Application Center - Maplesoft

# Converting truth tables into Boolean expressions

You can switch back to the summary page by clicking here.

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)

 > kezd:=238,242:

 >

 >

 >

 >

 >

 >

 >

 >

 (2)

 Converting truth tables into Boolean expressions (2)

 (2)

 Logic variables and truth table of inputs: (2)

 (2)

 (2)

 (2)

 1    Output values [1, 1, 1, 0, 1, 1, 1, 0] (2)

 Boolean function is from MAXTERM = B*c+b (2)

 ______________________________________________________________________________ (2)

 2    Output values [1, 1, 1, 0, 1, 1, 1, 1] (2)

 Boolean function is from MINTERM = A+b+c (2)

 ______________________________________________________________________________ (2)

 3    Output values [1, 1, 1, 1, 0, 0, 0, 0] (2)

 Boolean function is from MAXTERM = a (2)

 ______________________________________________________________________________ (2)

 4    Output values [1, 1, 1, 1, 0, 0, 0, 1] (2)

 Boolean function is from MAXTERM = A*B*C+a (2)

 ______________________________________________________________________________ (2)

 5    Output values [1, 1, 1, 1, 0, 0, 1, 0] (2)

 Boolean function is from MAXTERM = A*B*c+a (2)

 ______________________________________________________________________________ (2)

Verification

 >

 >

 >

 A = 0    B = 0      C = 0     F = 1

 A = 0    B = 0      C = 1     F = 1

 A = 0    B = 1      C = 0     F = 1

 A = 0    B = 1      C = 1     F = 0

 A = 1    B = 0      C = 0     F = 1

 A = 1    B = 0      C = 1     F = 1

 A = 1    B = 1      C = 0     F = 1

 A = 1    B = 1      C = 1     F = 1

 >

 >

 >