We present general results for showing process equivalences applied to the finite control fragment of the pi-calculus decidable. Firstly a Finite Reachability Theorem states that up to finite name spaces and up to a static normalisation procedure, the set of reachable agent expressions is finite. Secondly a Boundedness Lemma shows that no potential computations are missed when name spaces are chosen large enough, but finite. We show how these results lead to decidability for a number of pi-calculus equivalences such as strong or weak, late or early bismulation equivalence. Furthermore, for strong late equivalence we show how our techniques can be used to adapt the well-known Paige-Tarjan algorithm. Strikingly this results in a single exponential running time not much worse than the running time for the case of for instance CCS. Our results considerably strengthens previous results on decidable equivalences for parameter-passing process calculi.