  The Andrews-Curtis conjecture claims that every balanced presentation of the
trivial group can be transformed into the trivial presentation by a finite
sequence of "elementary transformations" which are Nielsen transformations
together with an arbitrary conjugation of a relator. It is believed that the
Andrews-Curtis conjecture is false; however, not so many possible
counterexamples are known. It is not a trivial matter to verify whether the
conjecture holds for a given balanced presentation or not. The purpose of this
paper is to describe some non-deterministic methods, called Genetic Algorithms,
designed to test the validity of the Andrews-Curtis conjecture. Using such
algorithm we have been able to prove that all known (to us) balanced
presentations of the trivial group where the total length of the relators is at
most 12 satisfy the conjecture. In particular, the Andrews-Curtis conjecture
holds for the presentation <x,y|x y x = y x y, x^2 = y^3> which was one of the
well known potential counterexamples.
