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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4 votes
0 answers
132 views

Learning a regular language with a specified closure property

Consider an alphabet $\Sigma$, and a partial transformation function $f:S\to\Sigma^\ast$ defined on some subset $S\subseteq\Sigma^\ast$. Let $S_f$ denote the set of strings $s\in S$ such that $f^n(s)\...
10 votes
5 answers
283 views

Obscure characterizations of the regular languages

I've been collecting equivalent characterizations of the regular languages. Does anyone know of any I haven't yet found? Wikipedia has a bunch: https://en.wikipedia.org/wiki/Regular_language#...
11 votes
2 answers
284 views

Is there a simple characterization of regular languages closed under circular shifts?

A language $L$ is closed under circular shifts if, for every word $w = a_1 ... a_n$ and circular shift $w' = a_i ... a_n a_1 ... a_{i-1}$ of $w$, then $w \in L$ iff $w' \in L$. It is equivalent to ...
13 votes
1 answer
415 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 ...
3 votes
1 answer
167 views

Is it useful to "untangle" an NFA by converting to a regular expression and back

Consider the following recursive algorithm for converting a regular expression into a transition diagram for an NFA with epsilon-edges (freely, optionally traversible edges), one start state and one ...
0 votes
1 answer
77 views

What is the intution on the TTT algorithm for regular grammar inference?

This question is about the TTT algorithm for blackbox automata inference as defined in [1] and [2]. I am finding it difficult to understand all the innovations made by the algorithm. I understand how ...
2 votes
1 answer
111 views

What is the current state of the art on exact identification of DFAs with a maximum N states

This is a question about the blackbox grammar inference of deterministic finite state automata (DFAs). In particular I want to ask about when one can exactly identify the target DFA using queries to ...
7 votes
1 answer
253 views

Are regular expressions polynomially decomposable?

This question is related to my previous question (LINK). I would like to ask whether regular expressions can be polynomially decomposed in the following sense: A regular expression $\mathcal{R}$ is $...
10 votes
1 answer
332 views

The complexity of conversion from a regular expression to a nondeterminsitic automata and back after changing initial and final states

Suppose that a regular expression $\mathcal{R}$ over an alphabet $\Sigma$ is given. It is well-known that one can now construct a non-deterministic finite automaton $\mathcal{A}$ such that $\mathcal{R}...
6 votes
0 answers
72 views

Updating (minimal) DFA incrementally

Is there algorithm to incrementally update (minimal) DFA? Namely, having relatively large minimized DFA I want to update it incrementally using union and sudtraction with other (relatively small, ...
0 votes
0 answers
67 views

How to reduce a code down to its configuration

I have built a system where from atomic information of a UI code I could generate a framework specific code. Here is the concept https://github.com/imvetri/ui-editor. For example, the user of this ...
7 votes
1 answer
158 views

Complexity of the inevitability problem over monoids

I am interested in the complexity of following problem: Inevitability problem in monoids Input: two regular languages $K$, $L$ specified by finite monoids $M_K$ and $M_L$ (+ morphisms and accepting ...
7 votes
1 answer
349 views

Converting 2-ambiguous NFA to unambiguous NFA

This must be known, but somehow I can't locate a reference about this. Let $A$ be a nondeterministic finite automaton (NFA) over words of an alphabet $\Sigma$. I say that $A$ is unambigous if, for ...
1 vote
0 answers
36 views

Overlap operator for simple ( regex-like ) Patterns

( Introduction ) Some Notation lower case letters, $a, b, c$ will be used to denote single symbols Upper case letters, $P, Q, R$ will be used to denote string of symbols $a\!:\!S$ means a string ...
2 votes
0 answers
73 views

Languages free of concatenations of stars

Union-free regular languages, defined by regular expressions (using $*$, literals (alphabet symbols), and concatenation) excluding union (written $+$ or $\cup$) have been studied. Note that unions ...
3 votes
1 answer
144 views

What is the solution of this equation on regular languages?

I need to characterize this language: $$ L = \{ s \in \Sigma^* \, | \, \{s\} \cdot A_1 \subseteq B_1 \land \ldots \land \{s\} \cdot A_n \subseteq B_n \} $$ where $A_i, B_i$ are all regular languages. ...
5 votes
1 answer
116 views

Logical Equivalents of Finite State Transducers

There's a notion of "regular" function on words in automata theory that corresponds nicely to functions in WS1S/Büchi Arithmetic/the logic of words with a prefix and equal-length relation. ...
9 votes
2 answers
280 views

Defining regular language classes with disjoint union

Regular languages are typically defined using the operations of union, concatenation, and Kleene star. Likewise, there are restricted classes of regular languages defined via similar operations, for ...
9 votes
1 answer
433 views

Regular languages in lambda calculus

With Turing machines, by imposing certain restrictions on the form of the transition function, one can get a machine that accepts only regular languages. I am wondering what is the counterpart in ...
3 votes
1 answer
174 views

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 ...
11 votes
1 answer
175 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 ...
4 votes
0 answers
57 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 ...
8 votes
1 answer
449 views

What is an unambiguous language in the sense of Schützenberger?

I'm reading Thomas Wilke's survey on the connections between Temporal Logic and finite automata, finite semigroups and first-order logic. In Theorem 6 (by Kamp), the fragment $\mathrm{TL}[\mathsf{F},\...
11 votes
1 answer
288 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 ...
0 votes
0 answers
46 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 ...
14 votes
1 answer
3k views

minimizing size of regular expression

Suppose we have a regular language specified by a regex, for example, (ab|ac)* and we wish to find an equivalent regex with the minimal number of symbols, (a(b|c))*. Is there any efficient way to do ...
5 votes
2 answers
223 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
211 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}= \...
5 votes
1 answer
186 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, ...
25 votes
3 answers
4k views

Regular languages from category-theoretical point of view

I noticed that regular languages over the alphabet $\Sigma$ can be naturally thought of as a poset, and indeed a lattice. Moreover, concatenation together with the empty language $\epsilon$ defines a ...
6 votes
1 answer
187 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, ...
2 votes
0 answers
102 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 ...
2 votes
1 answer
252 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 $PSPACE$ ...
2 votes
1 answer
182 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
489 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{...
5 votes
1 answer
105 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 ...
14 votes
2 answers
911 views

Büchi automata with acceptance strategy

The problem Let $A=\langle \Sigma, Q, q_0,F,\Delta\rangle$ be a Büchi automaton, recognizing a language $L\subseteq\Sigma^\omega$. We assume that $A$ has an acceptance strategy in the following sense :...
13 votes
0 answers
216 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 ...
32 votes
5 answers
5k views

Counting words accepted by a regular grammar

Given a regular language (NFA, DFA, grammar, or regex), how can the number of accepting words in a given language be counted? Both "with exactly n letters" and "with at most n letters" are of ...
2 votes
1 answer
175 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
30 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(...
1 vote
0 answers
124 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 ...
3 votes
1 answer
140 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^* ...
4 votes
1 answer
130 views

Subsets of $\omega$-words which share certain factors and languages accepted by special (prefix-closed) automata

Let $\mathcal A$ be an automaton, then I define the following $\omega$-language accepted by $\mathcal A$: $$ L'(\mathcal A) := \{ \eta \in X^{\omega} : v \sqsubset \eta \mbox{ implies } v \in L(\...
6 votes
0 answers
206 views

Separation of the states of a deterministic omega-automaton by looping words taken from a regular language of non-empty words

Consider a deterministic transition structure having states in set $X$ and transition function $\rightarrow$, and an initial state $x \in X$. This structure is intended to be part of an automaton ...
6 votes
2 answers
2k 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 ...
7 votes
1 answer
383 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 ...
1 vote
1 answer
181 views

Example of a $U^\omega$ that is not Deterministic Büchi recognizable

Is there a regular language $U$, for which $U^\omega$ is not a Deterministic Büchi recognizable language. I have been thinking over it for some time, but have been unable to come up with an example.
3 votes
1 answer
112 views

Size bound on Büchi automaton for complement

For a given Büchi automaton $\mathcal A = (A, Q, \delta, q_0, F)$ we define a congruence on $A^{\ast}$ by $$ \begin{array}{llll} u \sim_{\mathcal A} v & :\Leftrightarrow & \mbox{for all }s,s' ...
6 votes
1 answer
234 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 ...