Minimal Model of Dilatonic Gravity and Cosmological Constant

We consider: minimal scalar-tensor model of gravity with Brans-Dicke factor $\omega (\mathrm{\Phi })\equiv 0$ and cosmological factor $\mathrm{\Pi }(\mathrm{\Phi })$; restrictions on it from gravitational experiments; qualitative analysis of new approach to cosmological constant problem based on the huge amount of action in Universe; determination of $\mathrm{\Pi }(\mathrm{\Phi })$ using time evolution of scale factor of Universe.