1. State university of New York at New Paltz Electrical and Computer Engineering Department Logic Synthesis Optimization Lect10: Two-level Logic Minimization By Dr. Yaser Khalifa Electrical and Computer Engineering Department State University of New York at New Paltz

2. Basic Definitions Definition: A Boolean circuit is a directed graph C(G,N) where G are the gates and N  ´G is the set of directed edges (nets) connecting the gates. Some of the vertices are designated: Inputs: I  G Outputs: O  G, I  O =  Each gate g is assigned a Boolean function fg which computes the output of the gate in terms of its inputs.

3. The fanin FI(g) of a gate g are all predecessor vertices of g: FI(g) = {g’ | (g’,g)  N} The fanout FO(g) of a gate g are all successor vertices of g: FO(g) = {g’ | (g,g’)  N} The cone CONE(g) of a gate g is the transitive fanin of g and g itself. The support SUPPORT(g) of a gate g are all inputs in its cone: SUPPORT(g) = CONE(g)  I

4. FI(6) = {2,4} FO(6) = {7,9} CONE(6) = {1,2,4,6} SUPPORT(6) = {1,2}

5. Definitions Implicant: single element of the ON-set or any group of elements that can be combined together in a K-map Prime implicant: implicant that cannot be combined with another implicant to eleminate a term Essential prime implicant: if an element of the ON-set is covered by a single prime implicant, it is an essential prime.

6. 6 Prime Implicants: A’D, CD, AC, BC’D’, A’BC’, ABD’

7. Essential Prime Implicants: A’D, AC

8. Minimum Cover = A’D, AC, BC’D’

9. 5 Prime Implicants: A’BC’, A’CD, AC’D, ABC, BD

10. Essential Prime Implicants: A’BC’, A’CD, AC’D, ABC Minimum Cover = A’BC’, A’CD, AC’D, ABC

11. Quine-McCluskey Method The QM procedure reduces the minterm expansion of a function to obtain the minimum sum of product. The procedure consists of two main steps described in the following slide.

12. 1.Eliminate as many litrals as possible from each term by systematically applying the theorem XY + XY’ = X. 2.Use a prime implicant chart to select a minimum set of prime implicants which when ORed together, are equal to the function being simplified and which contain a minimum number of literals.

13. f (a, b, c, d) =  m(0,1,2,5,6,7,8,9,10,14) Is represented by the following minterms Group 000000 10001 Group 1 20010 81000 50101 Group 2 60110 91001 101010 Group 370111 141110

15. What are the Prime Implicants?

16. Answer: BD’, AB, BC’, AD, B’CD What are the Essential Prime Implicants? Answer: BD’, BC’, AD, B’CD

17. Cyclic Implicant Chart Cyclic Implicant Chart is a chart which has two or more marks in every row/column. Example: f (a, b, c) =  m(0, 1, 2, 5, 6, 7)

18. Simplification of Incompletely Specified Functions In the process of finding prime implicants, treat the don’t cares terms as if they were required minterms. Do not worry if this results in an extra prime implicant When forming prime implicant chart do not include don’t care terms.