1. Illustrative Example p p Lookup Table for Digits of h g f e ) ( d c b
d c b a
h g f e
inputs
outputs 3 1 2 4 5 9 6 7 8 10 11 12 13 14 15 p
Circuit a e b f c g d h
Illustrate concepts with a simple example.
Inputs, outputs, target functions. h g f e ) ( d c b a cd + = )) ac bc

2. Example: Digits of p h Multilevel acyclic network: h g f e = b d a h g+ g ) ( g de a d + ) ( d a b c h + e
Usual approach: enforce a partial ordering.
Result is a multi-level acyclic network
(as obtained with Berkeley SIS simplify command.) ) ( d c b a cd +
Cost: 33
(literal count) f

3. Example: Digits of p e g h f Multilevel cyclic network: h g f e = ) (b h d c a f + ) ( bc d b a g + ) ( c b d a e h + ) ( cd d c a f +
Drop the requirement of a partial ordering.
Generally lower cost, since every function can potentially benefit from work done elsewhere.
Cyclic network: how do we ascertain if it is combinational?
Cost: 31
(literal count)
Is it combinational?

4. Example: Digits of p e g h f Inputs d,c,b,a = [0,0,0,0]: h g f e = ) (+
= 1
= 1 ) ( bc d b a g + = g
= 1 ) ( c b d a e h +
= 0
= 0 ) ( cd d c a f +
= 0
= 0
Try specific input assignments.
In each case, cyclic dependencies disappear and outputs are well-defined.
Outputs h,g,f,e = [0,0,1,1]
(3 is the first digit of p )

5. Example: Digits of p e g h f Inputs d,c,b,a = [1,1,1,1]: h g f e = ) (+ = + h f
= 1 ) ( bc d b a g +
= 1
= 1 ) ( c b d a e h + = h e
= 0 ) ( cd d c a f + = f
= 0
Outputs h,g,f,e = [0,0,1,1]
(3 is the 16th digit of p )

6. Analysis e g h f
There are cycles in a topological sense, but none are sensitized in an electrical sense.

7. Analysis e Timing Analysis: find the length of sensitized paths. ?
Combinationality Analysis:
ensure that there are no sensitized cycles.
Functional timing analysis: find the lengths of sensitized paths.
Combinationality analysis: ensure that no sensitized paths bite their own tail.

8. Synthesis Goal: optimize a multi-level network representation.
Strategy: introduce cycles in the substitute/minimize step.
network N1
network N2 h ) ( d c b a cd + = g )) ac f bc e ) ( cd d c a f + h = b e g bc
Multi-level logic synthesis: obtain the best multi-layer, structured representation (technology independent phase).
Substitute/minimize operation: express or re-express a function in terms of other functions.
Generate candidate networks.
Cost 40 (literal count)
Cost 31 (literal count)

9. Analysis Novel algorithm based on a “first-cut” approach. e
Observation:
for each input assignment, in every strongly-connected component at least one node must be fully defined independently of the others. g
We propose a new method for analysis, with advantages in the context of synthesis.
Recursive formulation, based on a “first-cut” approach. f h

10. Marginal Given a node function e g h f ) , ( Y X F the marginal Y F ¯
specifies, in terms of X, when F is fully defined independently of Y.
Marginal: Specifies when a function does not depend on variables.
Dual of boolean difference.
X: primary input variables
Y: internal variables

11. Marginal For example, consider e g h f ) ( b h d c a f + e = Suppose := b ) ( 1 h d c a f + e =
= 1

12. Marginal For example, consider e g h f ) ( b h d c a f + e = Suppose := d c a 0) ( b h f + e = = b

13. Marginal For example, consider e g h f ) ( b h d c a f + e =
For input assignments that satisfy b d c a h f e + = ) , (
e is fully defined, independently of f, h.
The marginal can be computed
efficiently (with BDD’s)

14. Analysis e First-Cut Analysis: Cut each node from the network,g h f
First-Cut Analysis:
Cut each node from the network,
and apply the algorithm recursively.

15. Analysis Necessary condition: e g h f Either h f e ¯ ) , ( or g f ¯ h

16. Analysis Necessary and sufficient condition: e g h f Either h f e ¯ ), (
and
is combinational or g f
and
is combinational h e g ) , ( or
and
is combinational or f h
and
is combinational

17. Analysis for Synthesis
Advantage of Recursive Formulation:
Attack problem by breaking network into components. 3 N 2 1
Optimal local solution (subject to constraints) is part of optimal global solution (subject to constraints).

18. Analysis for Synthesis
Exclude non-combinational components. e h f g
Design f, g component: f = ) ( g de a h + g = ) ( c b e a f h +
Not combinational.
Exclude all solutions with this component.

19. Analysis for Synthesis
Cache combinational components. e
Design e, f, g component: ) ( c b d a e h + bc g f = f g h
Combinational.
Focus on h.

20. Analysis for Synthesis
Cache combinational components. e ?
Design e, f, g component: ) ( c b d a e h + bc g f = f g h
Combinational.
Focus on h.

21. Analysis for Synthesis
Cache combinational components. e ?
Design e, f, g component: ) ( c b d a e h + bc g f = f g h
Combinational.
Focus on h.

22. Analysis for Synthesis
Cache combinational components. e ?
Design e, f, g component: ) ( c b d a e h + bc g f = f g h
Combinational.
Focus on h.

23. Analysis for Synthesis
Cache combinational components. e ?
Design e, f, g component: ) ( c b d a e h + bc g f = f g h
Combinational.
Focus on h.

24. Analysis for Synthesis
Cache combinational components. e ?
Design e, f, g component: ) ( c b d a e h + bc g f = f ) ( cd d c a f + h = g h
Combinational.
Combinational.
Focus on h.