Limited bisimulations for nondeterministic fuzzy transition systems

The limited version of bisimulation, called limited approximate bisimulation, has recently been introduced to fuzzy transition systems (NFTSs). This article extends limited approximate bisimulation to NFTSs, which are more general structures than FTSs, to introduce a notion of $k$-limited $\alpha$-bisimulation by using an approach of relational lifting, where $k$ is a natural number and $\alpha \in [0,1]$. To give the algorithmic characterization, a fixed point characterization of $k$-limited $\alpha$-bisimilarity is first provided. Then $k$-limited $\alpha$-bisimulation vector with $i$-th element being a $(k-i+1)$-limited $\alpha$-bisimulation is introduced to investigate conditions for two states to be $k$-limited $\alpha$-bisimilar, where $1\le i\le k+1$. Using these results, an $O(2k^2\mathrm{\mid }V{\mathrm{\mid }}^6\cdot {\mid \text{lra}\mid }^2)$ algorithm is designed for computing the degree of similarity between two states, where $\mathrm{\mid }V\mathrm{\mid }$ is the number of states of the NFTS and $\mid \text{lra}\mid$ is the greatest number of transitions from states. Finally, the relationship between $k$-limited $\alpha$-bisimilar and $\alpha$-bisimulation under $\tilde{S}$ is showed, and by which, a logical characterization of $k$-limited $\alpha$-bisimilarity is provided.