  In a recent paper, the author has shown how Interaction Graphs models for
linear logic can be used to obtain implicit characterisations of
non-deterministic complexity classes. In this paper, we show how this semantic
approach to Implicit Complexity Theory (ICC) can be used to characterise
deterministic and probabilistic models of computation. In doing so, we obtain
correspondences between group actions and both deterministic and probabilistic
hierarchies of complexity classes. As a particular case, we provide the first
implicit characterisations of the classes PLogspace (un-bounded error
probabilistic logarithmic space) and PPtime (unbounded error probabilistic
polynomial time)
