# Algorithmic Method for the Design of Sequential Circuits with the Use of Logic Elements
## 1 Introduction  
The quality and effectiveness of automation systems to a large extent depends on the technical characteristics of the systems applied in them. Therefore, special attention is paid to the design of electronics and automation systems, which includes several stages, starting with the development of a technical task and ending with the serial manufacture of entire devices. Many tasks are involved in designing, and the most significant among these is the analysis and synthesis of the device's schematic diagram. Its correct implementation considerably reduces the time to execute, prepare and verify the diagram, and also decreases the number of corrections to the technical documentation after testing the prototype, and the experimental and trial series. All of the above not only expedite the developed device production but also reduce the cost of its manufacture. That is why it seems expedient to present the issues related to the design of control systems. A digital circuit in which the current state of the outputs depends not only on the current state of the inputs but also on the sequence of previous input states, is called a sequential circuit. Generally, when the system has n inputs \((x_{1},x_{2}\,\ldots\,x_{n})\) and m outputs \((y_{1},y_{2}\,\ldots\,y_{m})\) then  
\[Y^{t}=f\big{(}x^{t},x^{t-1},\,x^{t-2},\ldots\big{)}\]  
where:  
\(Y^{t}=(y_{1},y_{2}\,\ldots\,y_{m})^{t}\)--the current state of the system outputs at time \"\(t\)\",  
\(X^{t}=(x_{1},x_{2}\,\ldots\,x_{n})^{t}\)--the current state of the system inputs and  
\(x^{t-1}\)--previous input states of the system (appearing in moments \(t-\tau_{i}\) ),wherein \(i=1,\,2,\,\ldots,\,0<\tau_{1}\,<\tau_{2}<\ldots\) and  
\(f\) is a Boolean function. A digital circuit in which the current state of the outputs \(Y^{t}\) depends only on the current state of the inputs \(X^{t}\) is the name of the combination circuit. \[Y^{t}=f\big{(}x^{t}\big{)}\]  
In sequential systems, the relationship between the states of the outputs and the states of the inputs is not clear. The same states of inputs may in these systems correspond to different states of outputs. The status of the outputs is decided by the states of the inputs, which precede the current state of the inputs, i.e., the history of inputs. The memory of the circuit is responsible for including this history in sequential circuits. The memory of the sequential circuit is represented by a k-element sequence of binary signals \((Q_{i})\) defining the states of its individual elements. States \((Q_{i})\) of memory elements make up the so-called internal state of the sequential circuit. \[A=(Q_{1},Q_{2}\,\ldots Q_{n})\]  
In the automation of technological processes, a large variety of control systems and the use of new components lead to the fact that the issues of synthesis are quite well developed but only for specific systems [1; 2; 3; 4; 5]. By formulating a description of the sequential system using the concept of the internal state, it can be concluded that the current state of system outputs depends both on the current state of its inputs and the current internal state, while the current internal state is the function of the previous state of inputs and previous internal state [6; 7]. Formally, the sequential system can be described by the following functions  
\[A^{t+\tau}=\delta(A^{t},\,X^{t})\]  
\[Y^{t}=\lambda\big{(}A^{t},\,X^{t}\big{)}\]  
where:  
\(Y^{t}=(y_{1},y_{2}\,\,\ldots y_{m})^{t}\)--current state of outputs in time \"\(t\)\",  
\(X^{t}=(x_{1},x_{2}\,\ldots x_{n})^{t}\)--current state of inputs of the system,  
\(A^{t}=(Q_{1},Q_{2}\,\ldots Q_{n})^{t}\)--current input state of the system and  
\(A^{t+\tau}\) is the next internal state. The quantity \(\tau\) symbolizes the delay introduced by the system implementing the functions \(\delta\) (in fact, with each signal \(Q\) of vector \(A\) may be related to a different delay time, but for general considerations it is not important).