Opaque superlanguages and sublanguages in discrete event systems

Opacity describes the inability for an external observer to know what happened in a system. Recently, opacity has been investigated in the framework of discrete event systems. In our previous paper, we define two types of opacities: strong opacity and weak opacity. Given a general observation mapping, a language is strongly opaque if all strings in the language are confused with some strings in another language and it is weakly opaque if some strings in the language are confused with some strings in another language. In this paper, we investigate properties of opacities. We show that opacities are closed under union, but may not be closed under intersection. Based on these properties, we discuss how to modify languages to satisfy the strong opacity, weak opacity, and no opacity by investigating the sublanguages and superlanguages that are strongly opaque, weakly opaque, and not opaque respectively. We find the largest sublanguages and smallest superlanguages. Examples are given to illustrate results.