On almost cylindrical languages and the decidability of the D0L and PWD0L primitivity problems

AbstractPrimitive words and their properties have always been of fundamental importance in the study of formal language theory. Head and Lando in Periodic D0L Languages proposed the idea of deciding whether or not a given D0L language has the property that every word in it is a primitive word. After reducing the general problem to the case in which h is injective, it will be shown that primitivity is decidable when ((A)h)∗ is an almost cylindrical set. Moreover, in this case, it is shown that the set of words which generate primitive sequences (given a particular D0L scheme) is an algorithmically constructible context-sensitive language. An undecidability result for the PWD0L primitivity problem and decidability results for cases of the RWD0L primitivity problem are also given

Systems of equations over a free monoid and Ehrenfeucht's conjecture

AbstractEhrenfeucht's conjecture states that every language L has a finite subset F such that, for any pair (g, h) of morphisms, g and h agree on every word of L if and only if they agree on every word of F. We show that it holds if and only if every infinite system of equations (with a finite number of unknowns) over a free monoid has an equivalent finite subsystem. It is shown that this holds true for rational (regular) systems of equations.The equivalence and inclusion problems for finite and rational systems of equations are shown to be decidable and, consequently, the validity of Ehrenfeucht's conjecture implies the decidability of the HDOL and DTOL sequence equivalence problems. The simplicity degree of a language is introduced and used to argue in support of Ehrenfeucht's conjecture