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| Algebraic insight into universal logic functions and implications for logical systemmodeling |
| XiaoboLi1,YongyiYan1,JumeiYue2,PengleiHao1,ShuaibingZhang1 |
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| (College of Information Engineering, Henan University of Science and Technology, Luoyang 471000, Henan, China;College of Agricultural Equipment Engineering, Henan University of Science and Technology, Luoyang 471000, Henan, China) |
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| 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 powerraising operator, power-descending operator, erasure operator, and appending operator. Finally, these results are applied to model and analyze finite statemachines and their networks, which demonstrate the practical value of the method and operators. |
| 关键词: Logical systems · Finite-valued systems · Finite state machines · Semi-tensor product of matrices · Algebraic method · Matrix approach · STP approach |
| DOI：https://doi.org/10.1007/s11768-025-00286-2 |
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| 基金项目:This work was supported in part by the National Natural Science Foundation of China under Grants 62073124 and U1804150. |
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| Algebraic insight into universal logic functions and implications for logical systemmodeling |
| Xiaobo Li1,Yongyi Yan1,Jumei Yue2,Penglei Hao1,Shuaibing Zhang1 |
| (College of Information Engineering, Henan University of Science and Technology, Luoyang 471000, Henan, China;College of Agricultural Equipment Engineering, Henan University of Science and Technology, Luoyang 471000, Henan, China) |
| Abstract: |
| 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 powerraising operator, power-descending operator, erasure operator, and appending operator. Finally, these results are applied to model and analyze finite statemachines and their networks, which demonstrate the practical value of the method and operators. |
| Key words: Logical systems · Finite-valued systems · Finite state machines · Semi-tensor product of matrices · Algebraic method · Matrix approach · STP approach |
