Converting truth tables into Boolean expressions 

Dr.Laczik B?lint HUNGARY, Technical University of Budapest 

The simple method for designing such a circuit is found the normal form of Boolean expressions. The worksheet demonstrate the applicaton of Maple for the simplification of Boolean expressions by Maxterm-Minterm Methode.
The logic system is shown in the next figure.
 

The number of input Boolean variables is  ( ). 

The variables A, B, C, .., and they negated a = not(A), b = not(B), c = not(C), ..., are defined in the array 

                                                        var :=[A,a,B,b,C,c, ...,] 

The full truth table of input variables is 

The number of different inputs is . The outputs [0,, ..., 0] and [1,,...,1] are not used, buth all of different permutations with length of 0 and 1 elements are practicable. The number of the different outputs is   . 

The binary form of the number n is the  n-th value  of the truth table! 

The pre-defined Boolean rules for the simplification are: 

The maxterm is sum of products of the input variables for logic value 1 . The minterm is product of sum's of the input variables for logic value 0. 

The maxterm is defined by very simple way as
                                                                  

The minterm derived by the following recursion:
 

The choose between  maxterm or minterm depend from the number of its operator. So 

In first step we substitute all of the subexpressions rules1 into an expression minterm S:
 

In second step we substitute all of the subexpressions into an expression S two time: 

The next step is the application of Boolean rules the so-called redundance law. If the expression for variables P and Q in form yield by 

                                                            expression = Q + P*Q 

then 

                                                           expression = Q + P*Q = Q*(1 + P) = Q 

For the simplification we define the matrix T with n columns and n rows where n = nops(S). In general 

If the denominator of element  is       and     then   the iist element of S is 

If the logic form is
                                                SS = P.q + P.Q + R = P.(q + Q) + R = P + R 

we apply the possible Boolean rule by matrice 

We can the number of operators SS to reduce by following loop: 

    The  values of input variables kezd are  ,  ,     

>

>

>

>

(1)

>

>

>

>

>

>

>

>

(2)

(2)

Verification 

>

>

>