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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13
votes
3answers
695 views

On the realisation of monoids as syntactic monoids of languages

Let $L \subseteq X^{\ast}$ be some language, then we define the syntactic congruence as $$ u \sim v :\Leftrightarrow \forall x, y\in X^{\ast} : xuy \in L \leftrightarrow xvy \in L $$ and the quotient ...
0
votes
0answers
32 views

Question on the construction of a deterministic Mueller automaton in the proof of McNaughton's theorem

My question is related to this classic article by R. McNaughton, its content is almost identically written on wikipedia. I will change the notation a little bit to make it more modern. Let $L_1, L_2 \...
5
votes
1answer
141 views

Useful equivalence relations on $X^{\ast}$ (like the Nerode and syntactic equivalence relations)

I want to have an overview of all the meaningful equivalence relations defined on $X^{\ast}$, in particular when the languages in question are regular. They typically arise in connection with the ...
0
votes
0answers
6 views

Why are regular expressions defined with union, concatenation and star operations? [migrated]

A regular expresssion is defined recursively as $a$ for some $a \in \Sigma$ is a regular expression, $\varepsilon$ is a regular expression, $\emptyset$ is a regular expression, $(R_1 \cup R_2)$ ...
10
votes
1answer
117 views

Do bounded-visit nondeterministic linear bounded automata recognize only regular languages?

Do bounded-visit nondeterministic linear bounded automata recognize only regular languages? By a nondeterministic linear bounded automaton (nLBA) I mean a single-tape nondeterministic Turing machine ...
14
votes
2answers
481 views

Regular versus TC0

According to the Complexity Zoo, $\mathsf{Reg} \subseteq \mathsf{NC^1}$ and we know that $\mathsf{Reg}$ cannot count so $\mathsf{TC^0} \not\subseteq \mathsf{Reg}$. However it doesn't say if $\mathsf{...
6
votes
2answers
201 views

FSM transducer sequential composition decidability

this is a followup/ sequel to this recent question which was answered, this one presumably significantly harder. consider a deterministic FSM transducer $F$ and its mapping $F(x)$ of an input word $x$....
10
votes
1answer
196 views

Is it decidable whether the output length of a transducer is bounded by the input length?

The transducers considered here are those Wikipedia calls finite state transducers. The behavior of a transducer $T$, that is, the relation it computes, is written $[T]$: a word $y$ is an output for $...
-2
votes
1answer
97 views

Writing a regular expression for character set $\Sigma = \{a,b,(,)\}$ that not have a parenthesis inside a parenthesis [closed]

Let character set $\Sigma=\{a,b,(,)\}$. I want to write a regular expression for the language $L$ that does not have a parenthesis inside a parenthesis. For example, $(abaab)(bbbaa) \in L$, while $(...
8
votes
0answers
82 views

Chomsky-Schützenberger for Deterministic CFLs

Is there a Chomsky-Schützenberger representation theorem for deterministic CFLs? Knowing precisely the class of morphisms under which DCFL is closed, such a theorem would probably be of the form: ...
2
votes
1answer
76 views

Deterministic Parity Automata require unbounded index

Deterministic parity automata $(Q, \Sigma, q_0, \Delta, c)$ are powerful enough to recognize all $\omega$-regular languages. However, the number of priorities they require for a language can become ...
0
votes
2answers
81 views

Example of R and G when $R \subseteq L(G)$ is undecidable [closed]

Could anybody provide an example of regular language R and context-free grammar G such that $R \subseteq L(G)$ is undecidable. Of course, if such language could be constructed. Thanks.
5
votes
1answer
149 views

Who introduced nondeterministic M-automata, and proved that the finite ones recognize the rational sets over M?

Let $M$ be a monoid. The family $\operatorname{RAT}(M)$ of rational sets over $M$ is defined inductively: If $L$ is a finite subset of $M$, then $L\in\operatorname{RAT}(M)$. If $K,L\in\operatorname{...
0
votes
1answer
190 views

Deciding whether a context-free language is regular [closed]

Does anyone know whether the following decision problem is decidable: Given a context-free language $L$, is $L$ regular? Here $L$ can be expressed, e.g., using a context-free grammar. Does anyone ...
22
votes
1answer
519 views

Deciding emptiness of intersection of regular languages in subquadratic time

Let $L_1,L_2$ be two regular languages given by NFAs $M_1,M_2$ as input. Assume we would like to check whether $L_1\cap L_2\neq \emptyset$. This can clearly be done by a quadratic algorithm which ...
3
votes
2answers
240 views

computing maximal bit density over a FSM

let $L$ be a regular language defined by a FSM over binary symbols $\{0,1\}$. consider a function $f(x)$ on words/ strings that computes "bit density", defined as the number of $1$'s in a word ("...
2
votes
1answer
135 views

minimal finite automata given in-words and out-words

this seems an interesting FSM optimization problem; have not seen it studied, wondering if it has been and/ or looking for other insight. given: two finite sets of words $S_{in}$ and $S_{out}$. ...
1
vote
1answer
75 views

What is the complexity of the set of closed boolean tautologies assuming well formed formulae?

Let's define two CFGs: S ::= 0 | 1 | (S+S) | (S*S) S' ::= 0 | 1 | S+S | S*S | (S) And two languages: M = { w | S generates w, and w evaluates to 1 } M'...
3
votes
0answers
133 views

Restricted-Input Automaton

In the classic setting, an automaton for a language $L$ is required to accept all words in $L$ and reject/get stuck on every word in $\Sigma^*\setminus L$. All of the related concepts are then ...
1
vote
1answer
109 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.
0
votes
2answers
96 views

finite automata under morphism [closed]

Given two (deterministic) finite automata $A, B$ over $\Sigma$, a mapping $h:\Sigma\rightarrow \Sigma'$ Naturally $h$ can be extended to a mapping in $\Sigma^*\rightarrow \Sigma'^*$ which is denoted ...
6
votes
1answer
153 views

Bounds on the size of NFA for $r$-skip $k$-distinct language

This question is about an extension of a language discussed in this question. We define the $r$-skip $k$-distinct language as follows: $$L_{r,k}=\{\sigma_1\sigma_2\cdots \sigma_{rk}\in\Sigma^{rk} | \...
20
votes
3answers
425 views

Complexity of intersection of regular languages as context-free grammars

Given regular expressions $R_1, \dots, R_n$, are there any non-trivial bounds on the size of the smallest context-free grammar for $R_1 \cap \cdots \cap R_n$?
15
votes
1answer
383 views

Can constant ambiguity reduce the state complexity of a regular languages?

We say that NFA $M$ is Constantly Ambiguous if there exist $k\in \mathbb{N}$ such that any word $w\in \Sigma^*$ is accepted by either $0$ or (exactly) $k$ paths. If automaton $M$ is constantly ...
4
votes
3answers
209 views

Regular languages under change of encoding

Consider a regular language $L$ with alphabet $\Sigma = \{0,1\}$. Can we say that the set of strings in $L$ (representing non-negative integers in binary encoding) when represented in some other ...
18
votes
0answers
276 views

Is there a regular tree language in which the average height of a tree of size $n$ is neither $\Theta(n)$ nor $\Theta(\sqrt{n})$?

We define a regular tree language as in the book TATA: It is the set of trees accepted by a non-deterministic finite tree automaton (Chapter 1) or, equivalently, the set of trees generated by a ...
11
votes
2answers
378 views

How small can a NFA be, compared to the minimal Unambiguous Finite Automaton (UFA) of the same regular language?

Unambiguous Finite Automatons (UFA) are special type of non-deterministic finite automatons (NFA). A NFA is called $unambiguous$ if every word $w\in \Sigma^*$ has at most one accepting path. This ...
18
votes
2answers
630 views

Bounds on the size of the smallest NFA for L_k-distinct

Consider the language $L_{k-distinct}$ consisting of all $k$-letter strings over $\Sigma$ such that no two letters are equal: $$ L_{k-distinct} :=\{w = \sigma_1\sigma_2...\sigma_k \mid \forall i\in[k]...
3
votes
0answers
100 views

Transfering properties from subsets of $X^*$ to subsets of $X^{\omega}$ by using the topology induces by Cantor space

A language $L \subseteq X^*$ is non-counting of order $n > 0$ iff for all $u,v, w \in X^*$ $$ uv^nw \in L \Leftrightarrow uv^{n+1} w \in L. $$ A $\omega$-language (set of infinite sequences) $L \...
0
votes
1answer
57 views

Finding containing sets, within sets of sets

Let $s = \{\sigma_1, \sigma_2 \ldots \sigma_n\}$, where $\sigma_i \in \Sigma$, denote a set of alphabet characters. And $s \in S$ where $S$ denotes a set of sets. Given a new set $s' = \{\sigma'_1, \...
3
votes
0answers
95 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(\...
0
votes
1answer
129 views

Does every regular language contains a strictly locally testable language?

Let $L$ be an infinite regular language, then does there exists a strictly locally testable infinite language $P$ such that $P \subseteq L$?
5
votes
0answers
190 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 ...
2
votes
0answers
144 views

Finding self-similar homomorphisms of a FSM transducer

Consider a special case of homomorphisms of FSM transducers (or "generalized sequential machines" in [1]). Let $F$ be a transducer accepting a language $L$, and let $h(x)$ be a homomorphism function ...
5
votes
3answers
3k views

Ambiguity in regular and context-free languages

I understand the following claims to be true: Two distinct derivations of a string in a given CFG may sometimes attribute the same parse tree to the string. When there are derivations of some string ...
4
votes
3answers
236 views

How to minimize a FSM transducer?

In contrast to FSM minimization which is well studied with various algorithms, theorems and has many practical applications, I'm looking for any nontrivial insight, analysis and references to the ...
7
votes
3answers
367 views

Algorithm for ranking members of a regular language?

A little while back, I was reading a paper that mentioned a method for computing an integer 'rank' for a particular string $s \in L$ where $L$ is some regular language. This rank uniquely determines ...
6
votes
0answers
137 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 ...
9
votes
3answers
337 views

(N)DFA with same initial/accepting state(s)

What is known about the class of languages recognized by finite automata having the same initial and accepting state? This is a proper subset of the regular languages (since every such language ...
1
vote
0answers
160 views

Providence of pumping lemmas for regular languages

I'm looking to track down who discovered the following pumping lemmas for regualar languages. (where $p$ is the pumping constant.) Reg($L) \rightarrow \exists p\forall w(\in L) \forall u_1u_2v(\in \...
12
votes
1answer
2k views

Distance between regular languages

I want to define a notion of "closeness" between two regular languages of finite words in $\Sigma^*$ (and/or infinite words in $\Sigma^\omega$). The basic idea is that we want two languages to be ...
3
votes
2answers
502 views

Context-free Grammar for a Context-free Language Intersecting a Regular Language (get the Maximum Number of Rules)

It is well known that the intersection of $L \cap R$ of a context-free Language $L$ and a regular Language $R$ is context-free. Each proof I have seen constructs a automaton (a PDA) that accepts $L$ ...
5
votes
0answers
156 views

The regularity of Markov chains with a threshold

(This question has been asked on math.se, with no response.) I am studying Paz's "Introduction to Probabilistic Automata" and there is an exercise I cannot solve: Ex. 11, p. 170: Let $\Sigma = \{...
18
votes
2answers
378 views

Relation between $AC^0$ and regular languages

Let $\mathsf{REG}$ be the class of all regular languages. It is known $\mathsf{AC}^0 \not\subset \mathsf{REG}$ and $\mathsf{REG} \not\subset \mathsf{AC}^0$. But is there any characterization for ...
1
vote
0answers
86 views

Automatic structures/functions: Is (Z,+) under a unary representation automatic?

The group $(\mathbb{Z}, +)$ is automatic (ala Khoussainov) when using the "standard" representation in a decimal base. But if I want to use a different representation of Z, encoding my integers with ...
14
votes
3answers
546 views

The significance of state complexity in automata and regular languages?

I'm reading "Concatenation of Regular Languages and Descriptional Complexity" by Galina Jiraskova, 2009 on the state complexity resulting from concatenation of two regular languages ( by Galina ...
4
votes
1answer
140 views

Getting an automaton from set of words in and out of a language [duplicate]

Possible Duplicate: Is finding the minimum regular expression an NP-complete problem? Let's suppose that I have an unknown language $\mathcal L$, I know only two (particularly large) sets of ...
8
votes
1answer
202 views

Regular expressions of families of regular expressions

I was reading about the Star Height Problem and noticed that Eggan's family of regular expressions follows a simple pattern which can be described by a regular expression. My question is: are there ...
9
votes
0answers
1k 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 ...
14
votes
1answer
1k views

Sufficient conditions for the regularity of a context-free language

It would be nice to collect a list of conditions that imply that a context-free language L is regular, i.e. conditions of the form: "if a given CFG/PDA has property P, then its languages is regular" ...