1. Cyclic Combinational Circuits and Other Novel Constructs Marrella splendensCyclic circuit (500 million year old Trilobite)(novel construct)

2. Combinational Circuits Logic GateBuilding Block:

3. Combinational Circuits Logic GateBuilding Block: feed-forward device

4. Combinational Circuits “AND” gate 0 0 0 1 Common Gate: 0 0 1 1 0 1 0 1

5. Combinational Circuits “OR” gate 0 0 1 1 0 1 0 1 0 1 1 1 Common Gate:

6. Combinational Circuits “XOR” gate 0 0 1 1 0 1 0 1 0 1 1 0 Common Gate:

7. A circuit with feedback (i.e., cycles) cannot be combinational. s r q NOR Conventional View

8. s r q NOR 0 0 ? A circuit with feedback (i.e., cycles) cannot be combinational. Conventional View

9. inputsoutputs The current outputs depend only on the current inputs. Combinational Circuits combinational logic

10. Combinational Circuits inputsoutputs The current outputs depend only on the current inputs. combinational logic gate

11. NAND OR AND NOR 1 0 0 1 1 1 1 0 1 0 0 1 Acyclic (i.e., feed-forward) circuits are always combinational. Combinational Circuits

12. Acyclic (i.e., feed-forward) circuits are always combinational. Are combinational circuits always acyclic? “Combinational networks can never have feedback loops.” “A combinational circuit is a directed acyclic graph (DAG)...” Combinational Circuits 1 0 0 1 1 1 NAND OR AND NOR 1 0 1 0 0 1

13. Acyclic (i.e., feed-forward) circuits are always combinational. Are combinational circuits always acyclic? “Combinational networks can never have feedback loops.” “A combinational circuit is a directed acyclic graph (DAG)...” Combinational Circuits Designers and EDA tools follow this practice.

14. Generally feed-forward (i.e., acyclic) structures. Combinational Circuits x y x y z z c s

15. Generally feed-forward (i.e., acyclic) structures. Combinational Circuits 0 1 0 1 1 1 0 1 1 0 1

16. Feedback How can we determine the output without knowing the current state?... feedback

17. Feedback How can we determine the output without knowing the current state?... ? ? ?

18. Example: outputs can be determined in spite of feedback. Feedback

19. 0 0 Example: outputs can be determined in spite of feedback. Feedback

20. 0 0 0 0 Example: outputs can be determined in spite of feedback. Feedback

21. Example: outputs can be determined in spite of feedback. Feedback

22. 1 1 Example: outputs can be determined in spite of feedback. Feedback

23. 1 1 1 1 There is feedback is a topological sense, but not in an electrical sense. Example: outputs can be determined in spite of feedback. Feedback

24. Admittedly, this circuit is useless... Example: outputs can be determined in spite of feedback. Feedback

25. Circuits with Cycles a b x c d x AND OR AND OR )))((( 1 fxcdxab 1 f 

26. x 0 0 0 a b c d AND OR AND OR x x 0 )))((( 1 fcdxab 1 f  0 Circuits with Cycles

27. x x x 0 0 a b c d AND OR AND OR 0 )))((( 1 fxcdab 1 f  Circuits with Cycles

28. x 1 x 1 x x a b c d AND OR AND OR 1 1 1 )))((( 1 fcdab 1 f  Circuits with Cycles

29. 1 1 x x x a b c d AND OR AND OR 1 ))((cdab 1 f  )( 2 abxcdf  Circuit is cyclic yet combinational; computes functions f 1 and f 2 with 6 gates. An acyclic circuit computing these functions requires 8 gates. Circuits with Cycles

30. A cyclic topology permits greater overlap in the computation of the two functions: x x a b c d AND OR AND OR There is no feedback in a functional sense. Circuit is cyclic yet combinational; computes functions f 1 and f 2 with 6 gates. An acyclic circuit computing these functions requires 8 gates. )( 2 abxcdf  Circuits with Cycles x))((cdab 1 f 

31. Prior Work (early era) Kautz and Huffman discussed the concept of feedback in logic circuits (in 1970 and 1971, respectively). McCaw and Rivest presented simple examples (in 1963 and 1977, respectively).

32. McCaw’s Circuit (1963) Cyclic, 4 AND/OR gates, 5 variables, 2 functions: OR AND

33. McCaw’s Circuit (1963) Cyclic, 4 AND/OR gates, 5 variables, 2 functions: outputs are well defined OR AND

34. McCaw’s Circuit (1963) Smallest possible equivalent acyclic circuit:5 AND/OR gates. ORAND OR AND

35. Prior Work (later era) Stok observed that designers sometimes introduce cycles among functional units (in 1992). Malik, Shiple and Du et al. proposed techniques for analyzing such circuits (in 1994,1996, and 1998 respectively).

36. Cyclic Circuits: Key Contributions Practice Theory Devised efficient techniques for analysis and synthesis. Formulated a precise model for analysis. Implemented the ideas and demonstrated they are applicable for a wide range of circuits. Provided constructions and lower bounds proving that cyclic designs can be more compact.