Regular languages as local functions with small alphabets

We extend the classical characterization (a.k.a. Medvedev theorem) of any regular language as the homomorphic image of a local language over an alphabet of cardinality depending on the size of the language recognizer. We allow strictly locally testable (slt) languages of degree greater than two, and instead of a homomorphism, we use a rational function of the local type. By encoding the automaton computations using comma-free codes, we prove that any regular language is the image computed by a length-preserving local function, which is defined on an alphabet that extends the terminal alphabet by just one additional letter. A binary alphabet suffices if the local function is not required to preserve the input length, or if the regular language has polynomial density. If, instead of a local function, a local relation is allowed, a binary input alphabet suffices for any regular language. From this, a new simpler proof is obtained of the already known extension of Medvedev theorem stating that any regular language is the homomorphic image of an slt language over an alphabet of double size.