Relative approximate bisimulations for fuzzy picture automata

In this paper, we firstly give the notion of relative ϵ-approximate bisimulations between two fuzzy two-dimensional on-line tessellation automata (F2OTAs) and elaborate that the relative error between two F2OTAs, accompanying with a relative ϵ-approximate bisimulation between them, is less than or equal to ϵ where $ϵ\in [0,1]$. Moreover, if relative ϵ-approximate bisimulations between two F2OTAs are restricted to be surjective functional, then some properties are drawn and by which relative ϵ-approximate bisimulations on a F2OTA are defined. We construct the factor F2OTA of a F2OTA with respect to a relative ϵ-approximate bisimulation on it and describe the relationship between this factor automaton and the original F2OTA. Whereafter, we novelly design two algorithms to compute all maximal relative ϵ-approximate bisimulations on a given F2OTA. Finally, we discuss some interesting properties of bisimulations on F2OTAs.