On locally reversible languages

There exist several works that study the class of reversible languages defined as the union closure of 0-reversible languages, their properties and suitable representations. In this work we define and study the class of locally reversible languages, defined as the union closure of k-reversible languages. We characterize the class and prove that it is a local (positive) variety of formal languages. We also extend the definition of quasi-reversible automata to deal with locally reversible languages and propose a polynomial algorithm to obtain, for any given locally k-reversible language, a quasi-k-reversible automaton.