Continuous automaton

About: Continuous automaton is a research topic. Over the lifetime, 947 publications have been published within this topic receiving 17417 citations.

Dynamic neighbourhood cellular automata

Abstract: We propose a defi nition of Cellular Automaton in which links between cells can change during the computation. This is done locally by each cell, which can reach the neighbours of its neighbours in a single computational step. We suggest that Dynamic Neighbourhood Cellular Automata can serve as a theoretical model for studying Algorithmic and Computational Complexity issues in the are of Ubiquitous Computing. We illustrate this approach by giving an optimal logarithmic time solution of the Firing Squad Synchronisation problem in our model, which is an exponential speed-up over classical Cellular Automata.

Automaton structure preserving morphisms with applications to decomposition and simulation

Abstract: Publisher Summary This chapter discusses automaton structure preserving morphisms with applications to decomposition and simulation. The relation that expresses the preservation of structure from a structured automaton M to a structured automaton M' is called a structure preserving morphism. Such a morphism preserves not only the transition and output functions of the abstract automaton but also the manner in which these functions arise out of the local coordinate functions in its structured representation. The chapter describes the truly homomorphic preservation of the transition function and the reduction of complexity for measures relevant to the simulation of structured automata. The introduction of the structure morphism concept makes possible improved prediction of the relative complexities of an automaton and its simulators. As the number of possible simulators is controlled by the simulation criterion, more efficient search methods can be developed using structure morphisms.

Local entropy patterns in continuous cellular automaton models

Abstract: The article presents the results of the simulation of a cellular automaton model whose cells assume continuous values. The described experiment examined the behaviour of local entropy and showed the occurrence of its three qualitatively different images with their character resembling Turing patterns. The hypothesis, according to which the behaviour of local entropy might be identified with an image created in a reaction-diffusion process, was presented. Problem The research study concerning the mathematical foundations of creating cultures (Zgrzebnicki, in press-a) considers a continuous cellular automaton model as a hypothetical framework. Although there have been many studies on the properties of discrete cellular automata published so far, the models whose cells assume any values limited only by floating point precision have been much less discussed in scientific works. The research presented herein assumed that the discussed automaton may change values of its cells according to any function transforming the values of neighbours in the neighbourhood of a certain radius, and that this radius and weight of the neighbours' contribution to a cell's change may also be arbitrarily defined for the purposes of a certain experiment. In comparison with discrete automata (Wolfram, 1984), such assumptions imply a much broader class of solutions. The conducted research is aimed at finding an answer to the question whether there are any general regularities or patterns describing the dynamics of the assumed model. Method Let there exist a matrix !, whose components are called cells. Let there exist a function ", which transforms matrix ! into matrix !’ in a way that: $% ′ = "()% * $*, ,) , where $ stands for the cell's value, while % and * mean coordinates, and r means a radius within which the neighbourhood of a certain cell has an influence on its value's transformation. Furthermore, to eliminate the problem of boundary values, let us assume that the boundaries of the matrix are glued together so that topology of the described space is identical with a torus. Figure 1: The value of cells established on the basis of the values of their neighbours located within the radius , = 3. Topology of glued boundaries guarantees continuity of solutions on the edges of the matrix. Finally, let us consider process 0 in which values of all the cells are changed so that a matrix obtained after the change becomes a source matrix for the next step: 0:! → !3 → !33 → !333 → ⋯

Some more on the basis finite automaton

Abstract: We consider in this paper the basis nite automaton and its some properties We shall also consider some properties of special binary relation dened on the sets of states of canonical automata for the given language and for its mirror image We shall also consider an algorithm of constructing the basis automaton dening the language which has a priory given variant of this relation