Efficient algorithms for computing bisimulations for nondeterministic fuzzy transition systems

Nondeterministic fuzzy transition systems (NFTSs) offer a robust framework for modeling and analyzing systems with inherent uncertainties and imprecision, which are prevalent in real-world scenarios. Wu et al. (2018) provided an algorithm for computing the crisp bisimilarity (the greatest crisp bisimulation) of a finite NFTS S = < S, A, S >, with a time complexity of order O (| S | 4 center dot |S|2) under the assumption that |S| >= |S|. Qiao et al. (2023) provided an algorithm for computing the fuzzy bisimilarity (the greatest fuzzy bisimulation) of a finite NFTS S under the G & ouml;del semantics, with a time complexity of order O (| S | 4 center dot |S|2 center dot l ) under the assumption that |S| >= |S|, where l is the number of fuzzy values used in S plus 1. In this work, we provide efficient algorithms for computing the partition corresponding to the crisp bisimilarity of a finite NFTS S, as well as the compact fuzzy partition corresponding to the fuzzy bisimilarity of S under the G & ouml;del semantics. Their time complexities are of the order O((size(S) log l + |S |) log (|S | + |S |)), where l is the number of fuzzy values used in S plus 2. When |S| >= |S|, this order is within O (| S | center dot|S| center dot log2 |S|). The reduction of time complexity from O (| S |4 center dot |S|2) and O (| S |4 center dot |S|2 center dot l ) to O (| S | center dot |S | center dot log2 |S |) is a significant contribution of this work.