36. Boolean Logic Simplification

The same combinational function can often be represented by different Boolean expressions. Logic optimization finds a simpler equivalent expression, reducing the required gates and logic levels.

Common methods include:

• Boolean algebra: Simplifies expressions using Boolean laws.
• Karnaugh Map (K-map): Groups related input combinations to eliminate unnecessary variables.
• Don’t-care conditions: Uses unused input combinations when they help simplify the function.

Boolean Algebra Laws and Rules:

SOP and POS Forms

Boolean functions are commonly written in two forms:

SOP: F = A' · C + A · B

POS: F = (A + C) · (A' + B)

In SOP, each product term represents a condition that makes F = 1. In POS, each sum term is derived from a condition where F = 0.

3-Variable K-Map Example

For inputs A, B, and C:

F(A,B,C) = Σm(1,3,6,7)

Here, Σm identifies the minterms where F = 1.

In a 3-variable K-map, A selects the row and BC selects the column. The columns follow Gray-code order:

00 → 01 → 11 → 10

This ensures that adjacent cells differ in only one variable.

 For SOP simplification, adjacent 1s are grouped in the largest possible groups of 1, 2, 4, 8, ... cells. A variable that changes within a group is eliminated; variables that remain constant form the simplified term.

• m1, m3 → A' · C
• m6, m7 → A · B

Each group is a condition that can make F = 1, so the terms are combined using OR (+):

F = A' · C + A · B

POS from the Same K-Map

For POS simplification, group the cells where F = 0:

F(A,B,C) = ΠM(0,2,4,5)

Here, ΠM identifies the maxterms where F = 0.

• m0, m2 → (A + C)
• m4, m5 → (A' + B)

Each group forms a sum term. The sum terms are combined using AND (·):

F = (A + C) · (A' + B)

Thus, the same function can be represented as:

SOP: F = A' · C + A · B

POS: F = (A + C) · (A' + B)

Optimized Logic Circuit

The simplified SOP expression can be implemented using logic gates:

F = A' · C + A · B

A parity bit generator generates an extra bit so the total number of 1s becomes either even or odd.

Even Parity

Peven = A ⊕ B ⊕ C

• Even number of data 1s → Peven = 0
• Odd number of data 1s → Peven = 1

Odd Parity

Podd = (A ⊕ B ⊕ C)'

• Even number of data 1s → Podd = 1
• Odd number of data 1s → Podd = 0

Example: ABC = 100

• Peven = 1 → total number of 1s = 2 → even parity
• Podd = 0 → total number of 1s = 1 → odd parity

An encoder converts an active input line into a binary code.

A priority encoder encodes only the highest-priority active input.

Priority: I3 > I2 > I1 > I0

V = I3 + I2 + I1 + I0
Y1 = I3 + I2     
Y0 = I3 + I2' · I1

In input columns, X means a lower-priority input does not affect the result. When V = 0, Y1_Y0 is invalid and may be treated as don't-care.

Example: 0111 → highest active input = I2 → Y1_Y0 = 10, V = 1

A decoder converts an n-bit input code into one selected output among up to 2ⁿ output lines.

• 2 input bits → 4 outputs     

• 3 input bits → 8 outputs

2-to-4 Active-HIGH Decoder

Y0 = A1'·A0'  
Y1 = A1'·A0  
Y2 = A1·A0'  
Y3 = A1·A0

Example: A1_A0 = 10 → Y2 = 1

Active-LOW Decoder

An active-LOW decoder selects one output by making it LOW while the other outputs remain HIGH.

Y0 = (A1'·A0')'  
Y1 = (A1'·A0)'  
Y2 = (A1·A0')'  
Y3 = (A1·A0)'

Active-LOW outputs are commonly used for chip-select and control signals.

A Multiplexer (MUX) is a data selector. It routes one of several data inputs to one output.

Many Data Inputs → One Output

A 4-to-1 MUX uses data inputs A, B, C, D and select lines S1, S0.

Y = A·S1'·S0' + B·S1'·S0 + C·S1·S0' + D·S1·S0

Example: S1_S0 = 10 → Y = C

A Demultiplexer (DEMUX) is a data distributor. It routes one input D to one selected output.

One Data Input → One Selected Output

Y0 = D·S1'·S0'  
Y1 = D·S1'·S0  
Y2 = D·S1·S0'  
Y3 = D·S1·S0

Example: S1_S0 = 10 → Y2 = D

Timing Note: Combinational outputs change after the propagation delay of the logic path.