Efficient algorithms for computing bisimulations for nondeterministic fuzzy transition systems

Abstract

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 $S=〈S,A,\delta 〉$, with a time complexity of order $O\left(\right|S{|}^4\cdot \left|\delta {|}^2\right)$ under the assumption that $\left|\delta \right|\ge \left|S\right|$. Qiao et al. (2023) provided an algorithm for computing the fuzzy bisimilarity (the greatest fuzzy bisimulation) of a finite NFTS $S$ under the Gödel semantics, with a time complexity of order $O\left(\right|S{|}^4\cdot |\delta {|}^2\cdot l)$ under the assumption that $\left|\delta \right|\ge \left|S\right|$, 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 Gödel semantics. Their time complexities are of the order $O\left(\right(\mathrm{size}\left(\delta \right)\log l+\left|S\right|)\log (\left|S\right|+\left|\delta \right|\left)\right)$, where l is the number of fuzzy values used in $S$ plus 2. When $\left|\delta \right|\ge \left|S\right|$, this order is within $O\left(\right|S|\cdot |\delta |\cdot {\log }^2|\delta \left|\right)$. The reduction of time complexity from $O\left(\right|S{|}^4\cdot \left|\delta {|}^2\right)$ and $O\left(\right|S{|}^4\cdot |\delta {|}^2\cdot l)$ to $O\left(\right|S|\cdot |\delta |\cdot {\log }^2|\delta \left|\right)$ is a significant contribution of this work.