Lecture11

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

CS 21 - Computer Organization and Assembly Language Programming Lecture 11 Boolean Algebra 2

University of the Philippines - Diliman College of Engineering Department of Computer Science

Canonical Form

Sum of Minterms

Outline

1

Canonical Form

2

Sum of Minterms

3

Product of Maxterms

4

Karnaugh Maps

Product of Maxterms

Karnaugh Maps

Canonical Form

Sum of Minterms

Outline

1

Canonical Form

2

Sum of Minterms

3

Product of Maxterms

4

Karnaugh Maps

Product of Maxterms

Karnaugh Maps

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

What is a Canonical Form?

An expression structure with which every function could be realized or expressed with.

Canonical Form

Sum of Minterms

Product of Maxterms

Canonical Forms in Boolean Algebra

Sum of Minterms Product of Maxterms

Karnaugh Maps

Canonical Form

Sum of Minterms

Product of Maxterms

Why are Canonical Forms inportant?

They enable us to make the jump from truth table to circuit/expression

Karnaugh Maps

Canonical Form

Sum of Minterms

Outline

1

Canonical Form

2

Sum of Minterms

3

Product of Maxterms

4

Karnaugh Maps

Product of Maxterms

Karnaugh Maps

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

What is a Minterm?

In an n-variable Boolean function, a product term in which each of the n variables appears once (in either its complemented or uncomplemented form).

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Minterm Example

Let’s say that our Boolean function has the input variables a, b, c and d. Examples of minterms: abcd a’b’c’d’ a’bcd Question: Given n variables, how many minterms are there?

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Indexing Minterms

The minterms of a function are frequently indexed by ”reading” the values of inputs as 1s or 0s: 1 when uncomplemented, 0 when complemented. For example, given a four variable function with input variables a, b, c and d: abcd = 1111 = m15 (or minterm 15) a’b’c’d’ = 00004 = m0 a’bcd = 0111 = m7

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Sum-of-Minterms Principle

Given an n-variable Boolean function, we could express it a sum of some of its minterms.

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Sum-of-Minterms Principle

Given an n-variable Boolean function, we could express it a sum of some of its minterms.

Main issue: Which minterms?

Canonical Form

Sum of Minterms

Product of Maxterms

Truth Table to Sum-of-Minterms

Scan the rows of the truth table

Which minterms?

Karnaugh Maps

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Truth Table to Sum-of-Minterms Rows in which the output is 1, get to be converted to the included minterm, with the values of the columns dictating the minterm configuration

Canonical Form

Sum of Minterms

Truth Table to Sum-of-Minterms

Minterm Indexes: 0, 1, 2, 3, 7

Product of Maxterms

Karnaugh Maps

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Truth Table to Sum-of-Minterms

Minterm Indexes: 0, 1, 2, 3, 7 Sum-of-Minterm Boolean Expression: m0 +m1 +m2 +m3 +m7 a’b’c’ + a’b’c + a’bc’ + a’bc + abc

Canonical Form

Sum of Minterms

Outline

1

Canonical Form

2

Sum of Minterms

3

Product of Maxterms

4

Karnaugh Maps

Product of Maxterms

Karnaugh Maps

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

What is a Maxterm?

In an n-variable Boolean function, a product term in which each of the n variables appears once (in either its complemented or uncomplemented form).

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Maxterm Example

Let’s say that our Boolean function has the input variables a, b, c and d. Examples of maxterms: (note the capitalization) A+B+C+D A’ + B’ + C’ + D’ A’ + B + C + D Question: Given n variables, how many maxterms are there?

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Product-of-Maxterms Principle

Given an n-variable Boolean function, we could express it a product of some of its maxterms.

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Product-of-Maxterms Principle

Given an n-variable Boolean function, we could express it a product of some of its maxterms.

Main issue: Which maxterms?

Canonical Form

Sum of Minterms

Product of Maxterms

Truth Table to Product-of-Maxterms Scan the rows of the truth table

Which maxterms?

Karnaugh Maps

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Truth Table to Product-of-Maxterms Rows in which the output is 0, get to be converted to the included maxterm, with the values of the columns dictating the minterm configuration

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Truth Table to Product-of-Maxterms IMPORTANT DIFFERENCE: unlike in the Sum-of-Minterm derivation, zero column value -> uncomplemented input and one column value -> complemented input value

Product-of-Maxterm Boolean Expression: (M4 )(M5 )(M6 ) (A’ + B + C)(A’ + B + C’)(A’ + B’ + C)

Canonical Form

Sum of Minterms

Outline

1

Canonical Form

2

Sum of Minterms

3

Product of Maxterms

4

Karnaugh Maps

Product of Maxterms

Karnaugh Maps

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Sum of Products and Product of Sums

Sum of Products form Excluding inverters, two levels Just like sum of minterms, but terms/”addends” need not be minterms Product of Sums form Excluding inverters, two levels Just like product of maxterms, but terms/”multiplicands” need not be maxterms

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Point to remember

Sum-of-Minterms and Product-of-Maxterms NOT minimal

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Karnaugh Map

K-maps for short First described by Maurice Karnaugh in the 1950s Modified truth table intended to allow minimal SOP and POS expressions to be obtained by visual inspection

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Setting up the Karnaugh Map

�Split� the variables into two sets: those that would become rows and those that would become columns

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Setting up the Karnaugh Map �Split� the variables into two sets: those that would become rows and those that would become columns When drawing the grid or the truth table make sure that all combinatorial values for the variables are represented

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Setting up the Karnaugh Map �Split� the variables into two sets: those that would become rows and those that would become columns When drawing the grid or the truth table make sure that all combinatorial values for the variables are represented Fill in the output values

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Deriving the Expressions from the Karnaugh Map

For SOP find squares and rectangles of 1s, making sure that all 1 are covered. Notes make sure that the size of a rectangle or square is a power of 2 edges of the grid wraparound

each square or rectangle is a term, with the unchanged variable throughout the shape being present in the term

Canonical Form

Sum of Minterms

Product of Maxterms

Sum of Products

Minimal SOP: a’ + bc’

Karnaugh Maps

Canonical Form

Sum of Minterms

Product of Maxterms

Karnaugh Maps

Deriving the Expressions from the Karnaugh Map

For POS find squares and rectangles of 0s, making sure that all 1 are covered. Notes make sure that the size of a rectangle or square is a power of 2 edges of the gris wraparound

each square or rectangle is a term, with the unchanged variable throughout the shape being present in the term

Canonical Form

Sum of Minterms

Product of Maxterms

Product of Sums

Minimal POS: (A’ + C)(A’ + B)

Karnaugh Maps

Canonical Form

Sum of Minterms

Product of Maxterms

So...

Any questions?

Karnaugh Maps