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

Combinational Circuits Logic GateBuilding Block:

Combinational Circuits Logic GateBuilding Block: feed-forward device

Combinational Circuits “AND” gate Common Gate:

Combinational Circuits “OR” gate Common Gate:

Combinational Circuits “XOR” gate Common Gate:

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 

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).

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

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

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

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).

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.