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Cyclic Combinational Circuits and Other Novel Constructs

Cyclic Combinational Circuits and Other Novel Constructs. Marrella splendens. Cyclic circuit. (500 million year old Trilobite). (novel construct). Combinational Circuits. Building Block:. Logic Gate. feed-forward device. Combinational Circuits. Building Block:. Logic Gate. 0. 0. 0.

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Cyclic Combinational Circuits and Other Novel Constructs

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  1. Cyclic Combinational Circuitsand Other Novel Constructs Marrella splendens Cyclic circuit (500 million year old Trilobite) (novel construct)

  2. Combinational Circuits Building Block: Logic Gate

  3. feed-forward device Combinational Circuits Building Block: Logic Gate

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

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

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

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

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

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

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

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

  12. 1 1 0 NAND 1 0 OR 0 0 1 AND AND 0 1 NOR 1 1 AND Combinational Circuits 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)...”

  13. Combinational Circuits 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)...” Designers and EDA tools follow this practice.

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

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

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

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

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

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

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

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

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

  25. AND OR AND OR AND OR Circuits with Cycles x a = + + + f b ( a x ( d c ( x f ))) 1 1 b x c d

  26. Circuits with Cycles 0 0 x AND a OR = + + + 0 0 f b ( a x ( d c ( x f ))) 1 1 b AND 0 x OR c AND d OR

  27. Circuits with Cycles 0 0 x AND a OR = + + + f b ( a x ( d c ( x f ))) 1 1 b AND 0 x OR c AND d OR

  28. Circuits with Cycles 1 x AND a OR = + + + 1 1 f b ( a x ( d c ( x f ))) 1 1 b AND 1 1 x OR c AND d OR

  29. Circuits with Cycles Circuit is cyclic yet combinational; computes functions f1 and f2 with 6 gates. 1 x AND An acyclic circuit computing these functions requires 8 gates. a OR = + + f b ( a x ( d c )) 1 b AND 1 1 x OR c AND = + + f d c ( x b a ) 2 d OR

  30. Circuits with Cycles Circuit is cyclic yet combinational; computes functions f1 and f2 with 6 gates. There is no feedback in a functional sense. x A cyclic topology permits greater overlapin the computation of the two functions: AND An acyclic circuit computing these functions requires 8 gates. a OR = + + f b ( a x ( d c )) 1 b AND x OR c AND = + + f d c ( x b a ) 2 d OR

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

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

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

  35. Prior Work (later era) • Stokobserved 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 Theory • Formulated a precise model for analysis. • Provided constructions and lower bounds proving thatcyclic designs can be more compact. Practice • Devised efficient techniques for analysis and synthesis. • Implemented the ideas and demonstrated they are applicable for a wide range of circuits.

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