The Fanout Structure of Switching Functions

1. The Fanout Structure of Switching Functions Author: John P. Hayes Speaker: Johnny Lee

2. Outline • Introduction • Notation and Background Material • Fanout-Free functions • Concluding Remarks

3. Introduction • Fanout-free circuits are easy to test and require very few test patterns • Properties of functions that can be implemented without using fanout are investigated

4. Outline • Introduction • Notation and Background Material • Fanout-Free functions • Concluding Remarks

5. Notation and Background Material

6. Notation and Background Material (cont’d)

7. Notation and Background Material (cont’d)

8. Notation and Background Material (cont’d)

9. Theorem 1 • f(X) is unate •  f(X) can be written as a SOPs expression in which no variable appears both complemented and uncomplemented •  If some prime implicant of f(X) contains xe, then no prime implicant contains xe’ •  f’(X) is unate

10. Outline • Introduction • Notation and Background Material • Fanout-Free functions • Concluding Remarks

11. Fanout-Free Functions • Definition 1. A single-output network N is a fanout-free network if every line in N is connected to an input line of at most one gate • Definition 2. The constant functions 0 and 1 and the 1-variable functions x and x’ are fanout-free functions.

12. Fanout-Free Functions (cont’d) • Theorem 2. A function is fanout-free  it can be realized by a fanout-free network • Theorem 3. Every fanout-free function is unate (proved by induction on |X|) • Unateness is not a sufficient condition for a function to be fanout-free

13. Fanout-Free Functions (cont’d) • The unate function f=x1x2+x2x3+x1x3 is not fanout-free x1 x2 f x3

14. Fanout-Free Functions (cont’d) • Definition 3. xi≠ xj. xi is adjacent to xj if f(ai)=f(aj) for some pair of constants ai and aj. It is denoted by =a • Adjacency is a reflexive and symmetric relation • Lemma 1. Adjacency is transitive • Adjacency is an equivalence relation

15. Theorem 4 Let the variables of f(X) be partitioned into blocks X1,X2,…,Xm by the adjacency relation. There exists a set of m elementary functions φ1(X1), φ2(X2), …, φm(Xm) and an m-variable function F such that f(X)=F(φ1(X1), φ2(X2), …, φm(Xm) )

16. Theorem 4 Proof

17. Procedure 1 • To determine if f(X) is fanout-free, and to find a fanout-free realization if one exists

18. Procedure 1 (cont’d)

19. Procedure 1 (cont’d)

20. Procedure 1 (cont’d) x1 φ11 x2 x3 φ21 φ13 x6 x4 φ12 φ22 f x5

21. Review of Theorem 4 φ1(X1) X1 G1 φ2(X2) N X2 G2 f(X) … φm(Xm) Xm Gm

22. Characterization of Fanout-Free Functions • Definition 4. xi≠xj. xi masks xj if f(ai,0j)=f(ai,1j) for some constant ai. (denoted by xi→xj) • Lemma 2. The masking operation is transitive • Lemma 3. xi≠xj. xi =axj  xi→xj and xj→xi

23. Characterization of Fanout-Free Functions (cont’d) • Definition 5. Let Xi and Xj be disjoint subsets of the variables of f(X). Xi→Xj if there is an assignment Ai to Xi and a variable xj∈Xj such that f(Ai)=f(Ai,0j)=f(Ai,1j) • Definition 6. f(X) has the masking property if for every proper subset Xi, Xi →(X-Xj) • Lemma 4. If f(X) has the masking property and |X|≥2, then at least two distinct variables of X are adjacent • Theorem 5. f(X) is fanout-free f(X) has the masking property

24. Outline • Introduction • Notation and Background Material • Fanout-Free functions • Concluding Remarks

25. Concluding Remarks • A procedure is proposed to determine if a function is fanout-free and realize a fanout-free function if one exists • Two relations, adjacency and masking, are used to characterize fanout-free functions