From Regular to Strictly Locally Testable Languages

A classical result (often credited to Y. Medvedev) states that every language recognized by a finite automaton is the homomorphic image of a local language, over a much larger so-called local alphabet, namely the alphabet of the edges of the transition graph. Local languages are characterized by the value k=2 of the sliding window width in the McNaughton and Paperts infinite hierarchy of strictly locally testable languages (k-slt). We generalize Medvedevs result in a new direction, studying the relationship between the width and the alphabetic ratio telling how much larger the local alphabet is. We prove that every regular language is the image of a k-slt language on an alphabet of doubled size, where the width logarithmically depends on the automaton size, and we exhibit regular languages for which any smaller alphabetic ratio is insufficient. More generally, we express the trade-off between alphabetic ratio and width as a mathematical relation derived from a careful encoding of the states. At last we mention some directions for theoretical development and application.