$L$-Fuzzy Multiset Recognizer: Determinization and Minimization

This research aims to investigate the determinization and minimization problems of a new class of fuzzy recognizers with input as multisets. We begin by introducing and examining the concepts of an $L$-fuzzy multiset recognizer ($L$-LMR), complete, accessible, and coaccessible $L$-LMR over a quantale $L$. We, then characterize introduced $L$-LMR with a multiset regular language (MRL). Next, we investigate the method of construction of a complete $L$-FMR from a given incomplete $L$-FMR, and show that if $K(\mathcal {R})$ is the MRL recognized by an incomplete $L$-FMR $\mathcal {R}$, then there is a completion $\mathcal {R}^{c}$ of $\mathcal {R}$ such that $K(\mathcal {R})=K(\mathcal {R}^{c})$. Moreover, we also illustrate these notions with examples. Further, we introduce a new method for the construction of an equivalent deterministic $L$-fuzzy multiset recognizer (DLMR) from a given $L$-fuzzy multiset recognizer via $L$-fuzzy subsets and accessible $L$-fuzzy subset construction. Furthermore, we characterize introduced DLMR with an $L$-fuzzy multiset regular languages ($L$-FMRL). In the last, we introduce the concepts related to minimal DLMR and then we obtain a minimal DLMR for a given $L$-FMRL.