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BDD Representation for Incompletely Specified Multiple-Output Logic Functions and Its Applications to Functional Decompo

BDD Representation for Incompletely Specified Multiple-Output Logic Functions and Its Applications to Functional Decomposition. T. Sasao and M. Matsuura Kyushu Institute of Technology Iizuka, Japan. Outline of the Talk. LUT Cascade Representation of Incompletely Specified Functions

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BDD Representation for Incompletely Specified Multiple-Output Logic Functions and Its Applications to Functional Decompo

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  1. BDD Representation for Incompletely Specified Multiple-Output Logic Functionsand Its Applications toFunctional Decomposition T. Sasao and M. Matsuura Kyushu Institute of Technology Iizuka, Japan

  2. Outline of the Talk • LUT Cascade • Representation of Incompletely Specified Functions • Functional Decomposition • Example • Experimental Results • Arithmetic Functions • Conclusions

  3. LUT Cascade • New type of ProgrammableLogic Devices. • Regular Structure. • Easy to Design and Modify. • Design directly from BDDs.

  4. LUT Cascade Chip 0.35 micron 3MetalCMOS 8-LUT64K (13-input 8-output)6-Tr SRAM 200MHz (Pipeline mode) 1.38W 9.8mm x 9.8mmMore than 99% of chip area is memory.

  5. Applications of LUT Cascade • Radix converters • Digital filters • Routing tables in the internet • Generation of elementary functions • Programmable logic controllers • Replacement of contentaddressable memory

  6. Existing Methods to Represent Incompletely Specified Functions • Ternary function that takes 3 values. • A pair of BDDs to represent 3 values. • Auxiliary variable that representsdon’t cares. • Unsuitable for decompositions of multiple-output functions.

  7. Existing Methods to Represent Multiple-Output Functions • SBDD (Shared BDD) • MTBDD (Multi-terminal BDD) • BDD_for_CF(BDD for Characteristic Function) • Suitable for cascade synthesis. • Presented at DAC2004

  8. Characteristic Function of Completely Specified Multiple-Output Function Multiple-output function: F=(f1(X), f2(X),…, fm(X)) Characteristic function

  9. Example of Characteristic Function x1 x2 y1 y2 c x1 x2 f1 f2 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 1 1 0 1 0 1 1 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0

  10. Incompletely Specified Function d: Don't care.

  11. Characteristic Function of Multiple-output Function Completely specified function: Incompletely specified function:

  12. 0 edge 1 edge Node in the BDD_for_CF yi yi yi 0 0

  13. Example of BDD_for_CF x1 x2 x3 y1 x4 y2 1

  14. Functional Decomposition Decomposition chart Bound set X1=(x1,x2) 00 01 10 11 x1 H X1 G x2 Free set 00 0 1 0 1 f x3 01 1 1 1 1 X2=(x3,x4) X2 x4 10 1 0 1 0 11 1 0 1 0 f =g(h(X1),X2)

  15. Functional Decomposition Using BDDs X1 Column Multiplicity =Width of BDD in X1 =The number of nodes in the lower part that are directly connected to the nodes inthe upper part. X2 0 1

  16. Functional Decomposition Using BDDs x1 x1 x2 x2 x2 x2 x3,x4 x3 x3 0 1 00 1 1 01 x4 1 0 10 1 0 11 0 1

  17. H Y1 X1 u G Y2 X2 Theorem • Let the variable ordering of BDD for CF be(X1, Y1, X2, Y2), and let W be the width of the BDD, where (X1, Y1) is bound set and edges to constant 0 from the output variables are ignored. Then,

  18. F5 F5 F6 F6 Merging Compatible Functions X1=(x1, x2) 00 01 10 11 00 01 10 11 00 0 0 d 1 00 0 0 1 1 01 1 1 d d 01 1 1 d d X2=(x3, x4) 10 d 1 0 d 10 1 1 0 0 11 0 d 0 0 11 0 0 0 0 F1 F2 F3 F4 F5 F6 F1 F2 Compatible Graph F3 F4

  19. Example: Reduction of Width (1) Width Width x1 1 1 x2 2 2 x3 4 3 1 11 3 1 2 3 4 y1 8 6 5 7 8 9 10 x4 4 4 y2 3 3 1 1 1 1

  20. Compatible Graph x1 x2 8 7 10 x3 y1 5 7 8 9 10 6 6 5 9 x4 y2 1

  21. 1 2 3 4 3 2 1 Example: Reduction of Width (2) Width Width x1 1 x2 2 x3 3 y1 5 7 8 9 10 6 5 13 12 9 6 x4 4 y2 6 3 1 1 1

  22. 1 2 3 4 3 2 1 Before vs. After Reduction Width Width x1 1 x2 2 x3 4 y1 8 x4 4 y2 3 1 1 1

  23. Existing Method of BDD for Incompletely Specified Function: Shiple, et al, DAC94. When two sub-functions are compatible, they are merge into one. h f g

  24. Our Method vs. Existing Method Width Width x1 1 1 x2 2 2 x3 3 3 y1 4 5 x4 3 3 y2 2 2 1 1 1 1

  25. LUT Cascade with Intermediate Outputs Cell Cell Cell Cell rails

  26. Incompletely Specified Arithmetic Functions • Residue number system (RNS)to binary converter • k-nary to binary converter • k-nary adder and multiplier • Binary coded k-nary inputs produce don't cares.

  27. Comparison of Maximum Widths

  28. Reduction of LUT Cascadeby Don’t Cares

  29. 5-7-11-13 RNS to Binary Converter Our method DC=0 12 2 12 2 8 6 8 7 (LSB) 2 12 2 12 4 8 8 6 (MSB) 2 12 3 Number of LUT inputs: 12 Number of rails: 8 6

  30. Summary • Method to represent incompletely specified multiple-output logic functions. • Method to reduce widths of BDDs. • Designed LUT cascades for k-nary arithmetic circuits. • The method is useful for the design of LUT cascade.

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