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Karnaugh Maps of Logical Systems and Applications in Digital Circuit Design

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Abstract

This paper extends the concept of Karnaugh map from a logical function to a logical system, thereby allowing both the state transition graph and state transition matrix of a logical system, as well as a digital circuit, to be constructed directly. It is found that the Karnaugh map of a logical system is actually a two-dimensional state transition table or, equivalently, a table-like state transition graph and is also a grid-type state transition matrix. For constructing the state transition graph and state transition matrix of a logical system, the computational complexities of the methods based on the proposed Karnaugh map are exponentially lower than those of the existing methods. The Karnaugh maps of logical systems are then used in the analysis and design of clocked sequential circuits and in the simplification of multioutput gate circuits. Some illustrative examples and simulations are presented to demonstrate the results and their applications.

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Notes

  1. Hereafter, row (resp. square) i refers to the row (resp. square) that is numbered as i.

  2. Hereafter, the ‘jth bit code’ means the ‘jth bit code from the left’.

  3. Hereafter, the logical values 1 and 0 are replaced by \(\delta _{2}^{1}\) and \(\delta _{2}^{2}\), respectively.

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Acknowledgements

The authors would like to thank the reviewers for their comments and suggestions. This work was supported by National Natural Science Foundation of China under Grant numbers 61741307 and 60774007, and Innovation Project of Hebei under grant number CXZZBS2018032.

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Authors and Affiliations

  1. Department of Intelligent Science and Technology, School of Artificial Intelligence, Hebei University of Technology, Tianjin, 300130, People’s Republic of China

    Cailu Wang & Yuegang Tao

  2. School of Automation, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Cailu Wang

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  1. Cailu Wang
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  2. Yuegang Tao
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Correspondence to Yuegang Tao.

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Wang, C., Tao, Y. Karnaugh Maps of Logical Systems and Applications in Digital Circuit Design. Circuits Syst Signal Process 39, 2245–2271 (2020). https://doi.org/10.1007/s00034-019-01214-x

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