Efficient algorithms for computing bisimulations for nondeterministic fuzzy transition systems

Fuzzy transition systems offer a robust framework for modeling and analyzing systems with inherent uncertainties and imprecision, which are prevalent in real-world scenarios. As their extension, nondeterministic fuzzy transition systems (NFTSs) have been studied in a considerable number of works. Wu et al. (2018) provided an algorithm for computing the greatest crisp bisimulation of a finite NFTS $\mathcal{S} = \langle S, A, \delta \rangle$, with a time complexity of order $O(|S|^4 \cdot |\delta|^2)$ under the assumption that $|\delta| \geq |S|$. Qiao {\em et al.} (2023) provided an algorithm for computing the greatest fuzzy bisimulation of a finite NFTS $\mathcal{S}$ under the G\"odel semantics, with a time complexity of order $O(|S|^4 \cdot |\delta|^2 \cdot l)$ under the assumption that $|\delta| \geq |S|$, where $l$ is the number of fuzzy values used in $\mathcal{S}$ plus 1. In this work, we provide efficient algorithms for computing the partition corresponding to the greatest crisp bisimulation of a finite NFTS $\mathcal{S}$, as well as the compact fuzzy partition corresponding to the greatest fuzzy bisimulation of $\mathcal{S}$ under the G\"odel semantics. Their time complexities are of the order $O((size(\delta) \log{l} + |S|) \log{(|S| + |\delta|)})$, where $l$ is the number of fuzzy values used in $\mathcal{S}$ plus 2. When $|\delta| \geq |S|$, this order is within $O(|S| \cdot |\delta| \cdot \log^2{|\delta|})$. The reduction of time complexity from $O(|S|^4 \cdot |\delta|^2)$ and $O(|S|^4 \cdot |\delta|^2 \cdot l)$ to $O(|S| \cdot |\delta| \cdot \log^2{|\delta|})$ is a significant contribution of this work. In addition, we introduce nondeterministic fuzzy labeled transition systems, which extend NFTSs with fuzzy state labels, and we define and provide results on simulations and bisimulations between them.