Nonlinear automata over a finite ring

Abstract

510.5+681.3The properties of the class of information-lossless automata represented by recurrent relations over a finite ring are investigated. For these automata, the structure of classes of equivalent states is investigated, problems of parametric identification and identification of the initial state are solved, and the variation in the behavior of such automata as a result of variation of their parameters or initial states is characterized.

On the complexity of analysis of automata over a finite ring

“…For automata over the ring K = + × ( , , ) K , problems of construction of classes of equivalent states for an automaton, identification of the initial state of an automaton, parametric identification, and also construction of sets of fixed points for automaton mappings are reduced to the solution of corresponding systems (linear or nonlinear) of equations (as a rule, with parameters) over the ring K (which is illustrated in [12][13][14][15] in investigating automata over the ring Z p k ).…”

“…, which allows us to propose the following scheme of solution of the system of equations (14). Scheme 2.…”

“…Scheme 2. Replacing elements of the ring K by elements of the set B in formula (14) and operations in the ring K by actions in the algebraic system B that are specified by equalities (15) and (16), we obtain the following system of equations:…”

“…In [12], linear automata and in [13] linear one-dimensional automata with a lag l are investigated. In [14], nonlinear automata are considered whose "nonlinearity" is characterized by the fact that a change in values of state variables and output variables is represented by the algebraic sum of quadratic and linear forms of state variables and a linear form of input variables. In [15], nonlinear reversible symmetric automata are investigated that are constructed on the basis of the Guckenheimer-Holms cycle and a free-running system, i.e., two chaotic dynamic symmetric model systems.…”

“…Despite the fact that a ring Z p k is chosen in [12][13][14][15], the obtained results can be naturally generalized to any finite commutative-associative ring with unity K = + × ( , , ) K (in what follows, the ring K for brevity). The present work is devoted to the analysis of models and methods underlying the construction of such generalizations.…”