New Canonical Form for Fast Boolean Matching in Logic Synthesis and Verification

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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

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Introduction

• Boolean Matching
• Functional equivalence under permutation and
  complementation of inputs
• Applications
• Logic verification
• LUT-based FPGA synthesis
• Technology Mapping
• Clustering
• Boolean Matching
• Covering

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Boolean Matching (Example)
x1
x2
x3
x4
N P
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Equivalence Classes
Boolean functions of n variables
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Prior Work (Canonical Form)

• Burch and Long, 1992
• Canonical form for complementation only
• Semi-Canonical form for complementation and
  permutation
• Debnath and Sasao and
• Ciric and Sechen
• Canonical form for matching under permutation
  only
• Hinsberger and Kolla, 1998 and
• Debnath and Sasao, 2004
• Canonical form for functions of up to seven
  variables under both complementation and
  permutation

T
,
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Signatures
x2
x3
x1
f
00110011
01010101
00001111
10010110
1st - sig
2nd - sig
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Symmetry Classes
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NP Representative of a class
(i)
(ii)
Theorem
(i), (ii) holds for fi
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NP Representative of a class
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CP - Transformation
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CP - Transformation
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Grouping the Symmetry Classes
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Resolving Groups
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Resolving Groups
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Resolving Groups
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Algorithm Summary
Given
Form groups of symmetry classes
Use 2nd 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
).
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Example
f
4
? 2
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Experimental Results

• A library including a large number of cells
• Generated large number of Boolean functions

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

• 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 1st, 2nd or higher-order signatures
  exactly when they are needed
• Future work
• Classification of Boolean functions into those
  that need
• only 1st and 2nd signatures
• Higher-order signatures
• Capture higher-order symmetry relations

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Prior Work

• Hinsberger and Kolla, 1998
• Debnath and Sasao, 2004

T
,