Minimization of AND-OR-EXOR Three Level Networks with AND gate Sharing

1. Minimization of AND-OR-EXOR Three Level Networks with AND gate Sharing HasnainHeickal (SH-223)

2. Overview • Introduction • AND-OR-EXOR networks • Objective • Preliminary Definitions • Properties of EX-SOPs • Minimization of EX-SOPs • Idea of Minimization • Summary • Reference

3. Introduction • Logic networks are usually designed using AND and OR gates (SOP). • AND-EXOR networks (EX-SOP) are • More compact. • Easily testable. • Fault tolerant

4. AND-OR-EXOR Networks • A two input EXOR gate is used. • AND gates can be shared or not shared. • If not shared an EX-SOP for a function F can be written as F = FaxorFb • If shared the EX-SOPcan be written asF = (Fa + Fs) xor (Fb + Fs)

5. Objective • Designing a AND-OR-EXOR three level network. • Minimizing the number of products. • We will discuss an exact algorithm for minimization.

6. Preliminary Definitions • τ(F) • Number of products in an expression F. • τ(ABC + A’BC + AC) = 3 • τ(SOP:f) • Number of product in a minimum SOP for f. • τ(SOP : (ABC + A’BC + AC)) = 2 because it can be minimized as BC + AC.

7. Preliminary Definitions • τ(EX-SOPNS:f) • Number of products in a minimum EX-SOP for f with no product sharing. • τ(EX-SOPPS:f) • Number of products in a minimum EX-SOP for f with product sharing. • A logic function f can represented asf = (fa + g) xor (fb + g)……………………(1) • τ(EX-SOPPS:f) = min{τ(SOP:g) + τ(SOP:fa) + τ(SOP:fb)} • τ(EX-SOPNS:f) = min{τ(SOP:fa) + τ(SOP:fb)} while considering g = 0

8. Properties of EX-SOPs On the Karnaugh map of a function, a cell that contains a 1(one) is called a 1-cell and a cell that contains a 0(zero) is called 0-cell. • Property 1: • In a K-map for an EX-SOP, any 1-cell must be covered by the loop(s) for exactly one SOP. • If a 0-cell is covered, then it must be covered by at least one loop from both SOPs. • Definition 6: • Let g(x) and h(x) be n variable functions. B = {0,1}, if for every a εBn g(a)=1 satisfies h(a)=1 then g h

9. Minimization of EX-SOPs • Let g represent the shared products of an EX-SOP of function f. The number of different products in a minimum EX-SOP for f with product sharing is denoted by τ(EX-SOPPS:f:g). • To compute τ(EX-SOPPS : f : g) using the Eq 1, g is fixed and we choose faand fbsuch that Eq 1 satisfies. Thus we haveτ(EX-SOPPS:f:g) = τ(SOP:g) + min{ τ(SOP:fa) + τ(SOP:fb) }

10. Minimization of EX-SOPs • Lemma 2: • The proof of the lemma is out of scope. • The proof can be found on the paper [1].

11. Idea of Minimization • The idea is for 5 of less number of variables. • We will try for all possible g and minimize the following Eq for all possible g. • We need to use K-map.

12. Example • Let us consider g = A’C’D. • Possible values of h are • A’BC’D • A’B’C’D • A’C’D • We have to find h that makes minimum.

13. Example • Lets first try with h = A’BC’D • So K-map for f v h will be

14. Example • Rules for EX-SOPNS • Loop 1-cell entries odd numberof times. • Loop 0-cell entries even numberof times. • From the K-map we can see • fa = B • fb = A’CD’ • = 2 • τ(SOP:g) = 1 • Τ(EX-SOPPS:f:g) = 3 • We need to do this for every h. fa fb

15. Choosing g • We can choose g using the following lemma : • To obtain minimum EX-SOP of f it is sufficient to consider only the prime implicants of f’ as shared product of candidate. • The proof of this lemma can also be found in the paper [1]. • To find the prime implicants of f’ we can also use K-map.

16. Drawbacks • Choosing g is very time consuming. • We can use “Lookup Tables” to optimize it. • Overall an NP equivalent problem.

17. Summary • We have seen the algorithm for minimizing AND-OR-EXOR three level networks. • We have seen the algorithm for 5 or less variables. • There exists algorithm for more variables.

18. References • D. Debnath and T. Sasao, “Minimization of AND-OR-EXOR three level networks with AND gate sharing.”

19. Thank You