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Faster Logic Manipulation for Large Designs

This paper explores a novel representation for manipulating millions of small Boolean functions (6-16 inputs) in complex hardware designs through Disjoint Support Decomposition (DSD). It introduces a DSD manipulation package that enhances runtime, memory efficiency, and quality of computations. The study includes experimental results demonstrating significant runtime improvements and discusses canonical forms and Boolean properties. The findings suggest that DSDs outperform traditional methods like Binary Decision Diagrams (BDDs) and truth tables (TTs) in practical applications, paving the way for further innovations in hardware design optimization.

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Faster Logic Manipulation for Large Designs

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  1. Faster Logic Manipulation for Large Designs Alan Mishchenko Robert Brayton University of California, Berkeley

  2. Outline • Motivation • Simple, local, iterative computation • Millions of 6-16 input functions (“small practical functions” = SPFs) • Runtime / memory / quality can be improved • Choosing the canonical form • Historic viewpoint • Disjoint-support decomposition (DSD) • DSD manager • Impact on computations • Experimental results • Statistics of DSD functions • Runtime improvements • Typical DSD structures • Conclusions

  3. Disjoint Support Decomposition • Primitive gates • Const0 • AND • XOR • PRIME • 2:1 MUX • Majority d e c a b f g h i k

  4. References • Canonical form • R. L. Ashenhurst, “The decomposition of switching functions”. Computation Lab, Harvard University, 1959, Vol. 29, pp. 74-116. • Computation from cofactors • V. Bertacco and M. Damiani, "Disjunctive decomposition of logic functions," Proc. ICCAD ‘97, pp. 78-82. • Computation from cofactors (corrections) • Y. Matsunaga, "An exact and efficient algorithm for disjunctive decomposition", Proc. SASIMI '98, pp. 44-50. • Alternative computations • T. Sasao and M. Matsuura, "DECOMPOS: An integrated system for functional decomposition," Proc. IWLS ’98, pp. 471-477. • S.-I. Minato and G. De Micheli, “Finding all simple disjunctive decompositions using irredundant sum-of-products forms”. Proc. ICCAD’98, pp. 111-117. • Boolean operations • S. Plaza and V. Bertacco, "Boolean operations on decomposed functions", Proc. IWLS ’05. • Applications in synthesis and mapping • A. Mishchenko, R. K. Brayton, and S. Chatterjee, "Boolean factoring and decomposition of logic networks", Proc. ICCAD'08, pp. 38-44.

  5. Timeline of (Canonical) Forms • Truth tables (TTs) (< 1980) • Sums-of-products (SOPs) (1980-1990) • Binary decision diagrams (BDDs) (1990-2000) • And-inverter graphs (AIGs) and truth tables (2000-2012) • Disjoint-support decompositions (DSDs) (> 2012) • For small practical functions (SPFs) only

  6. DSDs vs BDDs vs TTs for SPFs • TTs dominate BDDs in terms of memory and runtime • TTs and BDDs are equally (in)convenient for detecting Boolean properties • DSDs take less memory/runtime than BDDs/TTs for pratical functions of K inputs (8 < K < 16) • DSDs explicitly represent Boolean properties • Symmetry, unateness, NPN canonicity, decomposability, etc • Very important for practical applications!

  7. DSD Manager • Similar to BDD manager • Maintains canonical forms • Performs Boolean operations • Employs computed table • Different from BDD manager • Different data structure • Different normalization rules • More reusable computed table

  8. Primitives of DSD Manager • One constant 0 node • One primary input node n • Multi-input AND and XOR nodes with ordered fanins • Three-input MUX nodes • Multi-input PRIME nodes • Non-decomposable functions

  9. Canonicity of DSDs • Propagating inverters • AND(!a, !XOR(b, c))  AND(n, XOR(n, n)) • Collapsing operators • AND(a, AND(b, !AND(c, d))  AND(n, n, !AND(n, n)) • Ordering faninsof AND/XOR • Use support size • If there is a tie, AND precedes XOR precedes MUX precedes PRIME. • If there is a tie, a non-inverted fanin precedes a inverted fanin. • If there is a tie, the fanins’ fanins are ordered and compared in their selected order • If the recursive comparison fails to produce a unique order, the fanins’ DSD structures are isomorphic and therefore their order is immaterial. • Unifying variables • AND(a, XOR(b, c), MUX(d, e, f))  AND(n, XOR(n, n), MUX(n, n, n))

  10. Boolean Properties of SPFs(industrial benchmarks)

  11. Boolean Properties of SPFs(public benchmarks)

  12. LUT Structure Mapping Runtime

  13. Typical DSD Structures !a = NOT( a ) (ab) = AND( a, b ) [ab] = XOR( a, b ) <abc> = MUX( a , b , c )

  14. Conclusion • (Re)invented DSD • Canonical form, which exposes Boolean properties • Introduced a DSD package • An alternative to a BDD package for SPFs • Discussed preliminary experimental results • Exciting future work is waiting to be done!

  15. Abstract When logic transformations, such as circuit restructuring, technology mapping, and post-mapping optimization, are repeatedly applied to large hardware designs, millions of relatively small (6-16 input) Boolean functions have to be efficiently manipulated. This paper focuses on a novel representation of these small functions, in terms of their disjoint-support decomposition (DSD) structures. A new DSD manipulation package is developed, which allows for faster logic manipulation compared to known methods.

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