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Unification plays a central rule in Logic Programming. We ”soften” the unification process by admitting that two first order expressions can be ”similar” up to a certain degree and not necessarly identical. An extension of the classical unification theory is proposed accordingly. Indeed, in our approach, inspirated by the unification algorithm of Martelli-Montanari, the systems of equations go through a series of ”sound” transformations until a solvable form is found yielding a substitution that is proved to be a most general extended unifier for the given system of equations.