Lecture #6 EGR 277 – Digital Logic Reading Assignment: Chapter 3 in Digital Design, 3rd Edition by Mano Handout: Quine-McCluskey computer program (example). The program is only 27 kB in size! Minimization Methods: Recall that this chapter deals with three methods for minimizing Boolean expressions: 1) Boolean algebra 2) Karnaugh maps 3) Tabulation methods – primarily the Quine-McCluskey method Procedure (Quine-McCluskey method): 1. List all minterms in binary form by weight with the smallest weight at the top. Call this LIST #1. 2. Compare terms in adjacent weight groups. If two terms differ in only one bit position, put the term in LIST #2 with a “-“ in the position of the differing bit. List the minterms that were combined to form the new term next to it and check the minterms (they are now covered by the new term, so they are no longer needed and are not prime implicants).

Lecture #6 EGR 277 – Digital Logic 3. Continue in the same manner, comparing terms in LIST #2 to form LIST#3 (and checking off the terms that were combined), comparing terms in LIST#3 to form LIST#4, etc. Note: To relate this to Karnaugh maps: LIST #1 corresponds to all minterms LIST #2 corresponds to all possible groups of two minterms LIST #3 corresponds to all possible groups of four minterms LIST #4 corresponds to all possible groups of eight minterms etc. 4. All unchecked terms are prime implicants (PI). Number them PI1, PI2, PI3, etc. 5. Construct a prime implicant chart (PI Chart) listing all PI’s versus all minterms and place X’s in the chart to indicate which minterms are covered by each PI.

Lecture #6 EGR 277 – Digital Logic 6. Select a minimum number of PI’s to cover all minterms as follows: A) If a minterms is only covered by one PI, it is an essential PI. Save the PI off to the side to be a part of the final solution. Remove the PI and all minterms covered by it from the PI Chart. You might now draw a “reduced PI Chart”. B) Look for and eliminate any PI’s that are completely contained within another PI (i.e., the smaller PI can be eliminated if the larger PI contains all of the smaller PI’s minterms). You might now draw a “reduced PI Chart”. C) Repeat steps A and B until all minterms in the PI Chart have been covered by PI’s removed in steps A and proceed to step 7. If steps A and B could not be used to completely reduce the PI Chart, then the chart is cyclic and there are multiple possible equivalent solutions. Proceed to step D. D) Cyclic PI Charts: Randomly pick any of the remaining PI’s and treat it as essential (i.e., save it off to the side to be a part of the final solution). Remove the PI and all minterms covered by it from the PI Chart. You might now draw a “reduced PI Chart”. Go back to steps A and B to finish reducing the PI Chart. Note: Different random choices yield different solutions. You could go back and try other choices of PI’s to be treated as essential to find other solutions. 7. The function f can now be expressed by the sum of the PI’s that were set aside as essential to the solution. Treat the 1’s in a PI as uncomplemented variables, the 0’s as complemented variables, and the dashes (“-“) as unnecessary variables.

Lecture #6 EGR 277 – Digital Logic Example: Use the Quine-McCluskey method to find a minimal SOP expression for f(A,B,C,D) = (2,4,6,8,9,10,12,13,15). 1. LIST #1 (Minterms by weight) 2. LIST #2

Lecture #6 EGR 277 – Digital Logic 3. LIST #3 4. All unchecked terms are prime implicants (PI). Number them PI1, PI2, PI3, etc. 5. Construct a prime implicant chart (PI Chart) listing all PI’s versus all minterms and place X’s in the chart to indicate which minterms are covered by each PI. PI Chart:

Lecture #6 EGR 277 – Digital Logic 6A. Remove Essential PI’s: Reduced PI Chart: 6B. Remove Unnecessary PI’s: Reduced PI Chart: 6A. Remove “Essential” PI’s: (essential in the reduced chart) 6B. Remove Unnecessary PI’s: 7. Express F(A,B,C,D) as a sum of essential PI’s.

Lecture #6 EGR 277 – Digital Logic Cyclic PI Charts: Recall from K Maps that there are often multiple minimal solutions. When using the Quine-McCluskey method, functions with multiple minimal solutions result in “cyclic PI Charts (see Procedures - Step 6C). Example: Use the Quine-McCluskey method to find a minimal SOP expression for f(A,B,C,D) = (1, 3, 4, 6, 7, 9, 13, 15).

Lecture #6 EGR 277 – Digital Logic Example: (continued)

Lecture #6 EGR 277 – Digital Logic Implementing SOP expressions using NAND gates NAND gates are "universal gates" and thus any circuit could be constructed using only NAND gates. Motivation: To reduce the # of IC’s instead of # of gates. NAND gates are commonly represented using two symbols: NAND gates with their inputs tied together act like inverters. This is true of any number of inputs, so we often represent a NAND tied as an inverter as having only one input as shown below. Illustrate how NAND's could be used to replace NOT, OR, AND, NOR, and XOR gates.

Lecture #6 EGR 277 – Digital Logic SOP circuits can be easily converted to NAND circuits by "balancing bubbles". Examples: Implement each expression below using only 2-input (7400) NAND gates. 1. f(A,B,C) = AB' + BC' 2. f(A,B,C,D) = AB' + BC' + CD 3. f(A,B,C,D) = A + BC' + CD

Lecture #6 EGR 277 – Digital Logic Implementing POS expressions using NOR gates NOR gates are "universal gates" and thus any circuit could be constructed using only NOR gates. Motivation: To reduce the # of IC’s instead of # of gates. NOR gates are commonly represented using two symbols: NOR gates with their inputs tied together act like inverters. This is true of any number of inputs, so we often represent a NOR tied as an inverter as having only one input as shown below. Illustrate how NOR's could be used to replace NOT, OR, AND, NAND, and XOR gates.

Lecture #6 EGR 277 – Digital Logic SOP circuits can be easily converted to NOR circuits by "balancing bubbles". Examples: Implement each expression below using only 2-input (7402) NOR gates. 1. f(A,B,C) = (A + B’)(B + C’) 2. f(A,B,C,D) = (A + B’)(B + C’)(C + D) 3. f(A,B,C,D) = A(B + C’)(C + D)