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-1
votes
0answers
22 views

BNF grammar for given problem [on hold]

I have a problem (a) Give a grammar using BNF rules to construct a program in the language "witless". A witless program must follow the rules: The program must start and end with the word 'endstart' ...
3
votes
3answers
89 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
187 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
282 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
vote
0answers
39 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
vote
0answers
51 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
47 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 ...
0
votes
0answers
93 views

Are there generating functions of rational or integral solutions of Diophantine equation that

As we know,there are generating functions for c.e.languages which are some retricted rational or algebraic or transcendental functions dependent on the class of languages like regular ...
2
votes
0answers
68 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
vote
0answers
97 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.
11
votes
1answer
154 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
241 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
vote
1answer
100 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
121 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 ...
5
votes
1answer
73 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
109 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
114 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
655 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
votes
0answers
68 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 ...
2
votes
0answers
137 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
votes
0answers
76 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
268 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
103 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 $F$ ...
4
votes
2answers
171 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
votes
1answer
117 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
133 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
73 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
167 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
170 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
267 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
75 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
263 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
201 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
452 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
556 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
600 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 ...
-1
votes
2answers
400 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
385 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
votes
0answers
52 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?
4
votes
0answers
91 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
516 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 ...
0
votes
1answer
257 views

prove (a+b)*=a*(ba*)* [closed]

formal language and automata theory regular expessions (a+b)* =a*(ba*)* please answer I want the proof thank you
0
votes
1answer
115 views

Can decidability results for monadic second-order logic be extended to monadic higher-order logics?

Call a higher-order logic fully monadic if and only if all of its predicate constants (at any order) and higher-order variables (at any order) are monadic (and it has no function symbols). In ...
4
votes
1answer
486 views

Deciding equivalence of regular languages

Given two regular expressions $R$ and $S$ on an alphabet $\Sigma$ it is possible to decide their equivalence as follows: build two finite automata $M_R$ and $M_S$ such that $L(R) = L(M_R)$ and $L(S) ...
12
votes
1answer
845 views

What's the difference between ZFC+Grothendieck, ZFC+inaccessible cardinals and Tarski-Grothendieck set theory?

Say that "U" is the axiom that "For each set x, there exists a Grothendieck universe U such that x $\in$ U", where Grothendieck universes are defined in the usual way (or, if that's unclear, in this ...
2
votes
2answers
603 views

All properties of a mathematical object

This is primarily a question about related literature. I am looking for specific references, or terminology that I can use to search for references. Let A a well defined mathematical structure of ...
1
vote
2answers
260 views

Satisfiability problem for FOL[<,R]

Let FOL[<,R] be the fragment of first-order logic enriched with two relational symbols < and R and the first-order axioms that say: < is a strict partial order and R is an irreflexive and ...
4
votes
3answers
286 views

Existential quantification over regular predicates

A regular language over an alphabet $\Sigma$ is a subset of the set of all words over $\Sigma$ that can be accepted by some finite automaton. A regular language identifies a certain property of ...
1
vote
1answer
158 views

Is complement of LL(k) grammar context free?

Is complement of LL(k) grammar context free?
5
votes
0answers
195 views

Büchi automata with acceptance strategy

I have already asked this question on cstheory.stackexchange, but without success. Maybe it is too close to an "open problem", although it is not a famous one. Anyway I try here, I can always remove ...