About one approach to the study of the topic

“Synthesis of combinational circuits using Carnot maps”

MICHAEL DOLINSKY

Faculty of Mathematics and Programming Technologies

Francisk Skorina Gomel State University

Sovetskaya Str.104, 246019, Gomel, BELARUS

Abstract. This article describes the technology of teaching the theme “Synthesis of combinational circuits using Carnot

maps” of basic digital electronics course to first/second-year students based on the DL.GSU.BY website. The main

advantages of the technology include training adapted to the student, many years of experience in practical application

and effectiveness. The following issues are consistently considered in the article: the theoretical foundations of the

topic; library of standard components; a system of step-by-step learning to solve problems on Carnot Maps, a

technology for using these tasks.

Keywords: basic digital electronics, Carnot maps, website DL.GSU.BY, Gomel State University

Received: June 24, 2022. Revised: July 5, 2023. Accepted: August 9, 2023. Published: September 29, 2023.

1 Introduction

Blended learning has been actively developed

andbefore COVID, but with the advent of this disease it

has become critical. Works [1,2] substantiate the need

for a transition to blended learning, which combines the

possibilities of traditional classroom learning and new

information technologies. The works [3-8] describe

examples of such blended learning on various topics:

mathematics [3], English language [4], earth sciences

[5], basic medical knowledge [6], anatomy [7], and

chemistry [8].

The author has been using blended learning for

disciplines related to teaching programming and digital

electronics for many years at the Faculty of

Mathematics and Programming Technologies

(specialties: "Information Technology Software",

"Informatics and Programming Technologies",

"Applied Informatics") of Gomel State University

named by F.Skorina [9-11].

To increase the effectiveness of training, the

system for designing, simulation and debugging of

digital electronics circuits HLCCAD [12] and the

instrumental system for distance learning DL.GSU.BY

[13] both developed under the guidance of the author

are used. This work presents the author's approach to

the introduction of blended learning in the basics of

digital electronics.

2 Problem Formulation

In recent decades, the conditions for learning

have been rapidly changing: computer tools and

Internet technologies are being improved, the level of

training and motivation of students is decreasing on

average, and the requirements for knowledge and skills

of university graduates are growing at the same time.

This leads to the need to change the learning process, so

that, on the one hand, theoretical knowledge is given in

much more detail and in a much simpler language than

was previously accepted, on the other hand, not only

and not so much seminars with students were used to

consolidate the knowledge gained, but rather solving

practical problems. At the same time, a significant part

of such tasks can be performed in systems that simulate

projects developed by students. To involve poorly

prepared students in the educational process, it is useful

to use tasks of a control and training nature of various

forms, when, in the process of performing a system of

automatically checked tasks, the student implicitly

receives training information. The technical basis of

this approach to learning is the software packages

developed at F. Skorina GSU under the guidance of the

author.

This work is devoted to the description of such

a modified methodology for teaching the topic

"Synthesis of combinational circuits from truth tables”,

taking into account the trends described above. Before

this topic, students have already studied the topics

"Introduction to the subject" .

The topic "Introduction to the subject" is

specifically intended for mastering the HLCCAD

system, with which you can edit, simulate and debug

the functional diagrams of digital devices, as well as to

familiarize yourself with the basic logical operations

NOT, AND, OR, XOR and the corresponding basic

logic elements.

3 Problem Solution

3.1 Theoretical Background

The topic "Synthesis of combinational circuits

according to truth tables", on the one hand, consolidates

knowledge and skills in applying the basic logical

elements NOT, AND, OR, XOR to solve problems for

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DOI: 10.37394/232010.2023.20.9

Michael Dolinsky

E-ISSN: 2224-3410

Volume 20, 2023

the design of digital devices, on the other hand,

develops the skills of analysis (and mental simulation)

of circuits, composed of these basic logical elements,

and on the third hand introduces the method of

minimizing logical functions by means of Carnot maps.

Let us consider in more detail what a student

should know to be able to solve problems on the topic

“Carnot Maps”. Let's start with the formulation of the

conditions of the problems. It can be explicitly

represented by a truth table, like so:

Design a circuit that works according to the

following truth table

x4 x3 x2 x1 y4 y3 y2 y1

0 0 0 0 0 0 0 1

0 0 0 1 0 0 1 0

0 0 1 0 0 0 1 1

0 0 1 1 0 1 0 0

0 1 0 0 0 1 0 1

0 1 0 1 0 1 1 0

0 1 1 0 0 1 1 1

0 1 1 1 1 0 0 0

1 0 0 0 1 0 0 1

1 0 0 1 1 0 1 0

1 0 1 0 1 0 1 1

1 0 1 1 1 1 0 0

1 1 0 0 1 1 0 1

1 1 0 1 1 1 1 0

1 1 1 0 1 1 1 1

1 1 1 1 0 0 0 0

Figure 1. Truth table 1

Or represented by the description of the

circuit operation algorithm, for example:

The 3-bit input is a binary number. Design a

device that outputs the number of zero bits in the input

number.

And then the student must make a truth table

himself, for example like this:

x1 x2 x3 y1 y2

0 0 0 1 1

0 0 1 1 0

0 1 0 1 0

0 1 1 0 1

1 0 0 1 0

1 0 1 0 1

1 1 0 0 1

1 1 1 0 0

Figure 2. Truth table 2

The next step is for each output variables, enter

the ones in the Carnot Map. For example, for the output

y3 of the first task, the Carnot map filled with ones will

look as shown in Figure 3.

Figure 3. Carnot map

The next step is to cover all the ones with a

minimum amountm of ships of maximum area. Each of

these ships has an area equal to a power of two. They

are shown in Figure 4.

Figure 4. Covering ships

In the figure above, a vertical ship of four cells

“did not fit”.

When covering, the following considerations

should be taken into account:

- You only need to cover ones, the ship cannot

cover empty cells

- The Carnot map is considered glued around the

edges (left to right, top to bottom).

Accordingly, the ship can be located on the left and

right columns or on the top and bottom lines of the

Carnot Map.

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DOI: 10.37394/232010.2023.20.9

Michael Dolinsky

E-ISSN: 2224-3410

Volume 20, 2023

After completing the coverage by ships, it is

required to write out the corresponding logical

functions. For example, for the presented Karnot map

of the output y3, such a function looks like this:

y3=x3^x2 V x3^x1 V ^x3x2x1

Here the conjunction sign between the variables is

omitted to increase readability, the symbol V denotes

the logical operation “disjunction”, the symbol ^

denotes the logical operation “inversion”. And the

formula itself can be interpreted as follows: The

variable y3 gets the value 1 if x3=1 and x2=0 or if x3=1

and x1=0 or if x3=0 and x2=1 and x1=1, which is

consistent with the task truth table 1.

3.2. Schema validation technology

To check the correctness of the assignment, the

student must draw the resulting scheme in the

HLCCAD system, check its correctness by simulation

and send it to the DL.GSU.BY website for automatic

verification of the correctness of the designed scheme.

There, the project received from the student is launched

for simulating in the HLCCAD system on a set of tests

prepared by the author of the problem. If the scheme

gives correct answers on all tests, the task is accepted,

otherwise it is not, and the student is informed under

what input data his scheme gave incorrect answers. The

correct answers are also reported.

3.3 Library of standard components

Figure 5. Library of standard elements

Figure 5 shows the table of contents of the

library of standard logic elements that can be used in

circuit design.

Figure 6. NOT and NOT8

The elements NOT and NOT8 (Figure 6)

represent the inversion of one or 8 contacts (bit by bit).

We also note the presence in the HLCCAD

system of the ability to change the dimension of the

contacts to the required one, which is shown in Figure

Figure 7. Changing the dimension of NOT

By changing the value in the Dimension field,

you can get a 16-bit inverter (Figure 8).

Figure 8. NOT16

Elements XOR2, XOR3, XOR4 (figure 9)

represent, respectively, 2-, 3- and 4-bit XOR operations

Figure 9. XOR2, XOR3, XOR4

At the same time, it is possible to establish the

required number of contacts, for example 6, as shown in

Figure 10.

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Michael Dolinsky

E-ISSN: 2224-3410

Volume 20, 2023

Figure 10. Changed XOR dimensions

Similarly, AND2, AND3, AND4 are elements that

perform logical conjunction on 2, 3, and 4 bits, and

OR2, OR3, OR4 are elements that perform logical

disjunction on 2, 3, and 4 bits. In addition, for both

types, as well as for XOR, you can set an arbitrary

required number of input bits.

The elements XORu8, XORp8, ORu8, ORp8,

ANDu8, ANDp8 are not used by students when

working with Carnot maps, therefore they are not

described in this article. Nevertheless, one can easily

guess their purpose from their drawings (Figure 11).

Figure 11. Additional XOR-AND-OR elements

HLLCAD also allows you to invert input and

output contacts directly on the body of a standard

element. For example, 4AND-NOT can be obtained like

this (Model type DOT), which is shown in Figures 12

and 13.

Figure 12. Inverting the output pin

Figure 13. Inverting an input pin

To solve the task, students create a device body (Figure

14).

Figure 14. Device body

And then for each output contact draw a circuit.

For example, for the previously given function y3, it

might look like this (Figure 14):

Figure 14. Device diagram

When drawing a circuit, you can use both

visible lines and “invisible” ones by simply specifying

the same name on the corresponding output and input

pins. When drawing, you can also use branching, as

shown in Figure 15.

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DOI: 10.37394/232010.2023.20.9

Michael Dolinsky

E-ISSN: 2224-3410

Volume 20, 2023

Figure 15. Branching on the device diagram

However, students are advised not to abuse

such opportunities, so as not to complicate the

perception of schemes.

After the circuit drawing is completed, the

student can check the functionality of the circuit by

running the simulation and entering input values,

HLCCAD displays the resulting values on each input

and output pin. This allows the student to analyze the

correctness of the circuit, which is shown in Figure 16.

Figure 16. Device simulation

The symbol U means that as a result of the simulation it

was not possible to calculate the value on this contact.

In this case, this is expected, since the logical functions

for the outputs y1, y2, y4 have not yet been introduced.

3.4 Technology of step-by-step learning to solve

problems on Carnot maps

As noted, students have widely varying levels

of training and motivation. For some students, a brief

summary of the theory is enough, others require

additional explanations, others “gain knowledge” by

working on a problem in the HLCCAD system.

However, there is a fourth group of students who do not

have enough theory, additional explanations and work

in HLCCAD. It is for them (as well as for those who

missed the corresponding theoretical lesson) that a set

of tasks has been developed that ensures the

assimilation of the material by the students of this

group. Further, the tasks of this group are sequentially

presented and described.

Figure 17. Variable names

In this task (Figure 17), you need to correctly

label the variables in the Carnot Map. This is done by

clicking on the ? under each of which is a list of

variables to choose from (x1,x2,x3,x4). If the student

does not know the correct answer, he must click on the

“I don’t know” button in the lower right corner of the

task and then he will see the correct answers (Figure

18).

Figure 18. Correct answers of names

The student need to understand how they are

obtained, remember and click the “Start” button, to the

left of the “I don’t know” button. After that, perform

the task correctly. If the task is completed correctly, a

check mark appears on the screen (Figure 19):

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Michael Dolinsky

E-ISSN: 2224-3410

Volume 20, 2023

Figure 19. Correct execution checkbox

As long as she's gone, the task has been

completed incorrectly. This is followed by the task

“Borders of the Carnot Map” (Figure 20).

Figure 20. Carnot map borders

Instead of questions, it is required to enter the

desired combination of numbers 00 01 11 10, which are

characterized by the fact that each next combination

differs from the previous one in exactly one digit,

including the difference between the last and the first.

In this task, as in the previous and all subsequent tasks,

there is also a “Don’t know” button in order to see the

correct answers, and a “Start” button to return to the

independent completion of the task until a check mark

appears, meaning that the task has been completed

correctly.

This is followed by the task “The scope of the

function” (Figure 21), where it is required to

systematically (in the order of binary counting) list all

possible combinations of values of the input variables

by clicking on the questions and choosing 0 or 1 in the

corresponding position.

Figure 21. Specify all input combinations

The correct answers for this task are shown

below in Figure 22:

Figure 22. Valid input combinations

Students are encouraged to copy all tasks into

their notebooks, both for better memorization of the

material and for later use as hints when performing

individual practical tasks.

In the next task, you need to enter for a specific

output one from the truth table into the Carnot Map by

clicking on the questions and choosing between a space

(corresponding to zero) and one (Figure 23).

Figure 23. Enter ones

For example, for the task above, the correct answers are

(Figure 24).

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DOI: 10.37394/232010.2023.20.9

Michael Dolinsky

E-ISSN: 2224-3410

Volume 20, 2023

Figure 24. Correct answer to “Deposit ones”

Then similar tasks follow for the other three

output contacts. The next training task is to “Cover the

Carnot Map” (Figure 25).

Figure 25. Cover Carnot Map

The figures of ships in the upper part of the

screen can be moved with the mouse and rotated with

the arrows "Right" and "Left". It is required using the

minimum number of ships of the maximum area to

cover all ones (and only ones). The correct answer is

(Figure 26):

Figure 26. Proper coverage

Four ships were used:

Quadruple horizontal on the bottom line.

Square 2*2 on the bottom left.

Square 2 * 2, in the lower left and lower right corner.

Single ship covering cell 01-11.

This is followed by the same tasks for the rest

of the output variables.

Then comes the task “Logic functions” (Figure

27), in which it is required to enter in the input fields

the logical functions of the specified terms first

separately, and then their disjunction for the output

contact.

Figure 27. Enter Boolean Functions

Note that there are two types of hints here:

1) Color, while the input is correct it is green, the

first incorrect character is highlighted in red.

2) As in all tasks, you can click the “I don’t know”

button in the lower right corner of the task.

Then there are three similar tasks for three other

output contacts.

Next come the tasks that provide propaedeutics for

subsequent work in the HLCCAD system.

The first of them suggests that on the diagram in

which the connecting lines are drawn, put them in the

correct positions of the body (Figure 28). At the same

time, the diagram also shows the logical functions that

should be drawn.

Figure 28. Insert elements

The correct answer for this task can also be

seen by clicking the “I don’t know” button (Figure 29)

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DOI: 10.37394/232010.2023.20.9

Michael Dolinsky

E-ISSN: 2224-3410

Volume 20, 2023

Figure 29. Correctly entered elements

Three similar tasks follow for three other output

contacts.

The next task is called “Make a diagram like a

puzzle” (Figure 30).

Figure 30. Make a diagram as a puzzle

The correct scheme can be seen by clicking on

the leftmost button “Show answers” (Figure 31).

Figure 31. Completed puzzle

The “Return to task” button allows you to

return to the task.

Next are the last two tasks “Calculate the

scheme” (Figure 32). Each of them has input values on

contacts x1, x2, x3,x4. Student clicks on symbols ?

should replace them with 0 or 1 for all intermediate and

output values.

Figure 32. Calculate the scheme

For example, for the diagram above, the correct

answer is shown in Figure 33.

Figure 33. Correctly calculated scheme

3.5 Technology of using practical tasks

After presenting the theory on this topic at a

lecture, students are invited to solve the corresponding

tasks, for which they need to sequentially create Carnot

maps, perform coverage, write out logical functions,

draw and simulate the corresponding device in the

HLCCAD system and send it for verification in the

DL.GSU.BY. In case of errors, find and fix them using

the received tests and resubmit the solution for

verification. If successful, move on to the next problem.

If the student is not ready to independently

complete the tasks, he proceeds to the implementation

of the system of the above-described introductory tasks,

the successful completion of which also leads to the

design of the circuit in HLCCAD and sending it for

verification.

Students can also use these tasks during

practical exercises and self-study.

4 Conclusion

This article describes the methodology

developed by the author and repeatedly tested for

studying the topic "Synthesis of combinational circuits

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DOI: 10.37394/232010.2023.20.9

Michael Dolinsky

E-ISSN: 2224-3410

Volume 20, 2023

using Carnot maps" focused on working in groups of

students with fundamentally different levels of

motivation and preliminary training. A serious technical

basis of the methodology is the developed instrumental

system of distance learning (Distance Learning Belarus

- http://dl.gsu.by). The introduction of this teaching

methodology provided significant changes in the

quality of education, especially for the least prepared

and motivated category of students. At the same time,

the most prepared students are also satisfied with this

approach to learning.

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Michael Dolinsky

E-ISSN: 2224-3410

Volume 20, 2023

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The author contributed in the present research, at all

stages from the formulation of the problem to the

final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The author has no conflict of interest to declare that

is relevant to the content of this article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2023.20.9

Michael Dolinsky

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Volume 20, 2023