On the Ehrenfeucht conjecture on test sets and its dual version

Abstract

The Ehrenfeucht Conjecture on test sets states the following: Each language L over some finite alphabet contains a finite subset F (a "test set") such that for each pair (g,h) of homomorphisms it holds g(x) = h(x) for all × in F if and only if g(x) = h(x) for all × in L. In this paper we investigate the connections of this conjecture to its dual form where finite representation for any set of pairs of homomorphisms is stated. We also point out similarities and differences of these conjectures to well-known constructions of bases of linear vector-spaces.