Fuzzy grammar theory based on lattices

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.