Prognosis of ω-languages for the diagnosis of*-languages: A topological perspective

This article offers a novel perspective on the diagnosis of*-languages via a topological characterization of ω-languages. This allows for the different concepts that currently exist in diagnosis of discrete-event systems to be related to one another in a uniform setting and to study their complexity. For this purpose, we introduce the notion of prognosability of an ω-language, which in the classical setting corresponds to testing if a language is diagnosable and prediagnosable. We show that we can build a prognoser for some ω-language if this language is open and saturated, where openness is usually implied in the finitary setting. For both of these problems we present PSPACE algorithms, and establish that prognosability (i.e., whether or not a prognoser exists) for an ω-language is a PSPACE-complete problem. Our new characterization offers a novel point of view in the classical setting of diagnosis.