Related papers: Nondeterministic State Complexity of Positional Ad…

Using combinatorial properties of incomplete sets in a free monoid we construct a series of n-state deterministic automata with zero whose shortest synchronizing word has length n^2/4+n/2-1.

The problem of characterizing which automatic sets of integers are stable is here solved. Given a positive integer $d$ and a subset $A\subseteq \mathbb{Z}$ whose set of representations base $d$ is recognized by a finite automaton, a…

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. For a $k$-tuple of positive integers $\underline{\lambda} = (\lambda_{1}, \dots{} ,\lambda_{k})$ with $1 \le \lambda_{1} < \lambda_{2} < \dots{} < \lambda_{k}$, we…

This paper summarizes the fundamental expressiveness, closure, and decidability properties of various finite-state automata classes with multiple input tapes. It also includes an original algorithm for the intersection of one-way…

We provide a canonical form of mixed states in bipartite quantum systems in terms of a convex combination of a separable state and a, so-called, edge state. We construct entanglement witnesses for all edge states. We present a canonical…

Let k>2 be a fixed integer exponent and let \theta > 9/10. We show that a positive integer N can be represented as a non-trivial sum or difference of 3 k-th powers, using integers of size at most B, in O(B^{\theta}N^{1/10}) ways, providing…

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…

To better understand quantum computation we can search for its limits or no-gos, especially if analogous limits do not appear in classical computation. Classical computation easily implements and extensively employs the addition of two bit…

We present and prove a theorem answering the question "how many states does a minimal deterministic finite automaton (DFA) that recognizes the set of base-b numbers divisible by k have?"

Automata over infinite objects are a well-established model with applications in logic and formal verification. Traditionally, acceptance in such automata is defined based on the set of states visited infinitely often during a run. However,…

We associate in a canonical way a substitution to any abstract numeration system built on a regular language. In relationship with the growth order of the letters, we define the notion of two independent substitutions. Our main result is…

We consider blind, deterministic, finite automata equipped with a register which stores an element of a given monoid, and which is modified by right multiplication by monoid elements. We show that, for monoids M drawn from a large class…

This paper examines several measures of space complexity of variants of stack automata: non-erasing stack automata and checking stack automata. These measures capture the minimum stack size required to accept every word in the language of…

Recently, an infinite hierarchy of languages accepted by stateless deterministic pushdown automata has been established based on the number of pushdown symbols. However, the witness language for the n-th level of the hierarchy is over an…

We show that for any two distinct words $ s_1, s_2 $ over an arbitrary alphabets, there exists a deterministic finite automaton with $ O(\log^2 n) $ states that accepts $ s_1 $ and rejects $ s_2 $. This improves the previous upper bound of…

Using a novel rewriting problem, we show that several natural decision problems about finite automata are undecidable (i.e., recursively unsolvable). In contrast, we also prove three related problems are decidable. We apply one result to…

Soliton automata are mathematical models of soliton switching in chemical molecules. Several concepts of determinism for soliton automata have been defined. The concept of strong determinism has been investigated for the case in which only…

We introduce the category of dependency automata. A dependency automaton consists of two nondeterministic finite automata, with a relation between their states satisfying conditions. This category is equivalent to deterministic finite…

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set $A$ is the number of representations of a nonnegative integer $n$ as the sum of $k$ terms from $A$. Let $A(n)$…

We study the computational power of real-time finite automata that have been augmented with a vector of dimension k, and programmed to multiply this vector at each step by an appropriately selected $k \times k$ matrix. Only one entry of the…