Efficient algorithms for computing bisimulations for nondeterministic fuzzy transition systems
Linh Anh Nguyen
TL;DR
The paper tackles the problem of efficiently computing the greatest crisp and fuzzy bisimulations for finite NFTSs and extends the framework to NFLTSs with fuzzy state labels. By transforming NFTSs into fuzzy labeled graphs (FLGs), it leverages FLG-based bisimulation algorithms to achieve substantial reductions in time complexity, achieving O((size(δ) log l + |S|) log(|S|+|δ|)) and, under Gödel semantics with |δ| ≥ |S|, O(|S| · |δ| · log^2|δ|). It also introduces NFLTSs and provides efficient algorithms for computing greatest crisp/fuzzy simulations and bisimulations between NFLTSs with O((m+n)n) complexity, where m and n capture transition and state sizes. The work demonstrates practical scalability via a Python implementation (NFTS-impl) and highlights significant improvements over prior approaches, extending the utility of bisimulation analysis to fuzzy and nondeterministic models. Overall, the results deliver scalable verification tools for fuzzy nondeterministic models with broad applicability in uncertain systems modeling.
Abstract
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 $\mathcal{S} = \langle S, A, δ\rangle$, with a time complexity of order $O(|S|^4 \cdot |δ|^2)$ under the assumption that $|δ| \geq |S|$. Qiao {\em et al.} (2023) provided an algorithm for computing the greatest fuzzy bisimulation of a finite NFTS $\mathcal{S}$ under the Gödel semantics, with a time complexity of order $O(|S|^4 \cdot |δ|^2 \cdot l)$ under the assumption that $|δ| \geq |S|$, 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 Gödel semantics. Their time complexities are of the order $O((size(δ) \log{l} + |S|) \log{(|S| + |δ|)})$, where $l$ is the number of fuzzy values used in $\mathcal{S}$ plus 2. When $|δ| \geq |S|$, this order is within $O(|S| \cdot |δ| \cdot \log^2{|δ|})$. The reduction of time complexity from $O(|S|^4 \cdot |δ|^2)$ and $O(|S|^4 \cdot |δ|^2 \cdot l)$ to $O(|S| \cdot |δ| \cdot \log^2{|δ|})$ 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.