We consider a natural extension to the definition of M-automata which allows the automaton to make use of more of the structure of the monoid M, and by removing the reliance on an identity element, allows the definition of S-automata for S an arbitrary semigroup. In the case of monoids, the resulting automata are equivalent to valence automata with rational target sets which arise in the theory of regulated rewriting. We focus on the polycyclic monoids, and show that for polycyclic monoids of rank 2 or more they accept precisely the context-free languages. The case of the bicyclic monoid is also considered. In the process we prove a number of interesting results about rational subsets in polycyclic monoids; as a consequence we prove the decidability of the rational subset membership problem, and the closure of the class of rational subsets under intersection and complement. In the case of semigroups, we consider the important class of completely simple and completely 0-simple semigroups, obtaining a complete characterisation of the classes of languages corresponding to such semigroups, in terms of their maximal subgroups. In the process, we obtain a number of interesting results about rational subsets of Rees matrix semigroups.