Questions tagged [automata]

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

Automata (plural of automaton) are abstract models of machines. These machines take an input such as a string of characters and give some output, often an "accept" or "decline" value.

These machines are closely related to . Many languages can be categorized based on the type of automata that can accept them. For instance,

  • regular languages can be recognized by some deterministic finite automata.
  • context-free languages can be recognized by some pushdown automata.
  • recursively enumerable languages can be recognized by some Turing machine.

There are many different types of automata, each with slightly different definitions. A few examples are

  • Deterministic Finite Automata (DFA)
  • Nondeterministic Finite Automata (NFA)
  • Pushdown Automata
  • Infinite Automata
  • Alternating Automata

For instance, a deterministic finite automata $A$ has five parts:

  • a finite set of states $Q$
  • a finite set of input symbols called the alphabet $\Sigma$
  • a transition function $\delta: (Q\times \Sigma)\rightarrow Q$
  • an initial state $q_0 \in Q$
  • a set of accept states $F\subseteq Q$

The automaton $A$ takes in a string $\sigma$ of characters of $\Sigma$. The automaton starts at state $q_0$ and then for each letter of $\sigma$ it moves to another state according to the transition function $\delta$. Once the automaton has used every letter of $\sigma$, if its final state is in $F$, then we say that $A$ accepts $\sigma$.

For questions also about

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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.
Ogen
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What is the period of Langton's ant on a torus?

Langton's ant runs on an infinite white grid. At every white square, it turns right, flips the color of the square, and moves forward one square. At every black square, it turns left, flips the color of the square, and moves forward one square.…
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Why does this FSM accept binary numbers divisible by three?

This finite state machine (FSM) 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 is how the transitions get together because…
muffel
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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 hierarchy) of the nonrational phenomena... using rational…
David Lewis
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Interesting behavior of the $2$-tag system $\{a\rightarrow abc,b\rightarrow ab,c\rightarrow c\}$

Consider the following $2$-tag system: $$\begin{cases}a\rightarrow abc\\b\rightarrow ab\\c\rightarrow c\end{cases}$$ In brief, a $2$-tag system is a Turing-complete string substitution system wherein you check the current first letter against the…
Trevor
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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. Can someone help me out?
user68486
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Are the algebraic real numbers an automatic structure?

In the 1950's, Julius Büchi showed that $(\mathbb{N},S,+,0)$ is not merely a decidable structure as Presburger had shown, but an automatic structure, i.e. there is an encoding of the natural numbers (using finite binary strings) such that you can…
Keshav Srinivasan
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Can a regular grammar be ambiguous?

An ambiguous grammar is a context-free grammar for which there exists a string that has more than one leftmost derivation, while an unambiguous grammar is a context-free grammar for which every valid string has a unique leftmost derivation. A…
Perceptual
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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; change the current point to the unoccupied point one…
rlms
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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 that the set $\{(f(A),w) : w \in L(A)\}$ is…
Alberto
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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 finite-state automata are equivalent to one…
templatetypedef
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Automata | Prove that if $L$ is regular than $half(L)$ is regular too

I've see couple of approaches to this kind of questions yet I have no clue how to approach this one. Let L be regular language, and let $half(L)$ be: $half(L) = \{u \mid uv \in L\ s.t. |u|=|v|\}$. Prove that if $L$ is regular then $half(L)$ is…
Aviad
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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 (also: a pure-group language) if it is accepted by a…
Nobody
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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: $$\{s \in \{{\tt a}, {\tt b},{\tt c}\}^\ast : \mbox{every ${\tt a}$ in $s$ is immediately followed by a ${\tt b}$}\}$$ and $$\{s \in…
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pumping lemma: ww^R not regular

I'm trying to prove that $L = \{ww^R : w \in \{a,b\}^*\}$ ($w^R$ is the reverse of $w$) is not regular using the pumping lemma. Let $p$ be the pumping length and $s = a^pbba^p$. $x = \epsilon$, $y = a^p$, $z = bba^p \implies s = \epsilon a^p bba^p =…
David Robert Jones
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