Incompletely Specified Functions

Introduction

Incompletely specified functions, also known as can't happen conditions, is a situation that sometimes occurs when certain combinations of the variables of a function cannot occur. For these combinations we can select the value of the function to be 0 or 1; whichever leads to the more minimal solution. Related below is a situation where for certain combinations of the variables one does not care what the value of the function becomes (either 0 or 1). For these can't happen and don't care situations the Karnaugh map entry is X indicating that the particular cell can be taken either as 0 or 1.

Examples

A binary coded decimal counter, having four output lines, is connected to a logic network. It is required that the output of the network be logic 1 whenever there are two or more input lines at logic 1. Also, for the binary coded decimal number 0001, the output value is of no importance. A binary coded decimal number has values ranging from 0000 to 1001 (decimal 0 to 9) the values 1010 to 1111 (decimal 10 to 15) never occurs. Let the logic network have inputs A, B, C, D where A is connected to the most significant digit of the binary coded decimal number and D to the least significant. The output from the logic network will be:

Z = f(A, B, C, D) = (0011,0101,0110,0111,1001) =  (3,5,6,7,9).

With can't happen conditions: (1010, 1011, 1100, 1101, 1110, 1111) = (10, 11, 12, 13, 14, 15) 

and the don't care conditions: (0001)

Entering this on a Karnaugh map: 

The required function is therefore: Z = f(A, B, C, D) = BC + D

ncompletely Specified Functions

-       Incompletely specified function:

-       Output value is not known for every combination of input variables because all combinations cannot occur.

-       Truth table does not generate an output value for every possible combination of input values.

-       Min-terms or max-terms that are not used as part of the output function are called don�t care terms.

Examples

Conversion of BCD to Ex-3

Binary				Ex-3
W	X	Y	Z	A	B	C	D
0	0	0	0	0	0	1	1
0	0	0	1	0	1	0	0
0	0	1	0	0	1	0	1
�	�	�	�	�	�	�	�
1	0	0	1	1	1	0	0
1	0	1	0	Don�t care
1	0	1	1	Don�t care
�	�	�	�	�	�	�	�
1	1	1	1	Don�t care

A = f(w,x,y,z) = Σ(5,6,7,8,9) + Σ d(10,11,12,13,14,15)

B = f(w,x,y,z) = Σ(1,2,3,4,9) + Σ d(10,11,12,13,14,15)

C = f(w,x,y,z) = Σ(0,3,4,7,8) + Σ d(10,11,12,13,14,15)

D = f(w,x,y,z) = Σ(0,2,4,6,8) + Σ d(10,11,12,13,14,15)

-       Don�t care terms can be assigned a value 0 or 1.

Rule for simplification with K-Map

If don�t care terms help in creating a larger group of min-terms, then we can assign it as a 1. If it does not help in creating a larger group, then we assign it as a 0.

Example

A = W + XZ + XY

B = X�Z + X�Y + XY�Z�

C = Y�Z� + YZ

D = Z�

A = (W� * (XZ)� * (XY)�)�

B = ((X�Z)� * (X�Y)� * (XY�Z�)�)�

C = ((YZ)� * (Y�Z�)�)�

D = Z�