1. New Canonical Form for Fast Boolean Matching in Logic Synthesis and Verification Afshin Abdollahi and Massoud Pedram Department of Electrical Engineering University of Southern California Los Angeles CA

2. Introduction Boolean Matching: Boolean Matching: Functional equivalence under permutation and complementation of inputs Applications Applications –Logic verification –LUT-based FPGA synthesis –Technology Mapping Clustering Clustering Boolean Matching Boolean Matching Covering Covering

3. Boolean Matching (Example) x1x1 x2x2 x3x3 x4x4 x1x1 x2x2 x3x3 x4x4 N P

4. Equivalence Classes Boolean functions of n variables

5. Prior Work (Canonical Form) [Burch and Long, 1992] [Burch and Long, 1992] –Canonical form for complementation only –Semi-Canonical form for complementation and permutation [Debnath and Sasao] and [Debnath and Sasao] and [Ciric and Sechen] –Canonical form for matching under permutation only [Hinsberger and Kolla, 1998] and [Hinsberger and Kolla, 1998] and [Debnath and Sasao, 2004] –Canonical form for functions of up to seven variables under both complementation and permutation 00110011001100110011001100110011 x2x2 01010101010101010101010101010101 x3x3 00110011001100110011001100110011 x2x2 01010101010101010101010101010101 x3x3 00001111000011110000111100001111 x1x1 f 11010010110100101101001011010010 00110011001100110011001100110011 x2x2 01010101010101010101010101010101 x3x3 00001111000011110000111100001111 x1x1 f 11010010110100101101001011010010 00110011001100110011001100110011 x3x3 01010101010101010101010101010101 x2x2 00001111000011110000111100001111 x1x1 f 10110100101101001011010010110100, T,

6. Signatures 00110011001100110011001100110011 x2x2 01010101010101010101010101010101 x3x3 00001111000011110000111100001111 x1x1 f 10010110100101101001011010010110 1 st - sig 2 nd - sig

7. Symmetry Classes

8. NP Representative of a class (i) (ii) (i), (ii) holds for f i Theorem:

9. NP Representative of a class

10. CP - Transformation x1x1 x2x2 x3x3 x3x3 x1x1 x2x2 x1x1 x2x2 x3x3

12. Grouping the Symmetry Classes

13. Resolving Groups

16. Algorithm Summary Form groups of symmetry classes Use 2 nd – signatures to resolve groups Recursively resolve the remaining groups Theorem: Function F (X ) produced by the above algorithm is the canonical form of function f (X ). Given:

17. Example 4  2 2 x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 f 00 01 10 11 x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 f 00 01 10 11 f

18. Experimental Results A library including a large number of cells A library including a large number of cells Generated large number of Boolean functions Generated large number of Boolean functions

19. Experimental Results Number of Inputs Run-Time (micro-seconds) Worst case Average Prior work [Debnath and Sasao, 2004]

20. Conclusions Presented a canonical form for general Boolean Matching problem under input variable complementation and permutation Presented a canonical form for general Boolean Matching problem under input variable complementation and permutation –Applicable to Boolean functions with large number of inputs –Handles simple symmetries efficiently –Utilizes 1 st, 2 nd or higher-order signatures exactly when they are needed Future work Future work –Classification of Boolean functions into those that need : only 1 st and 2 nd signatures only 1 st and 2 nd signatures Higher-order signatures Higher-order signatures –Capture higher-order symmetry relations

22. Prior Work [Hinsberger and Kolla, 1998] [Hinsberger and Kolla, 1998] [Debnath and Sasao, 2004] [Debnath and Sasao, 2004] 00110011001100110011001100110011 x2x2 01010101010101010101010101010101 x3x3 00110011001100110011001100110011 x2x2 01010101010101010101010101010101 x3x3 00001111000011110000111100001111 x1x1 f 11010010110100101101001011010010 00110011001100110011001100110011 x2x2 01010101010101010101010101010101 x3x3 00001111000011110000111100001111 x1x1 f 11010010110100101101001011010010 00110011001100110011001100110011 x3x3 01010101010101010101010101010101 x2x2 00001111000011110000111100001111 x1x1 f 10110100101101001011010010110100, T,