Related papers: Elementary fractal geometry. 4. Automata-generated…
We introduce a new geometric tool for analyzing groups of finite automata. To each finite automaton we associate a square complex. The square complex is covered by a product of two trees iff the automaton is bi-reversible. Using this method…
We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These…
Automata over infinite alphabets have emerged as a convenient computational model for processing structures involving data, such as nonces in cryptographic protocols or data values in XML documents. We introduce active learning methods for…
Distributed automata are finite-state machines that operate on finite directed graphs. Acting as synchronous distributed algorithms, they use their input graph as a network in which identical processors communicate for a possibly infinite…
Directed Algebraic Topology studies spaces equipped with a form of direction, to include models of non-reversible processes. In the present extension we also want to cover critical processes, indecomposable and unstoppable. The previous…
We cast new light on the existing models of one-way deterministic topological automata by introducing a fresh but general, convenient model, in which, as each input symbol is read, an interior system of an automaton, known as a…
We introduce a framework for constructing fractal trees via analytic generator fields, replacing discrete affine transformations and symbolic rewriting rules by the integration of smooth vector fields in an internal state space. In this…
A new method for constructing self-referential tilings of Euclidean space from a graph directed iterated function system, based on a combinatorial structure we call a pre-tree, is introduced. In the special case that we refer to as…
Cellular automata are a set of computational models in discrete space that have a discrete time evolution defined by neighbourhood rules. They are used to simulate many complex systems in physics and science in general. In this work,…
The concept of typed topological space is introduced, for which open sets in a topology on a finite set will be assigned types (from lattice). The neighborhood system of a point, the closure and the connectedness can be defined according to…
Finite automata are used to encode geometric figures, functions and can be used for image compression and processing. The original approach is to represent each point of a figure in $\mathbb{R}^n$ as a convolution of its $n$ coordinates…
An interesting class of automatic sequences emerges from iterated paperfolding. The sequences generate curves in the plane with an almost periodic structure. We generalize the results obtained by Davis and Knuth on the self-avoiding and…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part we discus the main structures…
A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…
We introduce flat automata for automatic generation of tokenizers. Flat automata are a simple representation of standard finite automata. Using the flat representation, automata can be easily constructed, combined and printed. Due to the…
Given a multi-valued function $\Phi$ on a topological space $X$ we study the properties of its fixed fractal, which is defined as the closure of the orbit $\Phi^\omega(Fix(\Phi))=\bigcup_{n\in\omega}\Phi^n(Fix(\Phi))$ of the set…
Cellular automata (CA), originally developed as computational models of natural processes, have become a central subject in the study of complex systems and generative visual forms. Among them, the Ulam-Warburton Cellular Automaton (UWCA)…
We extend the functorial approach to automata by Colcombet and Petri\c{s}an [arXiv:1712.07121] from the category of sets to any elementary topos with a natural number object and establish general Myhill-Nerode theorems in our setting. As a…
Cellular automata, CA for short are continuous maps defined on the set of configurations over a finite alphabet A that commutes with the shift. They are characterized by the existence of local function which determine by local behavior the…
Fractal geometry, defined by self-similar patterns across scales, is crucial for understanding natural structures. This work addresses the fractal inverse problem, which involves extracting fractal codes from images to explain these…