Groups and semigroups generated by Mealy automata were formally introduced in the early sixties. They revealed their full potential over the years, by contributing to important conjectures in group theory. In the current chapter, we intend to provide various combinatorial and dynamical tools to tackle some decision problems all related to some extent to the growth of automaton (semi)groups. In ...

Self-assembly, the process by which objects autonomously come together to form complex structures, is omnipresent in the physical world. Recent experiments in self-assembly demonstrate its potential for the parallel creation of a large number of nanostructures, including possibly computers. A systematic study of self-assembly as a mathematical process has been initiated by L. Adleman and E. Win...

In this article we give a new proof of the undecidability of the periodic domino problem. The main difference with the previous proofs is that this one does not start from a proof of the undecidability of the (general) domino problem but only from the existence of an aperiodic tileset. The formalism of Wang tiles was introduced in [Wan61] to study decision procedures for the ∀∃∀ fragment of the...

This work reports on some useful applications of the tile model to the speciication and execution of CCS-like process calculi. This activity is part of our ongoing research on the relation between tile logic and rewriting logic. 1 Overview Tile Logic 1;2 is a framework for modular descriptions of the dynamic evolution of concurrent systems, extending rewriting logic 3;4 (in the non-conditional ...

We study the minimal complexity of tilings of a plane with a given tile set. We note that any tile set admits either no tiling or some tiling with O(n) Kolmogorov complexity of its (n× n)squares. We construct tile sets for which this bound is tight: all (n × n)-squares in all tilings have complexity Ω(n). This adds a quantitative angle to classical results on non-recursivity of tilings – that w...

We study the problem of covering Rd by overlapping translates of a convex polytope, such that almost every point of Rd is covered exactly k times. Such a covering of Euclidean space by a discrete set of translations is called a k-tiling. The investigation of simple tilings by translations (which we call 1-tilings in this context) began with the work of Fedorov [5] and Minkowski [15], and was la...

The experimental design of many micropuncture studies requires clhanges in the perittll)lllar environrment. C'lassically, these clhanges are incluced by systemic infusion of suitable solutions. This technique lhas significant slhortcomings due to the tunavoidlable modification in the composition of the glomertular filtrate an(i to clhanges in the rate of glomerular filtration. Various approachi...

This paper describes the configuration of a floor-tile installation robot for commercial buildings. The research is motivated by the need to reduce the installation time and cost while guaranteeing consistent quality. In order to compete with human installation, a time of 24 seconds per installed tile has to be matched. The technical solution that is deemed feasible and capable of reducing this...

Several applications such as multiprojector displays and microscopy require the mosaicing of images (tiles) acquired by a camera as it traverses an unknown trajectory in 3D space. A homography relates the image coordinates of a point in each tile to those of a reference tile provided the 3D scene is planar. Our approach in such applications is to first perform pairwise alignment of the tiles th...

We consider the problem of building approximate squares in the standard Tile Assembly Model. Given any ε ∈ (0, 1 4 ] we show how to construct squares whose sides are within (1 ± ε)N of any given positive integer N using O ( log 1 ε log log 1 ε + log log εN log log log εN ) tile types. We prove a matching lower bound by showing that Ω ( log 1 ε log log 1 ε + log log εN log log log εN ) tile type...