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<p>This paper explores the algebraic essence of universal logic functions (ULFs) from an algebraic perspective. Under the framework of semi-tensor product of matrices, the “sequential nature” of ULFs is revealed. Utilizing the nature, a technique called universal transformation method is proposed, by which any ULF can be transformed into an equivalent expression with desired features that facilitate achieving specific objectives, such as modeling, analyzing and synthesizing universal logical systems. Furthermore, several useful logical operators are constructed in a mixed-dimensional situation, including power-raising operator, power-descending operator, erasure operator, and appending operator. Finally, these results are applied to model and analyze finite state machines and their networks, which demonstrate the practical value of the method and operators.</p>

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Algebraic insight into universal logic functions and implications for logical system modeling

  • 期刊论文

  • 发表日期：

    2025年9月19日

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  • Xiaobo Li,
  • Yongyi Yan,
  • Jumei Yue,
  • Penglei Hao,
  • Shuaibing Zhang

摘要

This paper explores the algebraic essence of universal logic functions (ULFs) from an algebraic perspective. Under the framework of semi-tensor product of matrices, the “sequential nature” of ULFs is revealed. Utilizing the nature, a technique called universal transformation method is proposed, by which any ULF can be transformed into an equivalent expression with desired features that facilitate achieving specific objectives, such as modeling, analyzing and synthesizing universal logical systems. Furthermore, several useful logical operators are constructed in a mixed-dimensional situation, including power-raising operator, power-descending operator, erasure operator, and appending operator. Finally, these results are applied to model and analyze finite state machines and their networks, which demonstrate the practical value of the method and operators.