Related papers: An efficient algorithm to decide periodicity of b-…
Compared with linear time invariant systems, linear periodic system can describe the periodic processes arising from nature and engineering more precisely. However, the time-varying system parameters increase the difficulty of the research…
Given a monoid $(M,\varepsilon,\cdot )$ it is shown that a subset $A\subseteq M$ is recognizable in the sense of automata theory if and only if the $\varphi $-rank of $x=x$ is zero in the first-order theory $\operatorname{Th}(M,\varepsilon…
In this note we study automata recognizing birecurrent sets. A set of words is birecurrent if the minimal partial DFA recognizing this set and the minimal partial DFA recognizing the reversal of this set are both strongly connected. This…
Kaprekar's Routine is an iteration process that, with each iteration, sorts the digits of a number in descending and ascending orders and uses their difference for the next iteration. In this paper, we successfully address the structure of…
In many simple integral domains, such as $\mathbb{Z}$ or $\mathbb{Z}[i]$, there is a straightforward procedure to determine if an element is prime by simply reducing to a direct check of finitely many potential divisors. Despite the fact…
A computable ring is a ring equipped with mechanical procedure to add and multiply elements. In most natural computable integral domains, there is a computational procedure to determine if a given element is prime/irreducible. However,…
We study first-order logic (FO) over the structure consisting of finite words over some alphabet $A$, together with the (non-contiguous) subword ordering. In terms of decidability of quantifier alternation fragments, this logic is…
In this note, we present a characterization of sets definable in Skolem arithmetic, i.e., the first-order theory of natural numbers with multiplication. This characterization allows us to prove the decidability of the theory. The idea is…
In this paper will be introduced large, probably complete family of complex base systems, which are 'proper' - for each point of the space there is a representation which is unique for all but some zero measure set. The condition defining…
We develop an algorithm for recognizing whether a character belongs to $\Sigma^m$. In order to apply it we just need to know that the ambient group is of type $\mathrm{FP}_m$ or of type $\mathrm{F}_2$ and that the word problem is solvable…
It is an open problem to characterize the class of languages recognized by quantum finite automata (QFA). We examine some necessary and some sufficient conditions for a (regular) language to be recognizable by a QFA. For a subclass of…
Most ideas about what an algorithm is are very similar. Basic operations are used for transforming objects. The evaluation of internal and external states by relations has impact on the further process. A more precise definition can lead to…
In this paper we discourse basises of representable algebras. This question lead to arithmetic problems. We prove algorithmical solvability of exponential-Diophantine equations in rings represented by matrices over fields of positive…
We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers a so that either (a, a^2) or (a,…
The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…
A number of engineering and scientific problems require representing and manipulating probability distributions over large alphabets, which we may think of as long vectors of reals summing to $1$. In some cases it is required to represent…
The purpose of this paper is to provide efficient algorithms that decide membership for classes of several Boolean hierarchies for which efficiency (or even decidability) were previously not known. We develop new forbidden-chain…
Series-parallel (SP) graphs are binary edge-labeled graphs with a designated source and target vertex, built using serial and parallel composition. A set of graphs is recognizable if membership depends only on its image under a homomorphism…
We provide an effective algorithm for determining whether an element of the outer automorphism group of a free group is fully irreducible. Our method produces a finite list which can be checked for periodic proper free factors.
We prove decidability results on the existence of constant subsequences of uniformly recurrent morphic sequences along arithmetic progressions. We use spectral properties of the subshifts they generate to give a first algorithm deciding…