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Fuzzy grammar theory based on lattices

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Abstract

In this paper, on the basis of breadth-first and depth-first ways, we establish a fundamental framework of fuzzy grammars based on lattices, which provides a necessary tool for the analysis of fuzzy automata. The relationship among finite automata with membership values in lattices (l-VFAs), lattice-valued regular grammars (l-RGs) and lattice-valued deterministic regular grammars (l-DRGs) is investigated. It is demonstrated that, based on each semantic way, l-VFAs and l-RGs are equivalent in the sense that they accept or generate the same classes of fuzzy languages. Furthermore, it is proved that l-VFAs, l-valued deterministic finite automata, l-RGs and l-DRGs are equivalent based on depth-first way. For any l-RG, the language based on breadth-first way coincides with the language based on depth-first way if and only if the truth-valued domain l is a distributive lattice.

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Acknowledgments

The authors would like to thank the anonymous referees for their careful reading of this paper and for a number of valuable comments which improved the quality of this paper. This work is supported by National Science Foundation of China (Grant No. 11071061) and 973 Program (2011CB311808).

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Authors and Affiliations

  1. College of Mathematics and Econometrics, Hunan University, Changsha, 410082, People’s Republic of China

    Jianhua Jin & Qingguo Li

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  1. Jianhua Jin
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  2. Qingguo Li
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Correspondence to Jianhua Jin.

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Jin, J., Li, Q. Fuzzy grammar theory based on lattices. Soft Comput 16, 1415–1426 (2012). https://doi.org/10.1007/s00500-012-0845-1

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