Computing Crisp Bisimulations for Fuzzy Structures

Fuzzy structures such as fuzzy automata, fuzzy transition systems, weighted social networks and fuzzy interpretations in fuzzy description logics have been widely studied. For such structures, bisimulation is a natural notion for characterizing indiscernibility between states or individuals. There are two kinds of bisimulations for fuzzy structures: crisp bisimulations and fuzzy bisimulations. While the latter fits to the fuzzy paradigm, the former has also attracted attention due to the application of crisp equivalence relations, for example, in minimizing structures. Bisimulations can be formulated for fuzzy labeled graphs and then adapted to other fuzzy structures. In this article, we present an efficient algorithm for computing the partition corresponding to the largest crisp bisimulation of a given finite fuzzy labeled graph. Its complexity is of order $O((m\log{l} + n)\log{n})$, where $n$, $m$ and $l$ are the number of vertices, the number of nonzero edges and the number of different fuzzy degrees of edges of the input graph, respectively. We also study a similar problem for the setting with counting successors, which corresponds to the case with qualified number restrictions in description logics and graded modalities in modal logics. In particular, we provide an efficient algorithm with the complexity $O((m\log{m} + n)\log{n})$ for the considered problem in that setting

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 $1\leq i\leq k+1$. Using these results, an O(2k^2|V|^6\cdot\left|\lra\right|^2) algorithm is designed for computing the degree of similarity between two states, where $|V|$ is the number of states of the NFTS and \left|\lra\right| is the greatest number of transitions from states. Finally, the relationship between $k$-limited $\alpha$-bisimilar and $\alpha$-bisimulation under $\widetilde{S}$ is showed, and by which, a logical characterization of $k$-limited $\alpha$-bisimilarity is provided

Bisimulations for Kripke models of Fuzzy Multimodal Logics

The main objective of the dissertation is to provide a detailed study of several different types of simulations and bisimulations for Kripke models of fuzzy multimodal logics. Two types of simulations (forward and backward) and five types of bisimulations (forward, backward, forward-backward, backward-forward and regular) are presented hereby. For each type of simulation and bisimulation, an algorithm is created to test the existence of the simulation or bisimulation and, if it exists, the algorithm computes the greatest one. The dissertation presents the application of bisimulations in the state reduction of fuzzy Kripke models, while preserving their semantic properties. Next, weak simulations and bisimulations were considered and the Hennessy-Milner property was examined. Finally, an algorithm was created to compute weak simulations and bisimulations for fuzzy Kripke models over locally finite algebras

Polynomial-time algorithms for computing distances of Fuzzy Transition Systems

Behaviour distances to measure the resemblance of two states in a (nondeterministic) fuzzy transition system have been proposed recently in literature. Such a distance, defined as a pseudo-ultrametric over the state space of the model, provides a quantitative analogue of bisimilarity. In this paper, we focus on the problem of computing these distances. We first extend the definition of the pseudo-ultrametric by introducing discount such that the discounting factor being equal to 1 captures the original definition. We then provide polynomial-time algorithms to calculate the behavioural distances, in both the non-discounted and the discounted setting. The algorithm is strongly polynomial in the former case

Bisimulations for Fuzzy Transition Systems revisited

Bisimulation is a well-known behavioral equivalence for discrete event systems and has recently been adopted and developed in fuzzy systems. In this paper, we propose a new bisimulation, i.e., the group-by-group fuzzy bisimulation, for fuzzy transition systems. It relaxes the fully matching requirement of the bisimulation definition proposed by Cao et al. and can equate more pairs of states which are deemed to be equivalent intuitively, but which cannot be equated in previous definitions. We carry out a systematic investigation on this new notion of bisimulation. In particular, a fixed point characterization of the group-by-group fuzzy bisimilarity is given, based on which, we provide a polynomial-time algorithm to check whether two states in a fuzzy transition system are group-by-group fuzzy bisimilar. Moreover, a modal logic, which is an extension of the Hennessy-Milner logic, is presented to completely characterize the group-by-group fuzzy bisimilarity