formal languages, grammars, automata theory

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17
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2answers
210 views

Deciding whether a unary context-sensitive language is regular

It is a well-known result that the question Does a context-free grammar generate a regular language? is undecidable. However, it becomes decidable on a unary alphabet, simply because in this ...
13
votes
3answers
697 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 \...
14
votes
2answers
346 views

Is there a well-defined division operation on finite automata?

Background: Given two deterministic finite automata A and B, we form the product C by letting the states in C be the cartesian product of states in A and states in B. Then, we choose transitions, ...
5
votes
2answers
103 views

Characterization of union of DCFLs

We know that DCFL is not closed under the union operation and CFL is closed under union and contains the union of DCFLs. Is there a characterization of finite unions of DCFLs?
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 ...
5
votes
0answers
63 views

Does PEG contain CFG?

Despite their considerable expressive power, all PEGs can be parsed in linear time using a tabular or memoizing parser (8). These properties strongly suggest that CFGs and PEGs define incomparable ...
0
votes
1answer
59 views

Semantic equivalence using a model of computation of two languages

I am relatively new to the field of language semantics. However i had a question pertaining to language equivalence (i did read about the question, however my approach is slightly different and hence ...
2
votes
3answers
165 views

Current research topics in tree automata

What are current research topics connected with tree automata? I'm particularly interested with connection between automatas, logics and databases. Kind regards, XYZ
31
votes
9answers
12k views

Real computers have only a finite number of states, so what is the relevance of Turing machines to real computers?

Real computers have limited memory and only a finite number of states. So they are essentially finite automata. Why do theoretical computer scientists use the Turing machines (and other equivalent ...
10
votes
1answer
236 views

Introduction to probabilistic automata

Where can I find an introduction to probabilistic automata and what they recognize (certain functions from words to $[0,1]$)? Is there a standard term for such functions which are recognized by ...
3
votes
0answers
56 views

What is the class of the languages recognized by PCREs?

I have been considering building a tool that would convert regexes between the various syntaxes (BRE, ERE, PCRE). It is obvious that PCREs are too strong for the is-regular problem to be decidable, ...
14
votes
1answer
330 views

Novel proof of pumping lemma for regular languages

Let $\mathcal{L}$ be the family of all languages over $\Sigma$ satisfying the pumping property of regular languages. Namely: for each $L\in\mathcal{L}$, there is an $N\in\mathbb{N}$ s.t. every word $w\...
1
vote
2answers
99 views

Are equalizers of regular functions always regular languages? (My guess is no because PCP, but…)

Edit: I originally defined a regular function as a function computable by a Mealy machine, but Denis pointed out that that was a weaker model than what I was thinking of. So to be more precise, by a "...
10
votes
3answers
186 views

Does there exist a hardest DCFL?

Greibach famously defined a language $H$, the so-called nondeterministic version of $D_2$, such that any CFL is an inverse morphic image of $H$. Does there exist a similar statement with DCFL, ...
7
votes
1answer
126 views

A word anticorrespondence problem

A problem instance is a finite list of 4-tuples $(\alpha_1, u_1, v_1, \beta_1), ..., (\alpha_N, u_N, v_N, \beta_N)$, where $\alpha_i, \beta_i \in X$ come from a finite set, and each $u_i,v_i \in A^*$ ...
4
votes
1answer
45 views

An exponentially-ambiguous weighted automaton without an equivalent polynomially-ambiguous automaton

A min-plus weighted automaton (WFA) is a nondeterministic automaton with a weight function that assigns each transition a weight in $\mathbb{N}$. The weights along a run are summed, and the value of a ...
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{...
0
votes
2answers
145 views

Are there any open problems concerning decidability? [duplicate]

I am learning computability theory. I am just interested to know some famous problems (Formally languages) whose decidability is in question.
4
votes
0answers
49 views

Is there research on “minimal” Turing-universal Markov algorithms?

The Markov algorithm is a simple model of computation. For other models of computation, such as Turing machines, cellular automata, tag systems, etc., there is research on the "minimal" instances of ...
1
vote
0answers
34 views

Grammars whose LR automata have singleton item sets?

The states in LR parsers correspond to sets of items (ie, sets of productions from the original grammar, with a "dot" marking how far into the rule the parser has gotten). In general, states ...
-2
votes
2answers
133 views

An algorithm that determines if regular language accepts all string of its alphabet [closed]

Let $L$ be a regular language with the alphabet $\Sigma$. I'm trying to find an algorithm to tell whether $L=\Sigma^{*}$, whether $L$ accepts all strings in its alphabet. I think this algorithm uses ...
-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 $(...
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 ...
4
votes
0answers
60 views

How much smaller can universal Turing machines get if they only need to be universal for a subclass?

Say that a Turing machine $U$ is universal for a class $\mathcal{C}$ of languages if for any language $L \in \mathcal{C}$, there is a word $w_L$ with: $$(\forall w)\quad w \in L \Leftrightarrow U(w_L, ...
2
votes
2answers
89 views

If the set of factors of an infinite word $\xi$ is regular, is this property stable under “shift's” of $\xi$?

Let $\xi$ be an infinite binary sequence, and denote by $T(\xi)$ the set of all factors (infixes) of $\xi$. Also if $w$ is some finite prefix of $\xi$, denote by $\xi/w$ the unique $\eta$ such that $w ...
10
votes
1answer
226 views

What is the state complexity of the copy language?

Let a number $n$ be given. Consider the following language $L_n = \{ \; ww \; \vert \; w \in \{0,1\}^{n} \; \}$. In words, $L_n$ is the set of copy strings of length $2n$. Consider the following ...
18
votes
2answers
385 views

“Embedding” a language in itself

Main/General Question Let $L$ be a language. Define the languages $L_i$ with $L_0 = L$ and $$L_i = \{xwy : xy \in L_{i-1}, w \in L\}$$ for $i \geq 1$. Consider $\hat{L} = \bigcup L_i$. So, we ...
5
votes
1answer
89 views

Is there any research or findings on creating parse forests on Earley parsers with Leo Joop Enhancements?

Using the Earley Algorithm we can use the Leo enhancement to create cached items for recognition. http://www.sciencedirect.com/science/article/pii/030439759190180A Scott's algorithm on building ...
9
votes
2answers
140 views

Separating lists of words

There is an open problem in formal languages known as the Separating Problem; which is briefly stated as given two distinct strings of length $n$, how large of a DFA is required to "separate" them, ...
1
vote
0answers
176 views

Complexity of DBA-recognizable Omega-Languages

Given an $\omega$-regular expression $r$, how difficult is it to decide if $L(r)$ is recognizable by some deterministic Büchi automaton? I know it is solvable in EXPTIME by converting the regular ...
3
votes
2answers
118 views

Translation of context-free parsing into SAT

Is there a published algorithm for translating a context-free parsing problem into SAT? That is, an algorithm that translates a context-free grammar and an input string into a set clauses that is ...
11
votes
2answers
213 views

Vector Addition Systems with finite “obstacles”

A Vector Addition System (VAS) is a finite set of actions $A \subset \mathbb{Z}^d$. $\mathbb{N}^d$ is the set of markings. A run is a non-empty word of markings $m_0 m_1\dots m_n$ s.t. $\forall i \in \...
9
votes
2answers
153 views

Minimizing Automata accepting $\omega$-words (i.e. infinite words)

What is the standard approach on minimizing Büchi-Automata (or also Müller-Automata)? Transfering the usual technique from finite words, i.e. setting two states to be equal if the words "running out" ...
7
votes
1answer
103 views

Extensional characterization of non-deterministic finite state transductions

I recently became aware of the rather appealing characterization of deterministic word-to-word transductions as word functions with bounded variation (see e.g. [1]). This coincides with the set of ...
5
votes
1answer
270 views

Closure properties of deterministic context-sensitive languages

There does seem to be a lot of information regarding the closure properties of both deterministic context-free and nondeterministic context-sensitive languages. However, the literature is almost mute ...
5
votes
2answers
124 views

Can real-time deterministic multicounter automata recognize the marked palindrome language?

Consider the marked palindrome language which is defined as MPAL=$\{ w\#w^r | w \in \{a,b\}^* \}$. It is easy to recognize MPAL using only a single stack. My question is whether MPAL can be ...
0
votes
0answers
121 views

Is there an algorithm to find whether 2 combinators form a Turing-complete system?

It is known that K = (λx.(λy.x)) and S = (λx.(λy.(λz.((x z) (y z))))) define a turing complete system, and we know procedures to ...
10
votes
1answer
776 views

Decide the existence of a string homomorphism

Consider the following problem: Given two strings x,y, decide whether there exists a string homomorphism f such that f(x)=y. It is easy to show that this problem is in $NP$. Are there other ...
14
votes
2answers
315 views

Does XOR automata (NXA) for finite languages benefit from cycles?

A non-deterministic Xor automata (NXA) is syntactically an NFA, but a word is said to be accepted by NXA if it has an odd number of accepting paths (instead of at least one accepting path in the NFA ...
5
votes
2answers
216 views

Bounds on size of self-concatenation of Finite Languages

Given a finite language $L$ with $|L|$ number of elements, what is $|L^i|$ (the language $L$ concatenated with itself $i$ times)? If there is no exact result, is there an upper/lower bound? Define ...
5
votes
1answer
147 views

What is the importance of linear languages?

What is the point of linear languages? They appear to be an intermediate set of languages in between regular and context-free languages, but do they have any useful or nice properties that either have ...
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 ...
11
votes
0answers
237 views

Can we approximate the number of words accepted by an NFA?

Let $M$ be an acyclic NFA. Since $M$ is acyclic, $L(M)$ is finite. In a related question, it was suggested that exact counting of the number of words accepted by $M$ is $\#P$-Complete. The second ...
5
votes
1answer
151 views

What language $L \in NCM$ has $\overline{L} \not \in NCM$?

$NCM$, the class of non-deterministic reversal-bounded counter machines, has a lot of interesting dependability and closure properties. It's known that, unlike the deterministic version, NCM is not ...
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} | \...
0
votes
0answers
128 views

In what complexity classes other than $NP$ are these problems related to unary languages?

If I remember correctly saw this reduction in a paper can't find at the moment. Consider the following NP-complete variation of the Subset Sum problem. Given a set of positive integers $S=\{x_1,\...
4
votes
1answer
229 views

What characterizations exist for the grammars that can express subsets of the context-free languages?

It is well known that CFGs and PDAs are equivalent, and there has been extensive research about the relationship between deterministic pushdowns and $LR(1)$ grammars, as $DCFL$ is a subset of $LR(1)$. ...