6.9.1 Automata theory

Problem: Nonation \( a:=b(\bmod n) \) means that it is necessary to pun number \( a \) equal to the remainder of dividing number \( b \) by number \( n \). [a] denotes the integer part of number \( a \), i.e., the largest integer not exceeding \( a \). Number \( n \) is given in the table. Let's consider automaton \( A=(S, X, Y, \delta, \lambda) \), where \( S=\left\{s_{0}, s_{1}, \ldots, s_{n-1}\right\} \), \( X=\left\{x_{0}, x_{1}, x_{2}\right\}, Y=\left\{y_{0}, y_{1}\right\} \), in function \( \delta: S \times X \rightarrow S \) and \( \lambda: S \times X \rightarrow Y \) are defined as follows: for all \( s_{i} \in S, c_{j} \in \mathrm{X}, \delta\left(s_{i}, x_{j}\right)=s_{l} \), where \( l:=i^{2}+j+N(\bmod n), \lambda\left(s_{i}, x_{j}\right)=y_{k} \), where \( k:=\left[\frac{i+j+N}{2}\right](\bmod 2) \). Tasks: a) write out set \( S \); b) write out tables for the functions \( \delta \) and \( \lambda \); c) plot the diagram of automaton \( A \); d) calculate \( \delta\left(s_{o}, p\right) \) and \( \lambda\left(s_{0}, p\right) \) for all words \( p \), given in the table; e) Minimize automaton \( A \). For the obtained minimized automaton \( B \) : - write out sets of states, input and output signals; - write out tables, specifying the functions of transitions and outputs of automaton \( B \); - plot the diagram of automaton \( B \).