Homomorphic Characterization of Tree Languages Based on Comma-Free Encoding

Abstract

A classic result in language theory is Medvedev’s theorem for trees, stating that any regular tree language can be defined by the projection of a local tree language, i.e., of a language defined by its tiles of height 2, a.k.a. di-grams. The simple proof of the statement is based on a local alphabet whose size (linearly) depends on the number of states of the tree automaton recognizing the original language. We prove a new extended version of Medvedev’s theorem for trees, by using a k-locally testable tree language defined by tiles of height $k\ge 2$ (k-grams). The size of the local alphabet is just the double of the original one, hence it is independent from the complexity of the tree automaton. This result relies on an encoding of the states traversed by a tree automaton, by means of binary comma-free codes carefully laid on tree paths. We thus generalize from words to trees our recent extended Medvedev’s theorem for word languages that was based on strictly locally testable word languages. By applying the result to the syntax trees of context-free grammars, we characterize them by k-locally testable, binary-labeled trees .