Questions tagged [regular-language]

Questions about the formal languages that can be described by regular expressions (in the sense of Kleene), or, equivalently, the languages that can be accepted by finite automata.

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3 votes
1 answer
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Subset of regular languages

I have a system that is deciding a subset of regular languages and am curious if anyone has seen this type before and if it has a name I could use to research more. Specifically consider the ...
  • 293
11 votes
1 answer
150 views

Check whether DFA accepts majority of words less than a cutoff with another DFA

Question Let $M$ be some DFA that reads integers in base $k$. Does there always exist some other DFA $M'$ that also reads integers in base $k$, where $M'(x)$ accepts if and only if $M$ accepts the ...
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3 votes
0 answers
47 views

Reference for cost of translating between regular language formalisms

It is well-known that regular languages can be defined equivalently via many formalisms, among which regular expressions, NFAs, finite monoids, Monadic Second-Order logic (MSO). The cost (say in size ...
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11 votes
1 answer
282 views

Existence of injective length-preserving rational function to a smaller alphabet

(This is a simpler rephrasing of an earlier question I have since deleted.) Definitions For this question, a finite-state transducer is like a standard NFA, except at each transition, the transducer ...
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0 votes
0 answers
44 views

Capturing a particular regular language with $O(m)$ states

In dx.doi.org/10.1006/inco.2001.3069 the authors defined $NID_m = \{ u\in \{0,1\}^* | \exists i : u_i \neq u_{i + m} \}$ and claimed it could be recognized by a NFA of size $O(m)$. The paper mentions ...
6 votes
1 answer
167 views

star height of star-free languages

I'm interested in the (restricted) star-height of star free-languages. Recalling the definitions: the star height $h(\mathtt{e})$ of a regular expression $\mathtt{e}$ is $0$ if $\mathtt{e}= \...
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5 votes
1 answer
152 views

Regular Expressions that converts into unambiguous automata

Brüggemann-Klein and Wood (1992) proved that a certain kind of regular expressions, that they call “Deterministic Regular expressions”, when converted into automata using the Glushkov's Construction, ...
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5 votes
2 answers
199 views

Reference request: An algebraic characterisation of LTL[XF]-definable word languages

I'm looking for a reference to the fact that LTL[XF]-definable languages (LTL where only the (strict) finally/future modality is allowed) correspond to the variety $\mathbf{R}$ (see: 1). A similar ...
6 votes
1 answer
178 views

Is the function $f(a_1 \dotsm a_n) = a_1(a_1a_2)(a_1a_2a_3)\ \dotsm\ (a_1 \dotsm a_n)$ regularity-preserving?

A function $f: A^* \to A^*$ is regularity-preserving if, for each regular language $L$ of $A^*$, the language $f^{-1}(L)$ is regular. I think I have a proof, as a consequence of more general results, ...
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2 votes
0 answers
88 views

Necessary and sufficient condition for an infinite tree to be context-free

A Buchi automaton is non-empty iff it accepts an infinite word of the form $uv^\omega$ (here $u,v$ are finite words). In other words, if $\{w\}$ is an $\omega$-regular language, then it is of that ...
  • 193
2 votes
1 answer
219 views

DFSA and NFSA intersection problem

Given $k$ deterministic FSAs of $n$ states the intersection of their languages is empty is decidable in $n^{o(k)}$ time is an open problem. For unbounded $k$ it is known the problem is $PSPACE$ ...
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2 votes
1 answer
131 views

NP-Completeness of Finding Minimum Automaton, in Gold's paper

I have been investigating "learning automatas", and I came across reference to Gold's papers several times: "Complexity Of Automaton Identification From Given Data", and "...
-1 votes
1 answer
321 views

What is the time complexity of computing intersection and union of Nondeterministic Finite Automata (NFAs)?

Assume that $\mathcal{A} = (Q_A, \Sigma, \Delta_A, q_{i_A}, F_A)$ and $\mathcal{B} = (Q_B, \Sigma, \Delta_B, q_{i_B}, F_B)$ are two NFAs. What is the worst-case time complexity of computing $\mathcal{...
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5 votes
1 answer
95 views

Nonterminal descriptional complexity of regular languages

Recently I became interested in grammar complexity of regular language. Prior to searching for literature, I tried to investigate it on my own, proving two lemmas from comment below. I am aware of an ...
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12 votes
0 answers
187 views

Regular languages accepted by an automaton with at most one transition per letter

I'm interested in the (very restricted) subset of regular languages for which there is an automaton having the following property: for every letter $a$ of the alphabet, the automaton has at most one ...
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2 votes
1 answer
157 views

What graphs on $\mathbb{N}$ can be encoded as regular languages?

Suppose I represent the natural number 0 by "x", and use the symbol "s" for successor so that I get the following encoding of $\alpha : \mathbb{N} \rightarrow V$ of natural numbers ...
1 vote
0 answers
29 views

Decidability of regular partition construction given its existence

Let $G = (N,T,P,S)$ be a context-free grammar where $T,N$ are sets of terminals and nonterminals respectively, $P$ contains all the productions of the grammar, and $S \in N$. If we know that $G$ is LL(...
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1 vote
0 answers
117 views

Bounds on the construction of regular expressions' intersection operator

There are references on the exponential worst-case of the intersection operator for regular expressions (see [1]). However, I was wondering if there are similar results for the construction process ...
user avatar
13 votes
1 answer
343 views

Planarity of planar finite automata intersection

It was shown that any regular language can be specified by planar $\varepsilon$-free nondeterministic finite automaton (Bezáková, Ivona, and Martin Pál. "Planar finite automata."). Is it ...
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3 votes
1 answer
132 views

Rewrite relations - proof of correctness

Let $T \subseteq \Sigma^* \times \Sigma^*$ be a regular relation. We define the obligatory rewrite relation over $T$ as follows: $$ R^{obl}(T) := N(T) \cdot (T \cdot N(T))^* $$ $$ N(T) := Id(\Sigma^* ...
7 votes
1 answer
323 views

Kleene Algebra for star-free regular expressions

TLDR: Is there a notion of Kleene Algebra for star-free regular expressions? Kleene Algebras are algebraic structures that are equivalent to regular expressions. A Kleene Algebra is an idempotent ...
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6 votes
2 answers
590 views

2DFA to 1DFA - Converting two way deterministic finite automata to one way deterministic finite automata

How can I convert a 2DFA to a normal DFA. Is there an algorithm/elegant way to do that ? I've been researching this for a few days but I coundn't find anything. Actually I want to implement that in ...
10 votes
1 answer
693 views

What class of languages is recognized by finite-state automata with $k$ heads?

A DFA or NFA reads through an input string with a single head, moving left-to-right. It seems natural to wonder about finite-state machines that have multiple heads, each of which moves through the ...
6 votes
1 answer
213 views

Complexity of DFA intersection in this specific case?

In general, the size of the DFA that recognizes the intersection of $n$ languages is exponential in $n$. However, in my case I am computing the intersection of a very restricted subset of possible ...
7 votes
1 answer
175 views

Arranging letters to make a word in a regular language

Fix a regular language $L$ on the alphabet $\{a, b\}$, and consider the following problem. I am given as input: some number $m \in \mathbb{N}$ of copies of the letter $a$, and some number $n \in \...
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0 votes
1 answer
126 views

Why are all finite languages regular? [closed]

It is said that "All finite languages are regular". But the Pumping Lemma says that, if a language is regular one can find a 'large-enough' word w such that it can be decomposed into w = xyz such ...
2 votes
1 answer
391 views

Oncina-Garcia RPNI algorithm for learning DFAs

The question refers to this paper: ftp://altea.dlsi.ua.es/people/oncina/articulos/asspr1992.pdf Given a sample of $p$ positive and $n$ negative strings, RPNI constructs a consistent DFA in time $O((p+...
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8 votes
0 answers
101 views

Does ${\bf CFLPAD}={\bf PPAD}$?

What happens if we define ${\bf PPAD}$ such that instead of a polytime Turing-machine/polysize circuit, a (non-)deterministic finite/push-down automaton encodes the problem? I asked a similar ...
  • 13.8k
1 vote
0 answers
47 views

Question About Turing Machine Computability [closed]

If p is a Turing machine then L(p) = {x | p(x) = yes}. Let A = {p | p is a Turing machine and L(p) is a finite set}. Is A computable? Justify your answer. So I'm trying to figure out how to solve ...
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14 votes
2 answers
713 views

Automata learning without counterexamples

In Angluin's automata learning framework, a student aims to learn a regular language $L\subseteq \Sigma^*$ by asking two types of questions to his teacher: Word queries: given $w\in \Sigma^*$, is $w\...
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9 votes
0 answers
264 views

Shortest string in the intersection of regular languages

Inspired by https://codegolf.stackexchange.com/questions/53310/shortest-universal-maze-exit-string Each of the 138,172 valid mazes can be represented as a DFA with 9 states (including starting and ...
0 votes
1 answer
675 views

What is the practical importance of making or using a Turing complete language? [closed]

I get what a Turing machine is and what language is a Turing-complete language but when someone introduces me to a new programming language (like Solidity) and says it is Turing complete, what am I ...
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1 vote
0 answers
63 views

Rational power series over $\mathbb N \cup \{\infty\}$, rationality of singular part

Let $\Sigma$ be a finite alphabet, and consider the formel power series over $\Sigma$ considered as non-commuting variables with coefficients in the semiring $\mathcal N := \mathbb N \cup \{\infty\}$ ...
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3 votes
3 answers
145 views

Example of monoid $M$ such that $\operatorname{RAT}(M) \not\subseteq \operatorname{REC}(M)$

Let $M$ be a monoid, the family of rational sets $\operatorname{RAT}(M)$ is defined as the smallest set containing the finite subsets, and closed under union, concatentaion and the star operation. The ...
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9 votes
1 answer
206 views

Generalisation of the statement that a monoid recognizes language iff syntactic monoid divides monoid

Let $A$ be a finite alphabet. For a given language $L \subseteq A^{\ast}$ the syntactic monoid $M(L)$ is a well-known notion in formal language theory. Furthermore, a monoid $M$ recognizes a language $...
  • 1,947
-4 votes
1 answer
146 views

Prove that L* is a regular language [closed]

Suppose that L is any language , not necessarily regular, whose alphabet is {0}; that is the strings of L consist of 0's only. Prove that L* is regular.
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2 votes
1 answer
236 views

Existence of an algorithm

I need to show that there exists a polynomial time algorithm that inputs a deterministic automata $A$, and decides if $A$ accepts a word w if and only if it also accepts any word obtained by permuting ...
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7 votes
1 answer
230 views

Finding a minimal DFA whose language has a desired intersection with another

Suppose I have regular languages $B \subseteq A$, with corresponding (known) minimal deterministic finite automata $M_A, M_B$. I would like to find another regular language $C$ such that $B = A \cap ...
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23 votes
2 answers
543 views

Testing whether letters can be scheduled to achieve a word in a regular language

I fix a regular language $L$ on an alphabet $\Sigma$, and I consider the following problem that I call letter scheduling for $L$. Informally, the input gives me $n$ letters and an interval for each ...
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11 votes
1 answer
570 views

Parameterized complexity of inclusion of regular languages

I am interested in the classic problem REGULAR LANGUAGE INCLUSION. Given a regular expression $E$, we denote by $L(E)$ the regular language associated to it. (Regular expressions are on a fixed ...
-3 votes
1 answer
67 views

Is there a non regular and regular language where the non regular is not a subset of the regular and the union is regular? [closed]

Does there exist languages $L1$, $L2$ where $L1$ is non regular, $L2$ is regular $L1\not\subset L2$ and $L1 \cup L2$ is regular?
2 votes
0 answers
123 views

Possibly small circuit complexity class containing REG?

What is the smallest well-known Boolean-circuit complexity class containing all the regular languages over the binary alphabet {0,1}? If we believe Theorem 2 in Koucký, Circuit Complexity of ...
user avatar
2 votes
1 answer
228 views

Deformation of finite regular languages [closed]

Let $L \subseteq \{0,1\}^n$ be any finite regular language s.t it has an acyclic DFA. Let $C$ be some class of acyclic DFAs. Let $\sigma \in S_n$ be a permutation on $n$ symbols. We can apply $\...
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1 vote
0 answers
157 views

Relation between MDPs and non-deterministic finite automatons

I'm confused as to the relation (computability-wise) between markov decision processes and NFAs. Are finite state MDPs expressible as regular grammars? If so, are markov decision processes thus ...
  • 121
6 votes
1 answer
452 views

Pumping Lemma over 2 way Automaton

Pumping Lemma essentially implies that for a NFA $N$ having $N_0$ states over a unary language $L_0$ any string belonging to $L_0$ can be obtained by 'pumping' one of the strings that have a length ...
12 votes
0 answers
305 views

Reference request: exponential growth rates of subsequence-closed languages are integers

This question is migrated from MathOverflow, where it did not receive any answers a year ago. For a language $L$ over the finite alphabet $\Sigma$, let $L_n$ denote the set of words in $L$ of length $...
9 votes
1 answer
378 views

Transition monoid membership for DFAs

Given a complete DFA $A=(Q, \Gamma, \delta, F)$, we can define a collection of functions $f_a$ for each $a\in \Gamma$and with $f_a:Q\rightarrow Q$, $f_a(q)=\delta(q, a)$. We can generalize this notion ...
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8 votes
1 answer
935 views

NFA to DFA Powerset Construction : A Partial determinization algorithm with trade-off between running time and size for the resulting automata?

Given a NFA $N$ and its equivalent DFA $D$ resulting from the total determinization of $N$ (using powerset construction, for example), the following properties hold for $N$, $D$ and for any word $w$ : ...
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7 votes
1 answer
207 views

Automata : Language Containment, Minimality & Graph Homomorphism

Given two DFAs $A$ and $B$ defined on the same alphabet, a (graph) homomorphism $h:A \rightarrow B$ from $A$ to $B$ is a mapping of the states of $A$ into the states of $B$ such that : if the state $...
  • 427
10 votes
1 answer
353 views

Complexity of checking if two words have an interleaving in a language

For a fixed language $L$ on some alphabet $A$, let us consider the following problem, that I call $L$-INTERLEAVING: Input: two words $u, v \in A^*$ Output: whether there exists an interleaving of $u$ ...
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