Related papers: On asymptotically automatic sequences
We study conditions under which integer sequences with independent, identically distributed gaps are asymptotically $k$-complete, meaning that every sufficiently large integer can be represented as the sum of exactly $k$ distinct elements…
Let A be a finite alphabet and let L contained in (A*)^n be an n-variable language over A. We say that L is regular if it is the language accepted by a synchronous n-tape finite state automaton, it is quasi-regular if it is accepted by an…
The paper studies different variants of almost periodicity notion. We introduce the class of eventually strongly almost periodic sequences where some suffix is strongly almost periodic (=uniformly recurrent). The class of almost periodic…
Model checking properties are often described by means of finite automata. Any particular such automaton divides the set of infinite trees into finitely many classes, according to which state has an infinite run. Building the full type…
We construct automata with input(s) in base $k$ recognizing some basic relations and study their number of states. We also consider some basic operations on $k$-automatic sequences $(h(i))_{i \geq 0}$ and discuss their state complexity. We…
The correlation measure is a testimony of the pseudorandomness of a sequence $\infw{s}$ and provides information about the independence of some parts of $\infw{s}$ and their shifts. Combined with the well-distribution measure, a sequence…
An asymptotically orthonormal sequence is a sequence which is 'nearly' orthonormal in the sense that it satisfies the Parseval equality up to two constants close to one. In this paper, we explore such sequences formed by normalized…
The equivalence of finite automata and regular expressions dates back to the seminal paper of Kleene on events in nerve nets and finite automata from 1956. In the present paper we tour a fragment of the literature and summarize results on…
In some particular cases we give criteria for morphic sequences to be almost periodic (=uniformly recurrent). Namely, we deal with fixed points of non-erasing morphisms and with automatic sequences. In both cases a polynomial-time algorithm…
We present in this paper a new method to deal with automatic sequences. This method allows us to prove a M\"obius-randomness-principle for automatic sequences from which we deduce the Sarnak conjecture for this class of sequences.…
The theory of finite automata concerns itself with words in a free monoid together with concatenation and without further structure. There are, however, important applications which use alphabets which are structured in some sense. We…
We provide a complete characterisation of automaticity of uniformly recurrent substitutive sequences in terms of the incidence matrix of the return substitution of the underlying purely substitutive sequence. This resolves a recent question…
Given a sequence of complex square matrices, $a_n$, consider the sequence of their partial products, defined by $p_n=p_{n-1}a_{n}$. What can be said about the asymptotics as $n\to\infty$ of the sequence $f(p_n)$, where $f$ is a continuous…
Sequence theories are an extension of theories of strings with an infinite alphabet of letters, together with a corresponding alphabet theory (e.g. linear integer arithmetic). Sequences are natural abstractions of extendable arrays, which…
In this article, $q$-regular sequences in the sense of Allouche and Shallit are analysed asymptotically. It is shown that the summatory function of a regular sequence can asymptotically be decomposed as a finite sum of periodic fluctuations…
We propose novel algorithms for sequence prediction based on ideas from stringology. These algorithms are time and space efficient and satisfy mistake bounds related to particular stringological complexity measures of the sequence. In this…
We present a general approach to the problem of determining tight asymptotic lower bounds for generalized central moments of the optimal alignment score of two independent sequences of i.i.d. random variables. At first, these are obtained…
We study those automatic sequences which are produced by an automaton whose underlying graph is the Cayley graph of a finite group. For $2$-automatic sequences, we find a characterization in terms of what we call homogeneity, and among…
Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward…
A large class of problems in sciences and engineering can be formulated as the general problem of constructing random intervals with pre-specified coverage probabilities for the mean. Wee propose a general approach for statistical inference…