Related papers: Finite-state transducers for substitution tilings
Sufficiently accurate finite state models, also called symbolic models or discrete abstractions, allow one to apply fully automated methods, originally developed for purely discrete systems, to formally reason about continuous and hybrid…
In this paper we investigate certain properties of semi-totalistic cellular automata (CA) on the well known quasi-periodic kite and dart two dimensional tiling of the plane presented by Roger Penrose. We show that, despite the irregularity…
We explore the connections between automata, groups, limit spaces of self-similar actions, and tilings. In particular, we show how a group acting ``nicely'' on a tree gives rise to a self-covering of a topological groupoid, and how the…
When $\mathbb{Z}^d$ is represented as a finite disjoint union of translated integer sublattices, the translated sublattices must possess some special properties. Such a representation is called a \emph{lattice tiling}. We develop a…
We introduce a formalism for handling general spaces of hierarchical tilings, a category that includes substitution tilings, Bratteli-Vershik systems, S-adic transformations, and multi-dimensional cut-and-stack transformations. We explore…
We give a set of tiles that enforces the sphinx tiling substitution system; the tiles are thus aperiodic.
A new family of decagonal quasiperiodic tilings are constructed by the use of generalized point substitution processes, which is a new substitution formalism developed by the author [N. Fujita, Acta Cryst. A 65, 342 (2009)]. These tilings…
This article introduces spotlight tiling, a type of covering which is similar to tiling. The distinguishing aspects of spotlight tiling are that the "tiles" have elastic size, and that the order of placement is significant. Spotlight…
Minimizing finite automata, proving trace equivalence of labelled transition systems or representing sofic subshifts involve very similar arguments, which suggests the possibility of a unified formalism. We propose finite states…
The exactly solvable four-vertex model on a square grid with the different boundary conditions is considered. The application of the Algebraic Bethe Ansatz method allows to calculate the partition function of the model. For the fixed…
We consider the previously defined notion of finite-state independence and we focus specifically on normal words. We characterize finite-state independence of normal words in three different ways, using three different kinds of asynchronous…
We discuss problems of simultaneous tiling. This means that we have an object (set, function) which tiles space with two or more different sets of translations. The most famous problem of this type is the Steinhaus problem which asks for a…
We investigate the (non)-existence of universal automata for some classes of automata, such as finite automata and pushdown automata, and in particular the influence of the representation and encoding function. An alternative approach,…
This paper describes an alternative method of generating fixed points of certain substitution systems. This method centres on taking infinite words consisting of one repeated letter per word. These infinite words are then interlaced to form…
Suppose a set of prototiles allows $N$ different substitution rules. In this paper we study tilings of $\mathbb{R}^d$ constructed from random application of the substitution rules. The space of all possible tilings obtained from all…
Finite State Machine is a popular modeling notation for various systems, especially software and electronic. Test paths can be automatically generated from the system model to test such systems using a suitable algorithm. This paper…
This paper introduces a general approach for synthesizing procedural models of the state-transitions of a given discrete system. The approach is general in that it accepts different target languages for modeling the state-transitions of a…
Finite automata were used to determine multiple addresses in number systems and to find topological properties of self-affine tiles and finite type fractals. We join these two lines of research by axiomatically defining automata which…
Tile displacement is a newly-recognized mechanism in DNA nanotechnology that exploits principles analogous to toehold-mediated strand displacement but within the context of self-assembled DNA origami tile arrays. Here, we formulate an…
Two-, three- and four-dimensional representations of Penrose tilings of the plane are described. The vertices that occur in these representations lie on lattices. Symmetries and methods of visualizing these representations are discussed.…