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A Group Algebraic Approach to NPN Classification of Boolean Functions

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Abstract

The classification of Boolean functions plays an underpinning role in logic design and synthesis of VLSI circuits. This paper considers a underpinning question in Boolean function classification: how many distinct classes are there for k-input Boolean functions. We exploit various group algebraic properties to efficiently compute the number of unique equivalent classes. We have reduced the computation complexity from 2mm! to (m + 1)!. We present our analysis for NPN classification of Boolean functions with up to ten variables and compute the number of NP and NPN equivalence classes for 3-10 variables. This is the first time to report the number of NP and NPN classifications for Boolean functions with 9-10 variables. We demonstrate the effectiveness of our method by both theoretical proofs and numeric experiments.

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Acknowledgment

We would like to thank the National Natural Science Foundation of China (Grant No. 61572109) and the Special Fund for Bagui Scholars of Guangxi for the technology support.

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Authors and Affiliations

  1. Big Data Research Center, School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Juling Zhang & Guowu Yang

  2. School of Computer Science and Engineering, University of Xinjiang Finance and Economics, Urumqi, 830000, China

    Juling Zhang

  3. Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, Guangxi University for Nationalities, Nanning, 530006, China

    Guowu Yang

  4. Synopsys Inc., Mountain View, CA, USA

    William N. N. Hung

  5. Key Laboratory of High Confidence Software Technologies, Ministry of Education, Peking University, Beijing, China

    Tian Liu

  6. ECE Department, Portland State University, Portland, OR, USA

    Xiaoyu Song & Marek A. Perkowski

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  1. Juling Zhang
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  2. Guowu Yang
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  3. William N. N. Hung
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  4. Tian Liu
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  5. Xiaoyu Song
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  6. Marek A. Perkowski
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Correspondence to Guowu Yang.

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Zhang, J., Yang, G., Hung, W.N.N. et al. A Group Algebraic Approach to NPN Classification of Boolean Functions. Theory Comput Syst 63, 1278–1297 (2019). https://doi.org/10.1007/s00224-018-9903-0

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  • DOI: https://doi.org/10.1007/s00224-018-9903-0

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