers
131 views

Number of states in a finite automaton

How many states are required by a deterministic finite automaton to store $m$ words each of length $n$? I came across $2^{mn}$ as the solution but there was no explanation.
4
votes
2answers
504 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
314 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
2answers
179 views

context free grammar problem

$L$ is the context free grammar over $\{a, b\}$ $S \rightarrow aSb \; | \;bR \; |\;Ra$ $R \rightarrow bR \;|\;aR\;|\;\epsilon$ Briefly describe this CFG with English sentences and prove your ...
4
votes
1answer
130 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
830 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
109 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
1k 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
397 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
676 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
121 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
2answers
62 views

Turing Machine Decidability

I have been working on this problem for few hours, but haven't been able to come up with a solution : Is the following problem decidable? Given a TM M, whether there is a w such that M enters each of ...
4
votes
1answer
767 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
82 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 ...
4
votes
0answers
125 views

Designing a turing machine for primality check.

I am designing some turing machines, so far I have made Binary Addition and subtraction. Now I've been thinking that what if turing machine can check if the number is prime or not. Lets suppose we ...
4
votes
0answers
82 views

This proof in my textbook involving the pumping lemma appears incorrect - is it?

It states Let $B$ be the language $\{0^n1^n2^n | n \geq 0\}$. We use the pumping lemma to prove that $B$ is not regular. The proof is by contradiction. Assume to the contrary that $B$ is regular. ...
4
votes
1answer
69 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
86 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 ...
3
votes
3answers
815 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
4answers
15k 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 ...
3
votes
2answers
71 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
135 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
494 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
votes
2answers
96 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
3answers
84 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
3answers
295 views

Matrix representation of Automata

Is anyone know if there is any tutorial for the matrix representation of automata?? I am taking a theoritical computer science in this semester and the professor uses the matrix in his lecture. I ...
3
votes
1answer
184 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
427 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, ...