Regular languages as images of local functions over small alphabets

Abstract

The characterization (a.k.a. Medvedev theorem) of regular languages as homomorphic letter-to-letter image of local languages, over an alphabet of cardinality depending on the recognizer size, is extended by using strictly locally testable (k-slt) languages, $k>1$, and a local rational function instead of a homomorphism.

By encoding DFA computations via comma-free codes, we prove that regular languages are the output of quasi-length-preserving local functions, defined on alphabets with one more letter than in the language. A binary alphabet suffices if the local function is allowed to shorten input length, or if the regular language has polynomial density.

If local relations are considered instead of functions, a binary input alphabet suffices for any regular language. A new simpler proof is then obtained of the extension of Medvedev's theorem stating that any regular language is the homomorphic image of an slt language over an alphabet of double size.