Based on work on realisability models for linear logic, the author recently proposed a new approach of implicit computational complexity. He showed how to characterise in this way a hierarchy of sub-linear space non-deterministic complexity classes by means of group actions. These classes, defined by means of two-way non-deterministic automata, range from regular languages to NLogspace (non-deterministic logarithmic space). In the present paper, we extend those results in two directions. First, we show how the same techniques can be used to characterise the sub-linear space complexity classes defined by both deterministic and probabilistic two-way multi-head automata. We thus obtain characterisations of deterministic complexity classes between regular languages and Logspace, as well as a hierarchy of probabilistic classes between stochastic languages and PLogspace (unbounded error probabilistic logarithmic space). Second, we exhibit a monoid action capturing polynomial time computation based on pushdown machines, characterising both Ptime and PPtime (unbounded error probabilistic polynomial time).