The Equivalence Problem for DF0L Languages and Power Series

The theory of free monoids lies at the crossroads of mathematics and theoretical computer science. Many problems studied in free monoid theory have their origins in theoretical computer science. On the other hand, their solutions often require deep methods from classical mathematics. D0L systems constitute a very natural framework for studying free monoid morphisms and their iterations. Infinite words generated by D0L systems are also widely studied in combinatorics of words. Already the basic problems concerning D0L systems have turned out to be very challenging. The D0L sequence equivalence problem was a celebrated open problem until it was solved by Culik II and Fris [2]. Different solutions for the problem are given in [3, 4, 8, 17]. The simplest solutions of D0L and DT0L sequence equivalence problems available presently rely upon Hilbert’s basis theorem and other ideas from commutative algebra. The problems studied in connection with D0L systems and their generalizations can be divided into two classes. First, there are problems in which the noncommutative structure of free monoids does not have a significant role. An example is the well-known open problem of deciding whether or not a given D0L sequence contains two consecutive terms of equal lengths. Such questions are