Nilpotency and periodic points in non-uniform cellular automata

Nilpotent cellular automata have the simplest possible dynamics: all initial configurations lead in bounded time into the unique fixed point of the system. We investigate nilpotency in the setup of one-dimensional non-uniform cellular automata (NUCA) where different cells may use different local rules. There are infinitely many cells in NUCA but only a finite number of different local rules. Changing the distribution of the local rules in the system may drastically change the dynamics. We prove that if the available local rules are such that every periodic distribution of the rules leads to nilpotent behavior then so do also all eventually periodic distributions. However, in some cases there may be non-periodic distributions that are not nilpotent even if all periodic distributions are nilpotent. We demonstrate such a possibility using aperiodic Wang tile sets. We also investigate temporally periodic points in NUCA. In contrast to classical uniform cellular automata, there are NUCA—even reversible equicontinuous ones—that do not have any temporally periodic points. We prove the undecidability of this property: there is no algorithm to determine if a NUCA with a given finite distribution of local rules has a periodic point.

Verification of a brick Wang tiling algorithm

We have implemented a certified Wang tiling program for tiling a rectangle region using a brick corner Wang tile set. A brick corner Wang tile set is a special Wang tile set introduced by A. Derouet-Jourdan et al. in computer graphics in 2015 to model wall patterns texture. We have implemented a tiling algorithm using Coq proof assistant and proved its correctness. This correctness assures the existence of a tiling of any brick corner Wang tile set for any size of rectangle. The essential points of our proof are the existence of a tiling for a $2 \times 2$ rectangle and a simple induction process. Since the brick corner Wang tile is a class of infinite kinds of tile sets, it is not straightforward and there are many conditional branches to prove the correctness. The certification with Coq assures that there are no lack of conditions.

SYNTHESIZED ENRICHMENT FUNCTIONS FOR EXTENDED FINITE ELEMENT ANALYSES WITH FULLY RESOLVED MICROSTRUCTURE

Inspired by the first order numerical homogenization, we present a method for extracting continuous fluctuation fields from the Wang tile based compression of a material microstructure. The fluctuation fields are then used as enrichment basis in Extended Finite Element Method (XFEM) to reduce number of unknowns in problems with fully resolved microstructural geometry synthesized by means of the tiling concept. In addition, the XFEM basis functions are taken as reduced modes of a detailed discretization in order to circumvent the need for non-standard numerical quadratures. The methodology is illustrated with a scalar steady-state problem.

WANG TILE SIZE IN TERMS OF CIRCULAR PARTICLE DYNAMICS

The main advantage of the Wang tiling concept for material engineering is ability to create large material domains with a relatively small set of tiles. Such idea allows both a reduction of computational demands and preserving heterogeneity of a reconstructed media in comparison with traditional cell concepts. This work is dealing with a random heterogeneous material composed of monodisperse circular hard particles within a matrix. The Wang tile sets are generated via algorithm with molecular dynamics and adaptive boundaries approach. Even though previous works proved usefulness of the Wang tiling for material reconstruction, still plenty of questions remain unanswered. In here we would like to provide simulations with emphasis on the overall particle distribution and the ratio of hard disc number to tile size. The results and discussion should followers help with settings of both tile generations and the tiling algorithms when creating samples of various degree of heterogeneity.

Adaptive Boundaries of Wang Tiles for Heterogeneous Material Modelling

This contribution deals with algorithms for the generation of modified Wang tiles as a tool for the heterogeneous materials modelling. The proposed approach considers material domains only with 2D hard discs of both equal and different radii distributed within a matrix. Previous works showed potential of the Wang tile principles for reconstruction of heterogeneous materials. The main advantage of the tiling theory for material modelling is to stack large/infinite areas with relative small set of tiles with emphasis on a periodicity reduction in comparison with the traditional Periodic Unit Cell (PUC) concept. The basic units of the Wang Tiling are tiles with codes (colors) on edges. The algorithm for distribution of hard discs is based on the molecular dynamics to avoid particles overlapping. Unfortunately the nature of the Wang tiling together with molecular dynamics algorithms cause periodicity artefacts especially in tile corners of a composed material domain. In this paper a new algorithm with adaptive tile boundaries is presented in order to avoid edge and corner periodicity.

HYPERGRAPH AUTOMATA: A THEORETICAL MODEL FOR PATTERNED SELF-ASSEMBLY

Patterned self-assembly is a process whereby coloured tiles self-assemble to build a rectangular coloured pattern. We propose self-assembly (SA) hypergraph automata as an automata-theoretic model for patterned self-assembly. We investigate the computational power of SA-hypergraph automata and show that for every recognizable picture language, there exists an SA-hypergraph automaton that accepts this language. Conversely, we prove that for any restricted SA-hypergraph automaton, there exists a Wang Tile System, a model for recognizable picture languages, that accepts the same language. The advantage of SA-hypergraph automata over Wang automata, acceptors for the class of recognizable picture languages, is that they do not rely on an a priori defined scanning strategy.

THE FINITENESS PROBLEM FOR AUTOMATON SEMIGROUPS IS UNDECIDABLE

The finiteness problem for automaton groups and semigroups has been widely studied, several partial positive results are known. However, we prove that, in the most general case, the problem is undecidable. We study the case of automaton semigroups. Given a NW-deterministic Wang tile set, we construct a Mealy automaton, such that the plane admits a valid Wang tiling if and only if the Mealy automaton generates a infinite semigroup. The construction is similar to a construction by Kari for proving that the nilpotency problem for cellular automata is unsolvable. Moreover, Kari proves that the tiling of the plane is undecidable for NW-deterministic Wang tile set. It follows that the finiteness problem for automaton semigroups is undecidable.