Related papers: Finite-state transducers for substitution tilings
Checking infinite-state systems is frequently done by encoding infinite sets of states as regular languages. Computing such a regular representation of, say, the set of reachable states of a system requires acceleration techniques that can…
The modeling and control of networks over finite lattices are studied via the algebraic state space approach. Using the semi-tensor product of matrices, we obtain the algebraic state space representation of the dynamics of (control)…
In 2023, the quest for an aperiodic monotile was answered by the hat monotile. In this article, structures in this aperiodic tiling are discovered, which allow for a direct computation of the tiling, similar to well-known methods for the…
Rhombus Penrose tilings are tilings of the plane by two decorated rhombi such that the decoration match at the junction between two tiles (like in a jigsaw puzzle). In dynamical terms, they form a tiling space of finite type. If we remove…
This article considers application of genetic algorithms for finite machine synthesis. The resulting genetic finite state machines synthesis algorithm allows for creation of machines with less number of states and within shorter time. This…
Tilings and point sets arising from substitutions are classical mathematical models of quasicrystals. Their hierarchical structure allows one to obtain concrete answers regarding spectral questions tied to the underlying measures and…
We consider a class of finite state three-tape transducers which models the operation of shuffling and splitting words. We present them as automata over the so-called Shuffling Monoid. These automata can be seen as either shufflers or…
A combinatorial substitution is a map over tilings which allows to define sets of tilings with a strong hierarchical structure. In this paper, we show that such sets of tilings are sofic, that is, can be enforced by finitely many local…
One well studied way to construct quasicrystalline tilings is via inflate-and-subdivide (a.k.a. substitution) rules. These produce self-similar tilings--the Penrose, octagonal, and pinwheel tilings are famous examples. We present a…
The Tile Automata (TA) model describes self-assembly systems in which monomers can build structures and transition with an adjacent monomer to change their states. This paper shows that seeded TA is a non-committal intrinsically universal…
We study the construction of substitution tilings of the plane based on certain simplicial configurations of tangents of the deltoid with evenly distributed orientations. The random tiling ensembles are obtained as a result of tile…
Finite-State Transducers (FSTs) are effective models for string-to-string rewriting tasks, often providing the efficiency necessary for high-performance applications, but constructing transducers by hand is difficult. In this work, we…
Shape formation is one of the most thoroughly studied problems in programmable matter and swarm robotics. However, in many models, the class of shapes that can be formed is highly restricted due to the particles' limited memory. In the…
This short exposition presents an algorithm for an exact calculation of patch frequencies for the rhombic Penrose tiling. We recall a construction of Penrose tilings via dualisation, and by extending the known method for obtaining vertex…
After investigating by examples the unusual and striking elementary properties of the Penrose tilings and the Arnold cat map, we associate a finite symbolic dynamics with finite grammar rules to each of them. Instead of studying these…
Aperiodic tiling --- a form of complex global geometric structure arising through locally checkable, constant-time matching rules --- has long been closely tied to a wide range of physical, information-theoretic, and foundational…
We present a simple construction of hat tilings. The construction can be carried out by superimposing a triangular grid on a specially colored image and reading off the orientation of the tiles. We show that our construction produces valid…
This paper is concerned with realizing Lattes maps as subdivision maps of finite subdivision rules. The main result is that the Lattes maps in all but finitely many analytic conjugacy classes can be realized as subdivision maps of finite…
We define a model of advised computation by finite automata where the advice is provided on a separate tape. We consider several variants of the model where the advice is deterministic or randomized, the input tape head is allowed…
Some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied. A geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems, thus…