Related papers: Automata and automatic sequences
These informal notes, initially prepared a few years ago, look at various questions related to infinite processes in several parts of mathematics, with emphasis on examples.
We present several infinite series of synchronizing automata for which the minimum length of reset words is close to the square of the number of states. These automata are closely related to primitive digraphs with large exponent.
A process algebra is proposed, whose semantics maps a term to a nondeterministic finite automaton (NFA, for short). We prove a representability theorem: for each NFA $N$, there exists a process algebraic term $p$ such that its semantics is…
We study automaton structures, i.e. groups, monoids and semigroups generated by an automaton, which, in this context, means a deterministic finite-state letter-to-letter transducer. Instead of considering only complete automata, we…
We study the pseudorandomness of automatic sequences in terms of well-distribution and correlation measure of order 2. We detect non-random behavior which can be derived either from the functional equations satisfied by their generating…
We discuss an interesting sequence defined recursively; namely, sequence A105774 from the On-Line Encyclopedia of Integer Sequences, and study some of its properties. Our main tools are Fibonacci representation, finite automata, and the…
We construct the groups $<A,B,C \;| \, A^2,B^2,C^2,(ABC)^2>$ and $<A,B \;| \, A^2,B^4,(AB)^4>$, using 3-state automata over the alphabets $\{1,2,3\}$ and $\{1,2,3,4\}$. In addition, we show, how to define direct powers of $G$ by automaton…
These lecture notes are intended as a supplement to Moore and Mertens' The Nature of Computation or as a standalone resource, and are available to anyone who wants to use them. Comments are welcome, and please let me know if you use these…
Motivated by the search for idempotent cellular automata (CA), we study CA that act almost as the identity unless they read a fixed pattern $p$. We show that constant and symmetrical patterns always produce idempotent CA, and we…
We consider a category of all finite partial orderings with quotient maps as arrows and construct a Fra\"iss\'e sequence in this category. Then we use commonly known relations between partial orders and lattices to construct a sequence of…
This note contains a report of a proof by computer that the Fibonacci group F(2,9) is automatic. The automatic structure can be used to solve the word problem in the group. Furthermore, it can be seen directly from the word-acceptor that…
Quantum computing is concerned with computer technology based on the principles of quantum mechanics, with operations performed at the quantum level. Quantum computational models make it possible to analyze the resources required for…
We introduce a new tool, called the orbit automaton, that describes the action of an automaton group $G$ on the subtrees corresponding to the orbits of $G$ on levels of the tree. The connection between $G$ and the groups generated by the…
We establish new combinatorial transcendence criteria for continued fraction expansions. Let $\alpha = [0; a_1, a_2,...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients…
The infinite numbers of the set M of finite and infinite natural numbers are defined starting from the sequence 0\Phi, where 0 is the first natural number, \Phi is a succession of symbols S and xS is the successor of the natural number x.…
Eilenberg correspondence, based on the concept of syntactic monoids, relates varieties of regular languages with pseudovarieties of finite monoids. Various modifications of this correspondence related more general classes of regular…
Bruyere and Carton lifted the notion of finite automata reading infinite words to finite automata reading words with shape an arbitrary linear order L. Automata on finite words can be used to represent infinite structures, the so-called…
Infinite sequences are considered in the framework of classical logic from a new point of view.
We use fast-growing finite and infinite sequences of natural numbers and more complicated constructs to define models of hypercomputation and interpret non-arithmetic predicates, with the strongest extensions reaching full second order…
To study relationship between quantum finite automata and probabilistic finite automata, we introduce a notion of probabilistic reversible automata (PRA, or doubly stochastic automata). We find that there is a strong relationship between…