  In the 2nd version of this note we introduce the notion of viscosity solution
for a type of fully nonlinear parabolic path-dependent partial differential
equations (P-PDE). We then prove the comparison theorem (or maximum principle)
of this new type of equation which is the key property of this framework. To
overcome the well-known difficulty of non-compactness of the space of paths for
the maximization, we have introduced a new approach, called left frozen
maximization approach which permits us to obtain the comparison principle for
smooth as well as viscosity solutions of path-dependent PDE. A solution of a
backward stochastic differential equation and a G-martingale under a
G-expectation are typical examples of such type of solutions of P-PDE. The
maximum principle for viscosity solutions of classical PDE, called state
dependent PDE, is a special case.
