Computing Fuzzy Bisimulations for Fuzzy Structures Under the Gödel Semantics

Abstract

Bisimulation is a well-known notion in modal logic and the theory of labeled transition systems. It is used for characterizing indiscernibility between states and has important applications in minimizing structures, separating expressive powers of modal and related logics, as well as concept learning in description logics (DLs). Fuzzy bisimulation is a counterpart of bisimulation for dealing with fuzzy structures. In this article, we present an efficient algorithm with a complexity $O((m+n)n)$ for computing the greatest fuzzy bisimulation between two finite fuzzy interpretations in the fuzzy DL $\mathit {f}\!\mathcal {ALC}$ under the Godel semantics, where $n$ is the number of individuals and $m$ is the number of nonzero instances of roles in the given fuzzy interpretations. We also adapt our algorithm for computing fuzzy bisimulations and simulations between fuzzy finite automata, as well as for dealing with other fuzzy DLs. The resulting algorithms are much more efficient than the previously known ones, as they reduce the complexity from $O(n^5)$ to $O((m+n)n)$ .