A Colorful and Robust Measure for FDFAs

We define a measure on families of DFAs (FDFAs) that we show to be robust in the sense that two FDFAs for the same language are guaranteed to agree on this measure. This measure tightly relates to the Wagner-Hierarchy (that defines the complexity of omega regular languages). Inspired by the recently introduced natural colors of infinite words, we define natural colors for finite words (prefixes of periods of infinite words). From this semantic definition we derive the Colorful FDFA a novel canonical model for $\omega$-regular languages that also assigns correct colors for finite and infinite words. From the colorful FDFA, for languages that can be recognized by deterministic B\"uchi or coB\"uchi automata, we generate a canonical DBA or DCA termed the Black $\&$ White Automaton, thus complementing the recent result on canonical good for games coB\"uchi automata for coB\"uchi languages.