The study of formal languages (sets of strings or trees over an alphabet), rewriting systems and algorithms, recognition automata/algorithms, and related questions.

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1answer
50 views

Deterministic Finite Automata question [closed]

I am very new to finite automata, and I came across an issue in my professors lecture slides which I think is wrong, and I'd wonder if any of you could confirm: Alphabet: {1} Automata Surely the ...
0
votes
0answers
143 views

Is the positive existential theory undecidable?

Could you tell if the positive existential theory of $\mathbb{C}[e^{\mu x} \mid \mu \in \mathbb{C}]$ is undecidable in the language $\{+, \cdot , \frac{d}{dx} , 0, 1, e^x\}$ ? How can we prove the ...
4
votes
1answer
290 views

Show that the positive existential theory is undecidable

To show that the positive existential theory of $\mathbb{C}[t, e^{\lambda t} \mid \lambda \in \mathbb{C}]$ in the language $\{+, \cdot , ' , 0 , 1, t\}$ is undecidable we have to prove the following: ...
0
votes
0answers
95 views

Probability of substring given string production probabilities

I originally posted this question on the Math StackExchange, but have not received answers there and thought it might be more appropriate to post it here. Let $\Sigma$ be an alphabet and let $y = x_1 ...
1
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0answers
30 views

Relation between indexed languages (OI-macro or context-free tree) and scattered context languages

I'm not sure about the relation between indexed languages (generated by indexed grammars--Aho) and scattered context languages (generated by scattered context grammars--J Hopcroft). I think that ...
1
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1answer
75 views

Terminology for set of infinite strings with a certain prefix

Let $\mathcal{A}$ be a finite alphabet, and let $C$ be the Cantor space $\mathcal{A}^\omega$ under the product topology. Given a finite string $s \in \mathcal{A}^*$, let $C(s)$ be the set of all ...
1
vote
1answer
130 views

The automorphism groups of smallest grammars of a language string are isomorphic

Let $s \in \Sigma^*$ be a formal language string. Consider the automorphism group of $s$, defined to be the set of all permutations of positions of $s$ that leave $s$ fixed. For instance $G(abab) = ...
7
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0answers
95 views

Is there a ``Ladner's Theorem" for the PH-vs-PSPACE scenario?

Like a statement of the kind, ``If the Polynomial Hierarchy (PH) $\neq$ PSPACE then there exists $L \in PSPACE \backslash PH$ which is not PSPACE-complete"? Or is there something else that states ...
6
votes
2answers
147 views

Is this variant of the balanced bracket language context free?

Consider the language generated by the following context free grammar: $$ S \to SS \quad S \to () \quad S \to (S) \quad S \to [] \quad S \to [S] $$ There is a one-to-one correspondence between this ...
0
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0answers
83 views

Counting path generating sentences in a specific formal language

Given a formal grammar of a language or an Turing machine of the language, can we count the path that generating sentences of the language? For example, we know that if the grammar is context-free ...
6
votes
2answers
181 views

Formal languages with non-unique interpretations of terms

In mathematical logic and model theory, one considers interpretations of syntactic expressions: terms without free variables are interpreted as elements of some structure, formulas without free ...
3
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3answers
124 views

Are there any results on well-quasi-ordering of languages?

There are a number of papers that I can find about well-quasi-orders in formal lnaguage theory, by Kunc, de Luca, D'Alessandro, and Varricchio, among others. I am interested, however, in well-quasi ...
4
votes
2answers
444 views

Inherent ambiguity of the context-sensitive language $L = {a^ib^ic^id^je^jf^j \bigcap a^ib^jc^id^je^if^j} $ or $a^nb^nc^nd^ne^nf^n$

What is the definition of ambiguity of context-sensitive grammar?This is relevant to the definition of inherent ambiguity of context-sensitive language.And any proof for the inherent ambiguity of ...
7
votes
1answer
330 views

Is there a name for infinite words containing every finite words?

Apparently, the closest thing I've found would be normal number http://mathworld.wolfram.com/NormalNumber.html But requiring that every finite words occurs is weaker than this property. So I'm ...
1
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0answers
45 views

Question about link between non-terminals of grammars and variables of Diophantine equations

If we change the right arrow in the rewriting rules of grammar into equators , changes all terminals into x and keep the non-terminals unchanged,we get system of equations.In some cases,those ...
1
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0answers
56 views

Is it possible to classify all the inequivalent classes of first order sentence with quantifier rank fixed

It is known that for symbols with finite many relations, the number of inequivalent class of first order sentence with quantifier rank $m$ is finite. But is it possible to list (classify) them? At ...
0
votes
0answers
60 views

Two sets of strings resulted from a set represented by binary and decimal code are in the same class of languages?

Two sets of strings resulted from a set represented by binary and decimal code are in the same class of languages as regular languages,context-free languages,context-sentive languages ,computable ...
3
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0answers
159 views

Complexity of counting words of given length in regular or context-free language

Let $L$ be a regular or context-free language over alphabet $\{0,1\}$. What is the complexity of counting words of length $n$ in $L$? Is it possible to efficiently find if for given $n$ all words ...
1
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0answers
118 views

Examples of languages that are in P and are not in CFL [closed]

Any examples of languages that are in P(polynomial time to recognize it) and are not in CFL(context-free language)?The more the better.
13
votes
1answer
438 views

Is there an unambiguous CFL whose complement is not context-free?

I'm doing a little bit of research about context-free languages. A question that's popped up is whether or not there exists an unambiguous context-free language whose complement is not a context-free ...
2
votes
1answer
289 views

Expressive power of first-order category theory

Given the signature $\lbrace \mathsf{dom}, \mathsf{cod}, \mathsf{id},\circ \rbrace$ and the axioms of category theory – which are expressible in the signature's first-order (FO) language – ...
1
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1answer
141 views

Every infinite C.E.language is infinite or finite union of regular languages including at least one infinite regular language?

Is Every infinite C.E.language infinite or finite union of regular languages including at least one infinite regular language? And is every infinite C.E.language that is not indexed language(that may ...
6
votes
1answer
132 views

Separating infinite words sharing factors by automata

Two infinite words $\xi, \eta \in X^{\omega}$ are separated by an (Büchi-)automaton if it accepts one but not the other. Denote by $F_n(\xi)$ the factors of length $n$ of an infinite word $\xi$ and ...
6
votes
1answer
95 views

Generalising the adherence operator and its closure properties with regard to regular (rational) languages

Let $X$ be an alphabet and denote by $X^{\omega}$ the set of all infinite sequences (i.e. words) in $X$. A subset $L \subseteq X^{\omega}$ is called $\omega$-regular if it is acceptable by some ...
2
votes
1answer
123 views

Representability of sets of infinite sequences sharing common prefixes and factors (i.e. infixes)

Here we are concerned with the space $X^{\omega}$ of infinite sequences. Denote by $F_n(\xi)$ the set of factors (consecutive finite subsequences) of length $n$ and consider the set $$ K_n(\xi) = ...
5
votes
1answer
160 views

What prefix and factors determine a ultimately periodic word uniquely

Let $\xi$ be an ultimately periodic sequence, i.e. there exists finite sequences $p, q \in X^*$ such that $\xi = pq^{\omega}$. Does there exists a $n > 0$ such that the prefix of length $n$ and all ...
0
votes
1answer
693 views

Is there any danger far from home? (Edited & Revised Version) [closed]

The notion of formal proof is defined by finite sequences ($<\omega$ - sequences) of sentences. In some sense if a sentence $\sigma$ is (finitely) provable from the theory $T$ it is very "near" to ...
0
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0answers
95 views

Proof of conjecture that permutation-free automata restrict the possible states visitable from a stringset sharing prefixes and infixes

An automaton $\mathcal A = (X, Q, \delta, q_0)$ is called permutation-free iff no word $w \in X^*$ induces a nontrivial permutation of a subset of the states of $\mathcal A$. More formally for any $R ...
1
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0answers
235 views

Comparing two metrics on the space of infinite sequences and relating open and closed sets

Let $X = \{ 0, 1 \}$ and $X^{\mathbb N_0} = \{ x_0 x_1 x_2 \ldots : x_i \in X \}$ be the space of all infinite sequences, then a metric could be defined on it $$ d(u,v) := \frac{1}{2^r} \mbox{ with } ...
3
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0answers
101 views

The Kleene theorem

By the Myhill–Nerode theorem a language $L$ is accepted by a finite automaton iff it consists of classes of a finite congruence. By the Kleene theorem $L$ is accepted by a finite automaton iff it ...
4
votes
2answers
288 views

Concatenation of strings [closed]

We have two strings (i. e., finite tuples) $A$ and $B$. We have to find if for some positive integers $n$ and $m$, the string $A$ concatenated $n$ times equals the string $B$ concatenated $m$ times or ...
3
votes
1answer
124 views

Subsets of $\omega$-regular lanuages accepted by automata with special acceptance condition

Let $\mathcal A = (X, Q, \delta, q_0, F)$ be a deterministic finite automata with the following acceptance condition on infinite words: The automata accepts $\xi \in X^{\omega}$ with respect to $F$ ...
5
votes
2answers
192 views

Ordering on words

What are the known computation-friendly well-orderings on words from $A^*$, where $A$ is a finite alphabet, except the standard weightlex and syllable-order?
2
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1answer
216 views

Proof that the $\omega$-language consisting of all words containing every finite word as a factor is not rational/regular

Let $\eta$ be an $\omega$-word over $X = \{0,1\}$ and let $F_k(\eta)$ denote the factors of $\eta$ of length $k$. Define the following $\omega$-languages $$ L_k := \{ \xi : F_k(\xi) = X^k \} = \{ \xi ...
4
votes
1answer
146 views

Progress on group languages characterizations

Def. A group language is a recognizable language whose syntactic monoid is a group. q1. Is it known a "nice" combinatorial characterization of group languages ? q1.1. If no, is it well understood ...
2
votes
1answer
116 views

Varieties of rational languages and (pseudo-)varieties of finite monoids, question regarding closure property

Let $\mathcal Rat(A)$ denote the class of rational (or regular) languages over the alphabet $A$, a subset $\mathcal V(A) \subseteq \mathcal Rat(A)$ is called a variety of (rational) languages iff ...
5
votes
1answer
202 views

Is it decidable whether the support of a rational $\mathbb{Z}$-series is a regular language?

Let $S \in \mathbb{Z}\langle\langle A\rangle\rangle$ be a rational series in noncommutative variables. The support of $S$ is the set of all words $u \in A^*$ such that $(S, u) \not= 0$. It is ...
2
votes
2answers
188 views

Generating function of factorable binary words

A word $w$ on the alphabet $A := \{0, 1\}$ is factorable if \begin{equation} w = u^k \mbox{ where } u \in A^* \mbox{ and } k \geq 2. \end{equation} Let $L$ be the language of the set of ...
7
votes
1answer
357 views

Constructing Metrics for specific Topological Spaces, and Refinements of the Cantor-Space in particular

I have a Problem in general, given some some Topological Space $(X, \tau)$ from which I know it is metrisable, how can I find a metric (that is at best in some sence constructive and easy, at the very ...
1
vote
1answer
81 views

Simple asymptotic combinatorics - how many words are there in a certain weight category? [closed]

Given the set of all binary strings of length n, I am looking at the "middle" of these strings, weight-wise. Namely, I am trying to calculate how many words are there whose weight is between n/2 - ...
1
vote
3answers
277 views

Is there an recursively axiomatized system with infinitely many proofs for some propositions or a proposition [closed]

Is there any recursively axiomized system with infinitely many proofs for some propositions or a proposition? So we will have at least one proposition which is deduced from the recursively axiomatic ...
0
votes
0answers
403 views

Extended definition of unambiguous language and the existence of unambiguous grammar

Let's extend the unambiguity of language and grammar as follows: a language $L$ is unambiguous if there is a grammar that generates every word in $L$ in a unique way, the grammar may be of type 0 or ...
5
votes
1answer
580 views

List of open problems of formal languages [closed]

As we know, there are some open problems of formal languages. I am wondering if there is a somehow complete list of open problem of formal languages. If there isn't such a list, can we make it one as ...
20
votes
2answers
606 views

congruence on words: having the same (scattered) subwords of length at most n

For a fixed finite alphabet $A=\{a,b,...\}$, write $x \sim_n y$ if the two words $x$ and $y$ have the same (scattered) subwords of length at most $n$. The relation $\sim_n$ is a congruence of finite ...
3
votes
3answers
1k views

L-systems and Sierpinski Triangle

I was just shocked when I saw these consecutive outcomes of an L-system converging to the Sierpinski triangle (shown in the picture below). I'm interested to know how could one arrange the rules of ...
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votes
2answers
592 views

Any grammar for the language $L =a^p$, $p$ is prime number of $\mathbb{N}$

Any grammar for the language $$L =a^p,\text{ $p$ is prime and }p\in \mathbb{N}?$$ Is such a grammar related to any question of number theory like RH or the conjecture of twin primes?
5
votes
1answer
513 views

How to work out a grammar if we know the language?

How could we work out a grammar if we know the language? How could we work out a grammar if we know the language that is restricted to a special kind like CFL or CSL? For example, we know ...
2
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0answers
67 views

Is any CFL intersection,union of CFLs that are not inherently ambiguous?

Is any CFL intersection,union of CFLs that are not inherently ambiguous?
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0answers
104 views

Properties of classical automata preserved in Büchi automata

Given two NFW $A$ and $B$, we regarded $A$ and $B$ as Büchi automata. We can show that the containment property is not preserved in Büchi automata. That is, we can construct a example: $L(A) ...
7
votes
5answers
904 views

Generating function of a regular language

It is well known that the generating function of a regular language $L$, i.e. $\sum n_kz^k$ where $n_k$ is the number of words of length $k$ in $L$, is rational, i.e. a quotient of two polynomials ...