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We survey recent results on the so-called Ehrenfeucht Conjecture which states: For each language L over a finite alphabet z' there exists a finite subset F of f. such that for each pair (g, h) of morphisms on Z* the equation g(x) = h(x) holds for all x in L if and only if it holds for all x in F. We point out that the conjecture is closely related to the theory of equations in free monoids. We also state a surprising consequence of the conjecture: If it holds (even noneffectively) for al! DOLdoi:10.1016/0304-3975(84)90004-5 fatcat:4vpzemym5zff7o2yrspf53vkyy