On test sets and the Ehrenfeucht conjecture

  • Karel CulikII
  • Juhani Karhumäki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 140)


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.


Algebraic System Finite Subset Equivalence Problem Inclusion Problem Finite System 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1982

Authors and Affiliations

  • Karel CulikII
    • 1
  • Juhani Karhumäki
    • 2
  1. 1.Department of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Department of MathematicsUniversity of TurkuTurkuFinland

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