Topology of Language Classes

The implications of a specific pseudometric on the collection of languages over a finite alphabet are explored. In distinction from an approach in (Calude et al., 2009) that relates to collections of infinite or bi-infinite sequences, the present work is based on an adaptation of the “Besicovitch” pseudometric introduced by Besicovitch (1932) and elaborated in (Cattaneo et al., 1997) in the context of cellular automata. Using this pseudometric to form a metric quotient space, we study its properties and draw conclusions about the location of certain well-understood families of languages in the language space. We find that topologies, both on the space of formal languages itself and upon quotient spaces derived from pseudometrics on the language space, may offer insights into the relationships, and in particular the distance, between languages over a common alphabet.