1. Cyclic Combinational Circuits Theory Marc D. Riedel California Institute of Technology Marrella splendensCyclic circuit (500 million year old Trilobite)(novel construct)

2. Theory Prove that cyclic implementations can have fewer gates than equivalent acyclic ones. cyclic circuit acyclic circuit (optimal) functions, n variables, m fan-in gates d gates n more than gates n

3. Rivest’s Circuit Example due to Rivest:

4. 0 0 Rivest’s Circuit

5. 0 0 0 Example due to Rivest:

6. Rivest’s Circuit Example due to Rivest: 0 0 0

7. 1 1 Rivest’s Circuit

8. 1 1 1 Example due to Rivest: Rivest’s Circuit

9. Example due to Rivest: 1 1 Rivest’s Circuit 1

10. Example due to Rivest: 3 inputs, 6 fan-in two gates. 6 distinct functions, each dependent on all 3 variables. Addition: OR Multiplication: AND

11. A feedback circuit with fewer gates than any equivalent feed-forward circuit. Rivest’s Circuit 3 inputs, 6 fan-in two gates. 6 distinct functions.

12. 6/7 Construction Cyclic Circuit: 6 functions, 3 variables, 6 fan-in 2 gates. ANDORANDORANDOR Acyclic Circuit: at least 7 fan-in 2 gates.

13. Rivest’s Circuit Individually, each function requires 2 fan-in two gates:

14. An equivalent feed-forward circuit requires 7 fan-in two gates.

15. Rivest’s Circuit n inputs 2n fan-in two gates, 2n distinct functions.

16. ... a Rivest’s Circuit gates An equivalent feed-forward circuit requires fan-in two gates.

17. Rivest’s Circuit n inputs 2n fan-in two gates, 2n distinct functions. A feedback circuit with the number of gates of any equivalent feed-forward circuit. 3 2

18. Fan-in Lower Bound An acyclic circuit with Question: must have at leasthow many gates?

19. Fan-in Lower Bound An acyclic circuit with Claim: must have at least

20. Fan-in Lower Bound At least one output function does not depend upon any other. To compute this function, consider a tree of fan-in d gates.

21. Fan-in Lower Bound At least one output function does not depend upon any other. To compute this function, consider a tree of fan-in d gates.

22. Fan-in Lower Bound At least one output function does not depend upon any other. To compute this function, consider a tree of fan-in d gates. in general, with k nodes,

23. Fan-in Lower Bound At least one output function does not depend upon any other. To compute this function, consider a tree of fan-in d gates. set then

24. Fan-in Lower Bound An acyclic circuit with Claim: must have at least

25. ... a Rivest’s Circuit gates An equivalent feed-forward circuit requires fan-in two gates.

26. fan-in 2 gates. 1  m m To implement a function of variables, we need at least Acyclic Lower Bound For the other functions, we need at least gates. 1  n 1  n functions, n variables, m fan-in gates 2 “first” function other functions   

27. fan-in gates 2 d Acyclic Lower Bound functions, n variables, m “first” function other functions    fan-in 2 gates.Require at least fan-in d gates. Require at least

28. Constructions/Lower Bounds variablesfan-in cyclicacyclic min. # gates functions 63267 25423 312635 today’s talk Best construction (asymptotic): n2n

29. 6/7 Construction (Rivest’s Circuit) Cyclic Circuit: 6 functions, 3 variables, 6 fan-in 2 gates. ANDORANDORANDOR Acyclic Circuit: at least 7 fan-in 2 gates.

30. Rivest’s Circuit n inputs 2n fan-in two gates, 2n distinct functions. A feedback circuit with the number of gates of any equivalent feed-forward circuit. 3 2

31. Questions 1.Is feedback more than a theoretical curiosity, even a general principle? 3 2 2.Can we improve upon the bound of ? 3.Can we optimize real circuits with feedback?

32. Prior Work Kautz first discussed the concept of feedback in logic circuits (1970). Huffman discussed feedback in threshold networks (1971). Rivest presented the first, and only viable, example of a combinational circuit with feedback (1977).

33. Prior Work F(X)F(X)G(X)G(X) e.g., add e.g., shift Stok discussed feedback at the level of functional units (1992). Malik (1994) and Shiple et al. (1996) proposed techniques for analysis. X G(F(X)) Y F(G(Y))

34. Questions 1.Is feedback more than a theoretical curiosity, even a general principle? 3 2 2.Can we improve upon the bound of ? 3.Can we optimize real circuits with feedback?

35. Key Contributions 2.Efficient symbolic algorithm for analysis (both functional and timing). 1.A family of feedback circuits that are asymptotically the size of equivalent feed-forward circuits. 2 1 3.A general methodology for synthesis.

36. Example not symmetrical 4 inputs 8 gates 8 distinct functions

37. Examples, multiple cycles, 3 inputs9 gates, 9 distinct functions

38. Example, 5 inputs 20 gates, 20 distinct functions. (“stacked” Rivest circuits)

39. Theory Exhibit a cyclic circuit that is optimal in terms of the number of gates, say with C(n) gates, for n variables. Prove a lower bound on the size of an acyclic circuit implementing the same functions, say A(n) gates. Strategy: Main Result:

40. 6/7 Construction (Rivest’s Circuit) Cyclic Circuit: 6 functions, 3 variables, 6 fan-in 2 gates. ANDORANDORANDOR Acyclic Circuit: at least 7 fan-in 2 gates.

41. 2/3 Construction Cyclic Circuit: 2 functions, 5 variables, 2 fan-in 4 gates. a b c c d e Acyclic Circuit: at least 3 fan-in 4 gates.

42. Single Cycle Example

43. 5 inputs,2 fan-in four gates, 2 outputs Each output function depends on all 5 input variables. Single Cycle

44. 5 inputs,2 fan-in four gates, 2 outputs. Each output function depends on all 5 input variables. Single Cycle Any acyclic implementation would require at least 3 such gates.

45. Acyclic Circuit: at least 7 fan-in 2 gates.

46. 2/3 Construction Cyclic Circuit: 2 functions, 5 variables, 2 fan-in 4 gates. a b c c d e Acyclic Circuit: at least 3 fan-in 4 gates.

47. 3/5 Construction Cyclic Circuit: 3 functions, 12 variables, 3 fan-in 6 gates. Acyclic Circuit: at least 5 fan-in 6 gates. a,c,e,g,i b,d,f,h,j a,b,k,l

48. ½ Example Generalization: family of feedback circuits ½ the size of equivalent feed-forward circuits. (sketch)