1. Ruei-Rung, Lee 1

2. Outline Introduction Background General Non-disjoint Decompositions Rewriting K-LUT Networks Conclusions and Future Work 2

3. Introduction Function decomposition Algebraic methods Rely on an SOP representation of the logic functions Boolean methods Truth-table H D c b a F c b a 3

4. Background Cut: A set of nodes(leaves) with respect to a node n 1) Each path from any PI to n passes through at least one leaf. 2) For each leaf, there is at least one path from a PI to n passing through the leaf and not through any other leaf. n c 4

5. Disjoint-support Decomposition Disjoint-support decomposition(DSD): A decomposition in which the set of nodes of the resulting Boolean network have disjoint supports. DSD is a tree. Bound set: The set of leaf variables of any sub-tree of the DSD Free set: The remaining variables a b1 b2 5 F b1 a

6. (a,b)-decomposition A function F has an (a,b)-decomposition if it can be written as F(a,b,c) =H(D(a,b),b,c) where D is a single- output function. D is called the divisor and H the dividend. H D c b a F c b a 6

7. Theorem 3.1 A function F(a, b, c) has an (a, b)-decomposition ⇔ each of the 2 |b| cofactors of F with respect b has a DSD structure in which the variables a are in a separate block. 7

8. Proof A function F(a, b, c) has an (a, b)-decomposition ⇒ each of the 2 |b| cofactors of F with respect b has a DSD structure in which the variables a are in a separate block. Suppose F has an (a,b)-decomposition F(x) =H(D(a,b),b,c). Let b j = b1¬b2¬b3b4 for k=4. This sets b = 1,0,0,1 F b j (a, c) = H(D(a,1,0,0,1),1,0,0,1, c) ≡ H b j (D b j (a),c). 8

9. Proof A function F(a, b, c) has an (a, b)-decomposition <= each of the 2 |b| cofactors of F with respect b has a DSD structure in which the variables a are in a separate block. F b j (a, c) =H j (D j (a),c) for some functions H j and D j. We want to find H(x, b, c) and D(a, b) such that F(a, b, c) =H(D(a,b),b,c). Let D(a, b) =, H(T, b, c) = F’(a, b, c) =H(D(a,b),b,c). F’ b j (a, c) = H j (D b j (a,b), c) = H j (D j (a), c) = F b j (a, c). 9

10. Example (a,b) = ((e, f,y),x), c=(g,h) F ¬x =¬ye+yf and F x =¬yg+yh D 0 =¬ye+yf and D 1 = y F ¬x =H 0 (D 0 )=D 0,F x =H 1 (D 1,g,h)=¬D 1 g+D 1 h D = ¬xD 0 + xD 1 = ¬x(¬ye+yf)+ x(y), H(T,x,g,h) = ¬xH 0 (T)+ xH 1 (T,g,h) = ¬xT + x(¬Tg+Th) F = H(D,x,g,h) = ¬xD+x(¬Dg+Dh). 10

11. Bound Set Independent Cofactors F(a, b, c) =H(D(a,b),b,c) Bound set independent cofactors(bsi-cofactors): A cofactor of F with respect to b is independent of a. Otherwise, bsd-cofactors. E.g. F = ab + ¬bc, F ¬b = c is independent of a. If F b j (a, c) is a bsi-cofactor, F b j (a, c) = H b j (D b j (a),c) D b j (a) can be any function. 11

12. Theorem 3.2 f ≅ g : f equals g up to complementation. Let F have an (a,b)-decomposition with an associated divisor D(a, b) = Then D’(a,b) = is also an (a,b)-divisor ⇔ D’ j (a) ≅ D j (a), ∀ j ∈ J, where J is the set of indices of the bsd- cofactors of F. 12

13. Proof Let F have an (a,b)-decomposition with an associated divisor D(a, b) = Then D’(a,b) = is also an (a,b)-divisor ⇒ D’ j (a) ≅ D j (a), ∀ j ∈ J, where J is the set of indices of the bsd- cofactors of F. F = H(D(a,b),b,c) = H’(D’(a,b),b,c) F b j = H b j (D b j (a),c) = H’ b j (D’ b j (a),c) ⇒ D’ b j (a) ≅ D b j (a) HbjHbj DbjDbj c a H’ b j DbjDbj c a D’ b j fbjfbj fbjfbj 13

14. Let F have an (a,b)-decomposition with an associated divisor D(a, b) = Then D’(a,b) = is also an (a,b)-divisor  D’ j (a) ≅ D j (a), ∀ j ∈ J, where J is the set of indices of the bsd- cofactors of F. H’ 1 Proof H1H1 D1D1 c b a H2H2 D2D2 c b a H1H1 D1D1 c b a H2H2 D2D2 c b a H’ 2 D’ 2 D’ 1 14

15. (a,b)-compatible Two functions, f 1, f 2, are said to be (a,b)-compatible if each has an (a,b)-divisor, and D 1 b j (a) ≅ D 2 b j (a), ∀ j ∈ J 1 ∩j 2, where J i is the set of indices of the bsd-cofactors of F i. H1H1 D1D1 c b a H2H2 D1D1 c b a D2D2 f1f1 f2f2 15

16. Theorem 3.3 There exists a common (a,b)-divisor of {F 1, …, F n } ⇔ {F 1, …, F n } is pair-wise (a,b)-compatible. 16

17. Proof There exists a common (a,b)-divisor of {F 1, …, F n } ⇒ {F 1, …, F n } is pair-wise (a,b)-compatible. F 1 (a,b,c) = H 1 (D(a,b),b,c), F 2 (a,b,c) = H 2 (D(a,b),b,c) D 1 = D 2 =D  D 1 b j (a) ≅ D 2 b j (a), F 1 and F 2 are (a,b)-compatible. 17

18. Proof There exists a common (a,b)-divisor of {F 1, …, F n } <= {F 1, …, F n } is pair-wise (a,b)-compatible. Suppose F1 and F2 are (a,b)-compatible. F 1 (a,b,c) = H 1 (D 1 (a,b),b,c), F 2 (a,b,c) = H 2 (D 2 (a,b),b,c) 1) Define D’ b j (a) = D 1 b j (a) [ ≅ D 2 b j (a)], if b j are bsd for F 1 and F 2 2) D’ b j (a) = D 1 b j (a), if b j is bsd for F 1 and bsi for F 2 3) D’ b j (a) = D 2 b j (a), if b j is bsi for F 1 and bsd for F 2 4) D’ b j (a) = 0, if b j is bsi for F 1 and F 2 D’(a,b) =, is an (a,b)-divisor of both F 1 and F 2 (by Theorem 3.2) 18

19. Rewriting K-LUT Networks Objective: To rewrite a local window of a K-LUT mapped network. Given n and a cut C(n), the problem is to find a decomposition of f n (c) composed of the minimum number of K (or less) input blocks. n c 19

20. Example Assume K = 4. Suppose x is chosen as the cofactoring variable. The cofactors are F ¬x =¬ye+yf and F x =¬yg+yh. The K-1-feasible bound-sets are {{ ∅ },{e},{f},{y},{e,f,y}}, and {{ ∅ },{g},{h},{y},{g,h,y}}. A common bound-set {e,f,y} of size K-1 exists. H D c b a 20

21. Experimental Results 21

22. Conclusions An algorithm for decomposition of logic functions. 1) based on cofactoring and disjoint-support decomposition. 2) faster than previous solutions relying on BDD based decomposition and Boolean satisfiability 3) 5.4% reduction in area 22

23. Future Work Exploring other data structures for cofactoring and DSD decomposition, to allow processing of functions with more than 16 inputs. 23

24. Thank You 24