Guentcho Skordev

Linear cellular automata, finite automata and Pascal's triangle

Abstract We address the question whether double sequences produced by one-dimensional linear cellular automata can also be generated by finite automata. A complete solution for binomial coefficients and Lucas′ numbers is given and some partial results for the general case are presented.

Automatic maps in exotic numeration systems

We generalize the classical notion of ab-automatic sequence for a sequence indexed by the natural numbers. We replace the integers by a semiring and use a numeration system consisting of the powers of a baseb and an appropriate set of digits. For example, we define (−3)-automatic sequences (indexed by the ordinary integers or by the rational integers) and (−1 +i)-automatic sequences (indexed by the Gaussian integers). We show how these new notions are related to the old ones, and we study both the number-theoretic and automata-theoretic properties that permit the replacement of one numeration system by another.

Automaticity of double sequences generated by one-dimensional linear cellular automata

Abstract We give a complete answer to the question whether a double sequence that is generated by a one-dimensional linear cellular automaton, and whose states are integers modulo m , is k -automatic or not.

Linear Cellular Automata, Substitutions, Hierarchical Iterated Function Systems and Attractors

Many constructions and algorithms from fractal geometry have become fundamental tools within the naturalism program of computer graphics. As the interest of computer graphics is shifting towards new frontiers such as scientific visualization some other basic concepts from fractal geometry come into focus. Cellular automata (CA) in particular are becoming a premier modelling and simulation tool in engineering and the basic sciences. CA live in a discrete world and are local in nature. They produce structure and patterns subject to a set of rules (look-up table) which determine the state of a growing cell from the state of its neighbors.

SELF-SIMILAR STRUCTURE OF RESCALED EVOLUTION SETS OF CELLULAR AUTOMATA I

The self-similarity properties of the orbits of a class of cellular automata are deciphered by matrix substitutions, hierarchical iterated function systems and appropriate scaling procedure.

Coarse-graining invariant patterns of one-dimensional two-state linear cellular automata

We consider one-dimensional two state cellular automata and their orbital patterns. We are particulary interested in initial states and their orbital pattern which are invariant under a certain coarse-graining operation. We show that invariant initial states are automatic. Moreover, we numerically study the complex nature of the initial states and their invariant orbits.

Remarks on permutive cellular automata

We prove that every two-dimensional permutive cellular automaton is conjugate to a one-sided shift with compact set of states.

Schur congruences, Carlitz sequences of polynomials and automaticity

Abstract We first generalize the Schur congruence for Legendre polynomials to sequences of polynomials that we call ‘d-Carlitz’. This notion is more general than a similar notion introduced by Carlitz. Then, we study automaticity properties of double sequences generated by these sequences of polynomials, thus generalizing previous results on the double sequences produced by one-dimensional linear cellular automata.

Self-generating sets, integers with missing blocks, and substitutions

We give a new construction of the Kimberling sequence defined by:(a)1 belongs to S; (b)if the positive integer x belongs to S, then 2x and 4x-1 belong to S; and (c)nothing else belongs to S, hence, S=12346781112141516...which is sequence A052499 in the Sloanes On-line Encyclopedia of Integer Sequences, by proving that this sequence is equal to sequence 1+A003754, the sequence of integers whose binary expansion does not contain the block of digits 00. We give a general framework for this sequence and similar sequences, in relation to automatic or morphic sequences and to non-standard numeration systems such as the lazy Fibonacci expansion.

On the fractal structure of the rescaled evolution set of Carlitz sequences of polynomials

Abstract Self-similarity properties of the coefficient patterns of the so-called m -Carlitz sequences of polynomials are considered. These properties are coded in an associated fractal set – the rescaled evolution set. We extend previous results on linear cellular automata with states in a finite field. Applications are given for the sequence of Legendre polynomials and sequences associated with the zero Bessel function.