The Use of Instrumental and Controlling Program Master of Logic for Solving the Basic Class Problems of Propositional Algebra: Didactic Aspect

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Figure 1 shows the main window of the program.

3. Using Master of Logic Program for Solving Problems

These are the examples of solving the mentioned problems with Master of Logic program.

3.1. Construction of Truth Table and Determination of Formula Class

Problem. For the formula

to build the truth table and to determine to what class it belongs.

Solution.

To solve this problem follow these steps:

1) to choose "Таблиця істинності" (Truth Table) item in the menu "Побудова" (Build) (Point 1 Figure 1) or press the button with a mouse, or the button combination "Ctrl+T";

2) to enter the formula with a panel set and/or a keyboard in the window that has appeared and to press "Enter" button or "OK" button. Problem solution with Master of Logic Program is at Figure 3;

Figure 2. Panel Set of Master of Logic Program

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Figure 3. Truth Table and Formula Class in Master of Logic Program

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3) to put a mark next to the item "Клас формули" (Formula Class) with left mouse button to determine the formula class (Point. 1 Figure 3).

3.2. Construction of Perfect Normal Forms
3.2.1. Construction of Perfect Normal Forms for a Formula

Problem. To build perfect conjunctive and disjunctive normal forms (PCNF and (PDNF) for the formula

Solution.

To solve this problem, follow these steps:

1) to choose items "ДКНФ\за формулою" (PCNF\by formula) and "ДДНФ\за формулою" (PDNF\by formula) in the menu "Побудова" (Build) (Point 1, Fig. 1) or to press , buttons with a mouse, or the corresponding button combinations: "Ctrl+K", "Ctrl+D";

2) to enter the formula and to press "Enter" button or "OK" button in the window that has appeared with a panel set (Figure 2) and/or a keyboard. The problem solution is demonstrated at Figure 4.

Figure 4. PCNF and PDNF for the formula in Master of Logic Program

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3.2.2. Construction of Perfect Normal Forms According to Truth Table

Problem. Truth table of some formula of propositional algebra has a form:

Table 1. Truth Table

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View current table in a new window

Tables index

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To build perfect conjunctive and disjunctive normal forms of formula.

Solution.

1) to choose items "ДКНФ\за таблицею" (PCNF\by table) and "ДДНФ\за таблицею" (PDNF\by table) in the menu "Побудова" or to press , , buttons or the corresponding button combinations: "Ctrl+Alt+K", "Ctrl+Alt+D";

2) to fill in the right column of the table with the values of formula truths in the window that has appeared (Figure 5) (the sets of variables values are filled in by the program automatically);

3) to press "OK" or "Enter" buttons.

The result of these steps is shown in Figure 5.

Figure 5. PCNF and PDNF according to truth table in Master of Logic Program

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3.3. Finding out the Equality of Formulas

To find out whether the formulas і are equivalent.

Solution.

To solve this problem it is necessary:

1) to choose the menu item "Рівносильність" (Equality) (Point 2, Figure 1), or to press the button , or key combination "Alt+R";

2) to enter the given formulas in the window that has appeared with a panel set and/or a keyboard and to press the buttons "OK" or "Enter". The solution of the problem is given at the bottom of the window (Figure 6);

3) if the formulas are not equivalent (as in this case), the button should be pressed to find out on what sets of truth values of proposed variables the formulas і have different values of truth.

Figure 6. Formula Equality in Master of Logic Program

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3.4. Finding out Logical Implication

Problem. To find our whether the formula is a logical consequence of formulas, , і .

Solution.

Steps to solve this problem by Master of Logic are as follows:

1) to choose the menu item "Слідування" (Following) (Point 3, Figure 1), or to press the button , or the button combination "Ctrl+Alt+S";

2) to enter the premise and conclusion in the window that has appeared with a panel set and/or a keyboard (to add a new premise you should either press an arrow down on the keyboard , or the button with a mouse, or the button );

3) to press the button "OK" or "Enter".

The answer found by the program (Figure 7) is found with the criterion of logical implication:

де – arbitrary formulas of propositional algebra

Figure 7. Logical Implication of Master of logic Program

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3.5. Solving systems of Boolean equations

Problem. To solve the system of Boolean equations:

Solution.

The procedure for finding the solution of this system by Master of Logic is as follows:

1) to choose the menu item "Система" (System) (Point 4, Figure 1), or to press the button , or the button combination "Alt+S";

2) to enter system equation for solving in the window that has appeared with a panel set and/or a keyboard (adding a new equation is performed by analogy with adding a premise in the previous problem );

3) to press the button "OK" or "Enter".

All system solutions are displayed at the bottom of the window (Figure 8).

Figure 8.System of Boolean Equations in Master of Logic Program

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3.6. Building a Minimal Normal Forms

One of the significant problems which is of great practical importance, particularly in the design of digital machines, logic circuits, electronic circuits is the problem of minimization of Boolean functions. In general, this problem is formulated as follows: to find the analytical expression of the given Boolean function in a form that contains the minimum number of variables. This problem is nor solved in the general formulation, however, it can be solved in the class of disjunctive or conjunctive normal forms. The construction of minimal normal forms is rather difficult task, which cannot be efficiently solved without a computer. The universal method of Quine–McCluskey adopted by the author for computer use is in the basis of building minimal normal forms mechanism being programmed in Master of Logic (See e.g. ^{[1, 3, 5, 8]}).

Problem. To build minimal conjunctive and disjunctive normal forms (CNF and DNF) for the formula

Solution..

To solve this problem in Master of Logic Program follow these steps:

1) to choose the item "Побудова КНФ і ДНФ" (Building CNF and DNF) in the menu "Мінімізація" (Minimization) (Point 5, Figure 1), or to press the button , or the button combination "Alt+N";

Figure 9. Minimization in Master of Logic Program

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2) to enter the formula in the window that has appeared with a panel set and/or a keyboard;

3) to press the buttons "OK" or "Enter".

The answer is at Figure 9.

4. Controlling Component of Master of Logic Program

The use of computers for the control of knowledge and skills is one of the modern and efficient way of checking current state of students' (pupils') knowledge as to the certain material.

Controlling component of Master of Logic program:

– helps to create files for independent and control papers for prepared sets of tasks from related themes of Mathematical Logic (Discrete Mathematics, etc) or for complex control of knowledge and skills and to exercise them;

– has a flexible and convenient mechanism for setting files of control tasks.

Corresponding file of control tasks should be formed for automated control measures with Master of Logic.

The program allows creating the tasks of all mentioned classes of propositional algebra problems.

To include a task to a control file, it is necessary to enter input data in Master of Logic widows that are meant for solving such problems and press the button “Додати у файл” (Add to file).

For example, a control work should include such a task: to build a truth table and perfect disjunctive normal form for the formula of propositional algebra

(1)

To do it, it is necessary to open the window for solving such class problems, to enter this formula (1) and press the button (Figure 10).

Figure 10. Minimization in Master of Logic Program

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As a result, input data of the problem will appear in the main window of creating the file of control tasks; as well as the sign of "-" (minus) and √ symbol will appear in corresponding columns (Figure 11).

Figure 11. Main Window of Forming the File of Control Tasks in Master of Logic Program

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The sing of "-" (minus) next to each symbol √ in the table means that the detail settings of the problem have not been approved yet (Figure 12); the number in brackets corresponds to the number of a problem in the file of control tasks.

Figure 12. Detailed Settings of a Task in Master of Logic Program

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To correct the general (global) settings of control tasks, it is necessary to press the button in the window of creating files (See Figure 11).

The general settings of control are the parameters of:

– time;

– assessment;

– the order of performing tasks;

– processing answers;

– protection from unauthorized access.

General settings of control are formed in the corresponding window (Figure 13).

Figure 13. General Settings of Control in Master of Logic Program

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