Karnaugh Maps Ellen Spertus MCS 111 September 2, 2003

2 Big Picture Any number can be represented as 0s and 1s Functions can be represented as a table Any table of 0s and 1s can be interpreted as a truth table Any truth table can be converted into a boolean function We can implement boolean functions with switches

3 Homework 1

4 Sum of products form Product refers to and (·) Sum refers to or (+)

5 Practice

6 Homework 3: Nand is universal “Universal” means that you can build any boolean function out of it You must be able to construct –and –or –not All you need is nand gates!

7 Optimizing formulas Why is it better to have simpler formulas? What makes one function simpler than another? Given a truth table, is there a way to automatically generate the simplest possible function?

8 Building a Karnaugh map

9 Using a Karnaugh map Circle each horizontal/vertical region of 1s. Convert each term into a boolean product (i.e., and the variables or their negations together). Build a sum of the products (i.e., or the products together).

10 Practice

11 Rules for Karnaugh maps Label each axis with a gray code, i.e., only change one bit at a time. Regions can stretch horizontally or vertically. Each side of a region must be a power of 2 (e.g., 1, 2, or 4). For simplest formula, choose ________region and make use of “don’t care”s. Regions may wrap around the edges. Practice with 4-input Karnaugh map…

12 Putting it together Definitions: –An integer greater than 1 is prime if it has no divisors besides 1 and itself. –An integer greater than 1 is composite if it is not prime. Let’s design a circuit that will tell whether its 3-digit binary input is composite or prime Result should be 0 for composite, 1 for prime.

13 Big Picture Any number can be represented as 0s and 1s. Any table of 0s and 1s can be interpreted as a truth table. We can implement boolean functions with switches. Any truth table can be converted into a boolean function. 

14 4-variable Karnaugh map

15

16 Karnaugh map for primes

17 Lab 1 Get familiar with your lab kit and logic gates Build a full adder Make use of a 4-bit adder The voice of experience says: –Read the directions carefully –Draw your wiring diagram properly

18 Wiring diagrams 1/2 The drawing is neat, with straight horizontal and vertical (not diagonal) lines. (Use rulers and templates.) Chips are drawn not as rectangles but in shapes that suggest their function. Chips are labeled with their part number (e.g., LS283). Signal names (e.g., A3) appear inside the chip, pin numbers outside.

19 Wiring diagrams 2/2 The connections of the upper-right pin to power and the lower-left pin to ground are not shown. Inputs and outputs are clearly labeled and grouped together. If colors are used, they should be used logically and consistently.

20