COMP541 Bonus Topic: Karnaugh Maps

1. COMP541Bonus Topic:Karnaugh Maps Montek Singh April 15, 2010

2. Topic • Karnaugh maps • classic graphical method for combinational synthesis • represent truth table graphically… • … draw ovals/”cubes” to cover all the ones • has been highly optimized and automated • Quine-McCluskey method • we will cover the “lite” version

3. Karnaugh Maps (K-Maps) • Boolean expressions can be minimized by combining terms • K-maps minimize equations graphically • PA + PA = P

4. K-map • Circle 1’s in adjacent squares • Exclude variables whose true and complement forms are in circle from the implicant • Y = AB

5. 3-input K-map

6. 3-input K-map Y = AB + BC

7. K-map Definitions • Complement: variable with a bar over it A, B, C • Sometimes complement with a prime A’, B’, C’ • Literal: variable or its complement A, A, B, B, C, C • Implicant: product of literals ABC, AC, BC • Prime implicant • implicant that cannot be grown further • has the least number of literals possible for the 1’s it covers

8. K-map Rules • Covering Requirement • Every 1 in a K-map must be circled at least once • Each term must be a “cube” • Each oval must span a power of 2 (i.e. 1, 2, 4) squares in each direction • Each oval must be as large as possible (prime implicant) • An oval may wrap around the edges of the K-map • A “don't care” (X) is circled only if it helps minimize the equation

9. 4-input K-map

10. 4-input K-map

11. 4-input K-map

12. The Maps Wrap (toroid)

13. K-maps with Don’t Cares

14. K-maps with Don’t Cares

15. K-maps with Don’t Cares

16. Use of K-Maps • Manual use only for small specs • For bigger problems, use automated tools • e.g., your Verilog synthesis tools • Still, K-Maps form the basis of all modern algorithms • although actual representation is graph-based instead of table-based (“binary decision diagrams”)