We study a model of quantum cosmology originating from a classical model of
gravitation where a self-interacting scalar field is coupled to gravity wi
th the metric undergoing a signature transition. We show that there are dua
l classical signature changing solutions, one at large scales and the other
at small scales. It is possible to fine-tune the physics in both scales wi
th an infinitesimal effective cosmological constant. (C) 1999 Published by
Elsevier Science B.V. All rights reserved.