A description of deterministic context-free languages by some kind of finitely generated right-congruences (the sets of relations generating such right-congruences are called controlled rewriting systems or c-systems) has been given previously. In the first part of this paper, we study several decision problems on c-systems, namely the confluence problem, the equivalence problem, the refinement problem, the class equivalence problem and the class inclusion problem. In the second part of the paper, we deduce, from the main theorem given previously, a characterisation of dcfls by means of finitely generated congruences (the sets of relations generating such congruences are now ordinary finite semi-Thue systems). We study three decision problems on finite semi-Thue systems, namely the class equivalence problem, the word problem for the syntactic congruence of one class and the partial confluence problem. All the problems are investigated in several classes of c-systems or semi-Thue systems. For every class we give an answer which may be “yes” (the problem is decidable), “no” (the problem is undecidable) or “eq” which means that the problem is recursively equivalent to the equivalence problem for dpda.