# DISCRETE-DETERMINED MODELS (Y-SCHEMES), Basic relations. - Modeling of systems

## DISCRETE-DETERMINED MODELS (U-SCHEMES)

The features of the discrete-deterministic approach at the stage of formalizing the process of functioning of systems will be considered using the example of the use of the theory of automata as a mathematical apparatus. The automata theory is a branch of theoretical cybernetics, in which mathematical models - automata are studied. On the basis of this theory, the system is represented in the form of an automaton that processes discrete information and changes its internal states only at permissible instants of time. The concept of machine varies depending on the nature of the systems studied, from the accepted level of abstraction and the appropriate degree of generality.

## Basic relationships.

The machine can be represented as some device (black box), which receives input signals and takes off the output and which can have some internal states. An automaton is called a finite automaton , in which the set of internal states and input signals (and, consequently, the set of output signals) are finite sets.

An abstract finite automata can be represented as a mathematical scheme ( F-scheme ), characterized by six elements: a finite set of input signals (the input alphabet) ; a finite set of Y output signals (output alphabet); a finite set of Z internal states (internal alphabet or state alphabet); initial state z 0 , z 0 eZ; function of the transitions (p {z , x) by the output function i p (z, x). The automaton given by the F-scheme : F = X , Y, φ, z 0 & gt ;, - operates in a discrete automatic time, the moments of which are cycles, that is, equal intervals of time adjoining each other, each of which corresponds to constant values ​​of the input and output signals and internal states. We denote the state, as well as the input and output signals corresponding to the i-th cycle for 1-0, 1, 2, through r (/), y (C). In this case, by assumption, r (0) = r O y a

The abstract state machine has one input and one output channel. At each instant t = 0, 1, 2, ... of the discrete time, the t '-automaton is in a definite state r (t) from the set Z of states of the automaton, and at the initial time t = 0 it is always in the initial state r (0) = 2 o . At the time t, being in the state r (t), the automaton is able to perceive on the input channel the signal x (t) eX and to give out on the output channel a signal y (t) = ( i) (i), x (/)], passing to the state r (/ + 1) = & lt; p [r < ), x (i)], z (t) eZy y (/) € Y <. Abstract abstract automaton realizes some mapping of the set of words of the input of the alphabet X to the set of words of the output alphabet Y. In other words, if the input of the finite automaton set to the initial state r 0 the letters of the input alphabet x (0), x (1), x (2), that is, the input word, then the letters of the output alphabet y (0), > (1), y (2) y ... forming an output word.

Thus, the operation of the finite automaton proceeds according to the following scheme: in each i-th cycle, a signal x (t) is fed to the input of the automaton in the state z (t), to which it reacts by a transition to (n-1 ) -th cycle to the new state r (/ + 1) and output of some output signal. The foregoing can be described by the following equations: for the P-automaton of the first kind, also called the Mile Automaton ,

for an F-automaton of the second kind

An automaton of the second kind, for which

t. ie, the output function does not depend on the input variable * (/), is called the Moore automaton.

Thus, equations (2.13) - (2.17), completely defining the/-automaton, are a particular case of equations (2.3) and (2.4), when the system S is deterministic and a discrete signal X is input to this single input.

In terms of the number of states, finite automata with memory and without memory are distinguished. Automata with memory have more than one state, and memoryless machines (combinational or logic circuits) have only one state. In this case, according to (2.14), the operation of the combination circuit consists in that it assigns to each input signal * (/) a definite output signal y (/), that is, it realizes a logical function of the form

This function is called a Boolean function if the alphabets I and Y that contain the values ​​of the signals xyu, consist of two letters.

By the nature of counting discrete time finite automata are divided into synchronous and asynchronous. In synchronous P-automata , the times at which the automaton reads input signals are determined by forced synchronization signals. After the next synchronizing signal with the read and in accordance with equations (2.13) - (2.17), a transition to a new state occurs and an output signal is output, after which the machine can perceive the next value of the input signal. Thus, the response of the machine to each value of the input signal ends in one clock cycle, the duration of which is determined by the interval between neighboring synchronizing signals. The asynchronous P-automaton reads the input signal continuously, and therefore, reacting to a sufficiently long input signal of constant value x , it can, as follows from (2.13) - (2.17), several times to change the state, giving out a corresponding number of output signals, until it goes into a stable one, which can no longer be changed by this input signal.

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