6
votes
1answer
122 views

Separating infinite words sharing factors by automata

Two infinite words $\xi, \eta \in X^{\omega}$ are separated by an (B├╝chi-)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) = ...
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
1answer
118 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
0answers
181 views

How should one generate a random set of mappings?

My motivation for this question comes from the study of synchronizing automata. There is a general consensus that random automata are synchronizing and have short synchronizing words. I am hoping ...
3
votes
2answers
804 views

Are context-free languages with context-free complements necessarily deterministic context-free?

Let $L \subseteq A^\star$ be a formal language over $A$ generated by a context-free grammar, and $L' = A^\star - L$ be the relative complement in $A^\star$. If $L$ and $L'$ are both context-free, are ...
7
votes
6answers
2k views

Regular languages and the pumping lemma

In certain dark corners of computer science and group theory, one often wants to prove that a language is not a regular language (ie a language accepted by a finite state automaton). The only ...