On Test Sets and the Ehrenfeucht Conjecture

Ehrenfeucht conjectured that each language over a finite alphabet σ possesses a test set, that is a finite subset F of L such that any two morphisms on σ* agreeing on each string of F also agree on each string of L. We give a sufficient condition for a language L to guarantee that it has a test set. We also show that the Ehrenfeucht conjecture holds true if and only if every (infinite) system of equations (with finite number of variables) over a finitely generated free, monoid has an equivalent finite subsystem. The equivalence and the inclusion problems for finite systems of equations are shown to be decidable. As an application we derive a result that for DOL languages the existence of a test set implies its effective existence. Consequently, the validity of the Ehrenfeucht conjecture for DOL languages implies the decidability of the HDOL sequence equivalence problem. Finally, we show that the Ehrenfeucht conjecture holds true for so-called positive DOL languages.