Depth-Bounded Fuzzy Bisimulation for Fuzzy Modal Logic

Abstract

We introduce depth-bounded fuzzy bisimulation between fuzzy Kripke models. Roughly speaking, a depth-bounded fuzzy bisimulation is a decreasing sequence of fuzzy binary relations whose infimum is a fuzzy bisimulation. We provide logical characterizations of depth-bounded fuzzy bisimulations between fuzzy Kripke models w.r.t. a fuzzy multimodal logic $\mathit{\text{fK}}$ over complete residuated lattices, including fuzzy invariance of formulas of $\mathit{\text{fK}}$ with a modal depth bounded by $n$ under the $n$th component of a depth-bounded fuzzy bisimulation, as well as the Hennessy-Milner property of depth-bounded fuzzy bisimulations. We also provide a polynomial-time algorithm for computing the $n$th component of the greatest depth-bounded fuzzy bisimulation between two finite fuzzy Kripke models when the underlying complete residuated lattice is linear.

Keywords:

• Bisimulation
• fuzzy bisimulation
• fuzzy modal logic
• residuated lattice

Acknowledgments

I. Micic and S. Stanimirovic acknowledge the support of the Science Fund of the Republic of Serbia, Grant no 7750185, Quantitative Automata Models: Fundamental Problems and Applications – QUAM, and the Ministry of Education, Science and Technological Development, Republic of Serbia, Contract No. 451-03-47/2023-01/200124.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.