The study investigates the minimal information required to enable adversarial language learning within Gold's model of language identification in the limit. It demonstrates that a single terminal bit attached to the end of each string is sufficient to identify every countable collection of infinite languages.
- A global construction using transfinite recursion allows one bit per string to identify all countable subcollections.
- The colorings can be chosen collection-independently, meaning a single preassigned coloring works for any subcollection.
- No global terminal coloring with a finite number of colors defined by a Borel map can identify all countable subcollections.
- Known trace-coloring constructions are Borel but require infinitely many colors when encoded as terminal colorings.
This result shows that an entire color trace can be compressed to one bit, although the necessary nonconstructivity is unavoidable for any bounded number of colors.